![Chapter 1
Introduction
The classical theory of thermodynamics deals with phenomenological relationships
between flows of energy in macroscopic systems. While originally motivated by a
need to understand and optimize the performance of steam engines in the 1800s [1],
the scope of thermodynamics has expanded through the past two centuries, finding
its place as a mathematical framework that pervades the branches of science
and industry. At its heart, the theory of classical thermodynamics captures the
mathematical relationships between physical quantities, such as heat and work,
subject to a set of underlying assumptions. So long as these assumptions are valid,
the classical theory of thermodynamics can be encompassed by four axiomatic rules
known as the laws of thermodynamics [2].
Despite the widespread successes of classical thermodynamics, there are a
number of shortcomings that limit its utility. Central to the focus of this thesis
is the assumption of systems being in a state of equilibrium. The restriction of
thermodynamic analysis to equilibrium precludes the treatment of many interesting
situations, such as biological systems, which are manifestly out of equilibrium.
Furthermore, through the subsequent development of a statistical theory of thermo-
dynamics, known as statistical mechanics, it became clear that the laws of classical
thermodynamics pertain to ensemble-average quantities.
For macroscopic systems, an appeal to the central limit theorem makes clear
that for N ∼ 1023 interacting molecules, the fluctuations in energy and energy
flows can be assumed to be vanishingly small (so long as the system is not
sufficiently close to a critical point). However, for small systems—often on the
scale of micro- or nano-meters—thermal fluctuations can no longer be ignored,
and have significant implications for the overall physical description of the system
of interest. Furthermore, ever since Clausius, in 1865, first articulated the law of
increase in entropy [3], what has come to be known as the second law has posed
a deep, unresolved question: how can microscopic equations of motion, that are
symmetric under time reversal, give rise to macroscopic behavior that is not?
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
S. J. Large, Dissipation and Control in Microscopic Nonequilibrium Systems,
Springer Theses, https://doi.org/10.1007/978-3-030-85825-4_1
1](https://image.slidesharecdn.com/23428428-250509151846-2d979635/85/Dissipation-And-Control-In-Microscopic-Nonequilibrium-Systems-Steven-J-Large-21-320.jpg)
![2 1 Introduction
Over the past 25 years, the development of stochastic thermodynamics has
ushered in a new era in thermodynamics, largely overcoming these limitations, and
deepening our understanding of the second law as a statistical relation (Sect. 2.3). At
its heart, stochastic thermodynamics seeks to understand how the laws of thermo-
dynamics manifest in microscopic strongly fluctuating systems that are potentially
far from equilibrium. The theoretical framework of stochastic thermodynamics
provides a consistent method of assigning physical quantities–such as work, heat,
and entropy–to fluctuating systems in contact with thermodynamic reservoirs, even
when those systems are far from equilibrium. These physical quantities can be
identified along a single trajectory or at the level of probability distributions, thus
permitting a diverse set of methods to understand the physics of thermodynamic
systems across all scales.
Given a set of state energies, the equilibrium ensemble of a system can be
determined without any knowledge of its dynamics. Thus, the entire theory of
equilibrium statistical mechanics can be built without the need to model the
microscopic dynamics. In nonequilibrium systems, however, the same is not true.
In general, the state of a particular nonequilibrium system depends not only on its
present conditions, but also on its history [4].
In addition to deriving a set of consistent laws of stochastic thermodynamics,
the study of fluctuating systems led to the development of an entirely new class
of results, known collectively as fluctuation theorems [5–11] (Sect. 2.4). These
mathematical identities place stringent constraints on the fluctuations in stochastic
systems, even far from thermodynamic equilibrium, and can be viewed as general-
izations of the second law of thermodynamics. In particular, the entropy production
fluctuation theorem can be used to derive the second law, and provides a much
deeper understanding of its physical origins. In fact, all fluctuation theorems–even
those not relating to entropy–give rise to second-law-like inequalities [12, 13].
This realization has led to a significant improvement in our understanding of the
physical origins of irreversibility in nonequilibrium systems. For instance, the
generalized Jarzynski equality [14] has shown how incorporating information—
in the form of feedback control—into the thermodynamics of fluctuating systems
can lead to sub-zero bounds on entropy production, a result which has been
experimentally verified [15]. The incorporation of information into the theory of
thermodynamics has effectively ‘exorcised’ the long-standing thought experiment
known as Maxwell’s demon [16]: by treating information as a physical quantity—
which was suggested by Landauer in 1961 [17]—the familiar form of the second
law is restored.
In tandem with the theoretical developments of stochastic thermodynamics, a
new set of experimental techniques were developed to directly probe and perturb
microscopic systems [18]. Early experiments were concerned with measuring the
fluctuations in microscopic systems, and primarily aimed at verifying the fluctuation
theorems. Examples include measurements of heat and work fluctuations in the tip
of an AFM [19], the rotational angle of a torsion pendulum [20], and in an electrical
resistor [21]. Furthermore, the development—and subsequent refinement—of opti-](https://image.slidesharecdn.com/23428428-250509151846-2d979635/85/Dissipation-And-Control-In-Microscopic-Nonequilibrium-Systems-Steven-J-Large-22-320.jpg)
![1.1 Molecular Machines 3
cal tweezers for single-molecule force spectroscopy allowed the direct verification
of theoretical predictions in biological systems, such as a DNA or RNA hairpin [22–
24].
Biological systems have served as a central focus for applying stochastic
thermodynamics. Ultimately, this comes as a result of the nonequilibrium nature
of microscopic biological systems. The essential importance of nonequilibrium
physics in our understanding of biology is summarized effectively by Erwin
Schrödinger in What is Life? where he famously equated the ‘decay into thermody-
namic equilibrium’ with death [25]. More recently, a great deal of effort has focused
on our understanding of molecular machines, which are a class of nanoscale objects
that operate out of equilibrium to perform useful tasks within biological cells.
These molecular machines manage to remain out of thermodynamic equilibrium
by siphoning free energy off of nonequilibrium environmental conditions, such as
out-of-equilibrium concentrations of chemical reactants and products. In fact, the
understanding of molecular machines has served as a central motivation for the
continued development of stochastic thermodynamics as a whole [26, 27].
Stochastic thermodynamics can help in furthering our understanding of the
physics of molecular machines, in particular, the fundamental physical constraints
such nonequilibrium systems face in vivo, and how these constraints affect the limits
of performance in molecular machines. Thus, there is a need to define a performance
metric in such systems that quantifies the loss of capacity in the system. The entropy
production, or dissipation, can serve such a role: entropy production represents a
fundamental loss of system capacity to perform useful work. Low-entropy states
have a higher capacity to perform work than high-entropy states; for instance a fully
extended polymer chain has the capacity to exert forces during its compaction. Many
recent significant advances, such as the thermodynamic uncertainty relation [28, 29]
and its generalizations [30–33], have been motivated as providing such fundamental
functional limitations on the performance of molecular machines out of equilibrium.
1.1 Molecular Machines
At the sub-cellular scale, biological systems exhibit a strikingly high degree of
organization which is inconsistent with an equilibrium state [26]. This organization
in maintained, in large part, through the concerted effort of a host of molecular
machines [34]. These nanoscale machines consume energy, typically in the form of
high-energy chemical bonds, to perform useful functions in the cell [27].
Physically, molecular machines are made up of several interacting soft-matter
components, which experience large thermal fluctuations. Over the past several
decades, molecular machines have been a subject of intense focus within the
biophysics community, and have been studied using a host of modern experimental
techniques, such as optical tweezers [35–38], or electrorotation [15, 39–41], to probe
and perturb individual molecular machines. Efforts to better understand the physics
of molecular machines promise to deepen our understanding of the fundamental](https://image.slidesharecdn.com/23428428-250509151846-2d979635/85/Dissipation-And-Control-In-Microscopic-Nonequilibrium-Systems-Steven-J-Large-23-320.jpg)
![4 1 Introduction
operational constraints facing evolved molecular machines, but also promise a
range of practical benefits, such as the design and synthesis of de novo molecular
machines, perhaps accelerating their use in next-generation nanomedicine [42, 43].
1.1.1 Kinesin
A canonical example of a molecular transport motor is kinesin [44]. The first of
many variants, kinesin-1, was discovered in 1985 in observations of the mobility of
organelles in the extruded cytoplasm from the giant axon of a squid [45–47]. Soon
after, this ‘axoplasmic motility’ was understood to be due to the effect of ‘a novel
force-generating molecule’ which has since come to be known as kinesin-1 [48].
Many distinct variations of the kinesin motor were found soon after by identifying
a common genetic motif in the genome of Drosophila melanogaster [49], and
subsequently found to occur, in one form or another, in virtually all forms of
eukaryotic cells [50].
Kinesin is a molecular motor that is responsible for the transport of cellular
cargoes throughout individual cells. Morphologically, kinesin consists of two motor
head domains, which are connected through a long stalk to two cargo binding
domains. By making use of the high-energy phosphate bond in the cellular energy
currency ATP, kinesin moves by its head domains processively walking along
a microtubule, typically in a ‘+’-end directed fashion, although certain kinesin
motors can, in fact, travel in the opposite direction [51]. In vivo, the cell ensures
directionality by maintaining an out-of-equilibrium concentration of ATP and ADP
molecules [44, 50].
Through the use of modern experimental techniques, kinesin has been studied
in great detail, informing much of our modern understanding of the mechanics
of nanoscale machines. For instance, early studies with optical trapping allowed
direct measurement of the stall force—the maximum opposing force under which
the kinesin can still move in the forward direction—of ∼6–8 pN [35–38]. When this
stall force is multiplied by the kinesin step size of ∼8 nm, the resulting work done by
the kinesin motor is ∼48–64 pN· nm, which, when compared with the free energy
∼80 pN·nm liberated by hydrolysis of a single ATP, indicates an efficiency of up
to ∼80% [52]. Additionally, single-particle fluorescence tracking has measured the
in vivo speeds of kinesin motors directly, clocking in at a maximum of ∼1 µm/s, or
125 (8-nm) steps/s [53].
In addition to providing an interesting biophysical model for understanding the
physics of molecular machines, kinesins are suspected to play a central role in
pathophysiology of many diseases. In particular, their role in transport implicates
them in many neuronal disorders, such as amyotrophic lateral sclerosis (ALS) and
Alzheimer’s, where neurons’ extremely polarized cell morphologies make proces-
sive transport motors—such as kinesin—vital to their healthy function [54, 55].](https://image.slidesharecdn.com/23428428-250509151846-2d979635/85/Dissipation-And-Control-In-Microscopic-Nonequilibrium-Systems-Steven-J-Large-24-320.jpg)
![1.1 Molecular Machines 5
1.1.2 ATP Synthase
With ATP serving as the primary chemical fuel source for many molecular machines
(such as kinesin), the production of ATP is essential to the operation of biological
cells. The molecular machine ATP synthase is responsible, in large part, for
the production and maintenance of cellular ATP stores. As with kinesin, several
variations of ATP synthase exist in different organisms, but its presence is ubiquitous
amongst both eukaryotic and prokaryotic cells [56, 57]. In eukaryotic cells, ATP
synthase motors are often located in the membranes of organelles, whereas in
prokaryotic cells they are found in the cell membrane. In fact, the internalization
of ATP synthase—and corresponding increase in cellular energy capacity—during
the emergence of eukaryotic cells in the evolutionary past is thought to have been
a central factor in the subsequent formation of higher-order structures, such as
complex multicellular organisms [58].
While there are several different forms of ATP synthase, the most well-studied is
FoF1 ATP synthase, which is found in eukaryotic mitochondria and chloroplasts
as well as prokaryotic cell membranes. In contrast, variants such as the VoV1
ATPase inhabit organelle membranes other than mitochondria, such as endosomes
and lysosomes [59, 60], while AoA1 ATP synthase appears in the membrane
of extremophilic archaea. VoV1 ATPase operates in the opposite direction of
FoF1, using ATP as an energy source to acidify the interior of organelles, and
contains a biochemical control mechanism, whereby the depletion of cellular
glucose causes dissociation of the Vo and V1 components, inactivating the complex
as an ATPase [60, 61]. Even within the class of FoF1 ATP synthase motors,
however, there is significant diversity across organisms. For instance, the number
of protein subunits in the Fo component can vary between organisms, and can have
a significant impact on its overall energetics [62].
The remarkable efficiencies achieved by ATP synthase make it a fascinating
example of energy transduction in out-of-equilibrium biological systems. The
ATP synthase motor provides a straightforward example of a molecular machine
that converts energy stored in out-of-equilibrium chemical concentrations ([H+]
difference across the membrane) into an essential cellular resource (ATP). This
has made ATP synthase a canonical system in which to model and understand
the fundamental physics of molecular machines and the inherent trade-offs in the
functional capabilities of biomolecular machines [63–66].
Throughout this thesis, the simple models we use to test theoretical models are
motivated by the physics of molecular machines, such as kinesin and ATP synthase.
These represent a minute fraction of the diversity of molecular machines that exist
within biological cells, performing a wide array of functions, and largely responsible
for maintaining the nonequilibrium conditions necessary for life. Fundamentally,
these nanoscale engines operate in the presence of strong thermal fluctuations and
out of thermodynamic equilibrium, and thus to understand their physics we need to
make use of nonequilibrium theories of statistical physics and stochastic processes.](https://image.slidesharecdn.com/23428428-250509151846-2d979635/85/Dissipation-And-Control-In-Microscopic-Nonequilibrium-Systems-Steven-J-Large-25-320.jpg)
![6 1 Introduction
1.2 Nonequilibrium Statistical Physics
For a physical system in contact with a thermal reservoir, at equilibrium the
microstates x are distributed in accordance with the Boltzmann distribution
π(x|λ) = e−βE(x|λ)+βF(λ)
, (1.1)
where β ≡ (kBT )−1 is the inverse temperature of the reservoir, E(x|λ) is the
energy of microstate x given a set of external parameters λ—such as pressure or
volume—and F(λ) is the equilibrium free energy of the system. The equilibrium
free energy is related to the partition function by F(λ) ≡ −kBT ln Z, where Z ≡
exp [−βE(x|λ)] dx normalizes the equilibrium distribution. Thus, to determine
the equilibrium distribution for any system of interest, we only need to know the
energies of each of the microstates.
In contrast, determining the nonequilibrium distribution of a system requires
knowledge of its dynamics. Furthermore, aside from special cases, like a nonequi-
librium steady state, determining the instantaneous nonequilibrium distribution over
microstates also requires knowledge of the previous history of the system. For
systems that we are generally interested in, such as a polymer in solution, there are
a very large number of individual molecules, N ≈ 1023 (counting all of the waters)
that can have non-negligible interactions with the system. The explicit modeling of
all degrees of freedom in such a system is analytically intractable, and even outside
of the realm of numerical calculations.
Fortunately, it is often the case that a relatively small number of these molecules
represent our system of interest. Thus, we seek a method of obtaining equations of
motion for the system of interest that do not require explicitly accounting for all the
molecules that make up the environment. Mathematically, such reduced descriptions
of the system dynamics can be justified by an assumption of weak interactions
between the system and its surroundings, or a separation of timescales between the
system and environment. Weak interactions between the system and its environment
are a primary assumption behind, for instance, Zwanzig’s projection operator
method [4], where the Liouville equation for the entire system is projected onto
a low-dimensional space that captures the dynamics of the system of interest, and
the effective interactions between the system and environment are captured through
an additional fluctuating force. Conversely, the assumption of a separation of time
scales enforces a typical axiom of irreversible thermodynamics: thermodynamic
reservoirs always remain at equilibrium [67]. In either case, the same equations of
motion are the result, which transform the typical differential equations of classical
mechanics to stochastic differential equations, with fluctuating components that
capture the effective interactions between a system and its surroundings.
A canonical example for the equation of motion of a stochastic system is the
diffusive motion of a small object, such as a pollen grain, in water. This phenomenon
was first documented by the botanist Robert Brown in 1827 while observing the
motion of pollen grains in water [68], however it wasn’t until the early twentieth](https://image.slidesharecdn.com/23428428-250509151846-2d979635/85/Dissipation-And-Control-In-Microscopic-Nonequilibrium-Systems-Steven-J-Large-26-320.jpg)
![1.2 Nonequilibrium Statistical Physics 7
century when Einstein [69]—and, independently, Smoluchowski [70]—provided
a clear and consistent interpretation of the seemingly random dynamics. Einstein
proposed that the random motion of the pollen grain was due to its repeated
interaction with the water molecules surrounding it. Furthermore, because the
motion of all of the individual water molecules was exceedingly complex, the
motion of the pollen grain could only be understood statistically. As such, Einstein’s
treatment of what is now called Brownian motion is often seen as the first treatment
of a physical phenomenon using stochastic modeling [71].
These two main points led Einstein to the famous diffusion equation, quantifying
the temporal evolution of the probability of observing the position of the pollen
grain at a particular point in space
∂p(x, t|x0, t0)
∂t
= D∇2
p(x, t|x0, t0) , (1.2)
where p(x, t|x0, t0) is the probability density of observing the pollen grain at
position x at time t given it was at x0 at time t0, ∇2 is the Laplacian operator,
and D is the diffusion coefficient. The diffusion coefficient D is related to the rate
at which the probability distribution spreads out in time. Specifically, the mean-
squared displacement (MSD) of a diffusive particle obeys the relationship
δx2
(t) = 2NdimDt , (1.3)
where the angle brackets · · · indicate an average over an ensemble of pollen
grains, all initialized at the same position at time t = 0, δx ≡ x − x is the
deviation of x from its average position, and Ndim is the number of dimensions in
which the particle is diffusing. For a pollen grain diffusing on the surface of water,
Ndim = 2.1
Not long after Einstein’s discovery of the diffusion equation, Paul Langevin,
a French physicist, presented an alternative treatment of the process of diffusion
using stochastic processes. Specifically, he claimed that the equation of motion of
an individual pollen grain, as per Newtonian mechanics, is given by the differential
equation
d2x
dt2
= −γ
dx
dt
+ F(t) (1.4)
1 There are, however, some subtleties involved in diffusion of a particle at an air-water interface.
Specifically, the diffusion coefficient predicted by the Einstein relation (1.7) may be modified due
to the interaction of the particle with the interface. For instance, an increase in drag forces due to
surface tension [72], the Marangoni effect and capillary forces [73], and even the elastic response
of the interface to fluctuation-induced surface-wave formation [74] can have significant impact on
the value of the diffusion coefficient.](https://image.slidesharecdn.com/23428428-250509151846-2d979635/85/Dissipation-And-Control-In-Microscopic-Nonequilibrium-Systems-Steven-J-Large-27-320.jpg)
![8 1 Introduction
where the first term on the RHS represents a frictional damping force Ffric(t) =
−γ v(t), with γ the viscous damping coefficient for the pollen grain in the fluid,
and F(t) is a random variable that represents a fluctuating force, arising from the
frequent impacts of water molecules with the pollen grain. While it would take
another 40 years to construct a mathematically rigorous derivation of the form of
the fluctuating force, we will simply state two properties of the fluctuating force
that turn out to be true:
F(t) = 0 (1.5a)
δF(t)δF(t
) = 2kBT γ δ(t − t
) . (1.5b)
The first of these properties (1.5a) implies that the fluctuating force does not,
on average, impart any net force on the pollen grain, while the second (1.5b)
suggests that the random forces at different times are uncorrelated. The constant
of proportionality in (1.5b) is fixed by the equipartition theorem [4].
Using (1.5a) and (1.5b), we can write the Langevin equation in its more familiar
form
d2x
dt
= −γ
dx
dt
+
2kBT γ ξ(t) , (1.6)
where now ξ(t) is a zero-mean ξ(t) = 0, delta-correlated white noise process with
ξ(t)ξ(t) = δ(t − t). Finally, we can connect the phenomenological constants γ
from the Langevin equation (1.6) and D from the diffusion equation (1.2) through
the Einstein relation
D =
kBT
γ
. (1.7)
Fundamentally, the diffusion equation (1.2) and Langevin equation (1.6) repre-
sent two different approaches to solving the same problem: what is the equation of
motion for a fluctuating particle in contact with a thermal reservoir? In each case,
the influence of the thermal reservoir comes into the mathematical description of
the system through a transport coefficient (γ or D). The diffusion equation solves
this problem by calculating the full probability distribution function p(x, t) as the
solution of the partial differential equation in (1.2), whereas the Langevin equation
provides a description in terms of a stochastic differential equation, where the
solution is a fluctuating trajectory consistent with the statistical properties of the
dynamics.
Figure 1.1 shows a comparison between trajectory-level and probability distribu-
tion descriptions of a stochastic system. For n = 1000 Langevin trajectories (left),
the empirical distribution over positions x at specific times match the solution to the
corresponding (right) diffusion equation (1.2).](https://image.slidesharecdn.com/23428428-250509151846-2d979635/85/Dissipation-And-Control-In-Microscopic-Nonequilibrium-Systems-Steven-J-Large-28-320.jpg)
![1.4 Contributions to This Thesis 9
0 1 2 3 4 5
Time t
−10
−5
0
5
10
Position
x
0.0 0.2 0.4
P(x|t)
Fig. 1.1 Trajectory-level and distribution-level simulations of diffusive motion give alterna-
tive descriptions of the same physical process. (left) A sample of n = 1000 individual trajectories
of a Langevin simulation of Brownian motion compared to the (right) time-dependent probability
distribution obtained by solving the diffusion equation (solid curves). Circles indicate the empirical
distribution of positions obtained from the ensemble of Langevin trajectories at the same times
(indicated by the red vertical bars on the left), showing agreement
1.3 Overview of This Thesis
The remainder of this thesis consists of three parts, discussing research that broadly
falls into three categories: experimental tests of nonequilibrium theory (Part I), dis-
sipation through the lens of control theory (Part II), and the nonequilibrium physics
of autonomous machines (Part III). The content for Part I presents and elaborates on
results published in [24], Part II discusses the theoretical work presented in [75, 76]
and elaborates upon previous discussions in [75] on lower bounds on dissipation
(Chap. 8). Finally, in Part III we discuss the implications of control-theoretical
models of nonequilibrium physics in understanding the microscopic physics of
molecular machines. Specifically, we investigate the relationship between excess
work and entropy production in strongly coupled nonequilibrium system from [77],
and derive a near-equilibrium phenomenological method of quantifying the energy
flows between the components of such systems presented in [78].
1.4 Contributions to This Thesis
Several chapters in this thesis contain results from collaborative efforts. The
content in Part I (Chaps. 3, 4, and 5) is drawn from material published in [24]
regarding experiments performed in collaboration with Sara Tafoya, Shixin Liu,
and Carlos Bustamante at the University of California, Berkeley. In this project,
Sara Tafoya performed the experiments and gathered the raw data, and all parties](https://image.slidesharecdn.com/23428428-250509151846-2d979635/85/Dissipation-And-Control-In-Microscopic-Nonequilibrium-Systems-Steven-J-Large-29-320.jpg)
![10 1 Introduction
in the collaboration participated in conceptualization of the project. I performed the
data curation and formal analysis of the resulting data, designed experiments, and
wrote the necessary software for the project, including both the data analysis and
visualization. In summarizing the results and discussing how to present them, all
parties in the collaboration took part.
Furthermore, Raphaël Chetrite helped with the formal analysis and mathematical
details of our derivations in Chap. 6 (which are published in [75]), particularly for
the exact solutions found in Sect. 6.4.1. Jannik Ehrich helped with conceptualization
of the project discussed in Chap. 9 and published in [77], performed some of the
formal analysis, and helped in writing up the results.
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4. R. Zwanzig, Nonequilibrium Statistical Mechanics. Oxford University Press, 125 (2001)
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Phys. Rev. Lett. 104, 090602 (2010)
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16. K. Maruyama, F. Nori, V. Vedral, Colloquium: the physics of Maxwell’s demon and informa-
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Rev. X 7, 021051 (2017)](https://image.slidesharecdn.com/23428428-250509151846-2d979635/85/Dissipation-And-Control-In-Microscopic-Nonequilibrium-Systems-Steven-J-Large-30-320.jpg)
![Chapter 2
Theoretical Background
2.1 Mathematical Preliminaries
We now develop the mathematical framework behind the stochastic equations of
motion governing microscopic, fluctuating systems. In particular we will derive
the three primary methods of describing stochastic dynamics: the master equation,
the Langevin equation, and the Fokker-Planck equation. Each of these descriptions
of stochastic dynamics is used throughout the remainder of the thesis. The master
equation is used in Chap. 9, the Langevin equation is used in Chaps. 6 and 10, and
the Fokker-Planck equation is used in Chap. 7.
2.1.1 Random Variables, Probability Distributions and
Characteristic Functions
The fundamental mathematical underpinning of stochastic thermodynamics and
nonequilibrium physics is probability theory. As a mathematical theory originally
axiomatized by Andrey Kolmogorov in the 1930s [1], probability theory has become
central to a number of fields of science and industry, from the fluctuating proteins
within a biological cell to the statistics of returns in an investment portfolio [2, 3]. At
its core, probability theory deals with the mathematics of uncertainty, quantifying
the likelihoods of random events.
The outcome of a random event X, such as a coin toss, is referred to as a random
variable because its outcome is uncertain. While we do not know what the exact
outcome of the event will be, we know what the possible outcomes x may be,
and can assign relative likelihoods to them. For instance, in tossing a fair coin, the
possible outcomes are heads or tails, and both are equally likely. The function that
quantifies the relative likelihoods for different outcomes of the random variable X
is called the probability px, where here px is the likelihood that the outcome of the
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
S. J. Large, Dissipation and Control in Microscopic Nonequilibrium Systems,
Springer Theses, https://doi.org/10.1007/978-3-030-85825-4_2
15](https://image.slidesharecdn.com/23428428-250509151846-2d979635/85/Dissipation-And-Control-In-Microscopic-Nonequilibrium-Systems-Steven-J-Large-34-320.jpg)
![16 2 Theoretical Background
random event will be X = x. While a full treatise on the theory of probability is
beyond the scope of this introduction,1 a central axiom of probability theory is that
the probability distribution is normalized,
x
px = 1 , (2.1)
where the summation is taken over all possible outcomes of the random event.
As defined in (2.1) the probability distribution is applicable to discrete random
variables, and often referred to as the probability mass function. However, when
the outcome of a random variable can be any value in a continuous domain, we
instead use the probability density function p(x), which satisfies the alternative
normalization condition
x
p(x)dx = 1 . (2.2)
Intuitively, the need for a generalized definition of the probability in continuous
spaces derives from the fact that there is no way of assigning the probability of
a single outcome X = x when x has a continuous domain, as each individual
outcome has zero probability.2 Instead, we must refer to the probability that the
outcome x lies within a range of possible values p(a x b) =
b
a p(x)dx [3].
For the majority of this thesis, we will work with the probability density function,
as most model systems we look at are continuous, but will distinguish between
the two distributions with the notation used above: in probability mass functions
the outcome appears as a subscript px, while for probability density functions the
outcome appears as an argument p(x).
While a mathematically complete description of a random variable requires
specification of the probability distribution, we can describe particular properties
of the random variable through its moments. The nth moment of the distribution is
Xn
=
x
xn
p(x)dx , (2.3)
where here the angle brackets · · · indicate an average over the distribution of x.
Here, the distribution being averaged over is unambiguous, however in cases where
it is not, the averaging distribution will be made explicitly clear from the notation,
often with a subscript or superscript.
From (2.3), the zeroth moment (n = 0) of any distribution simply gives the
normalization condition (2.2), and the first moment (n = 1) is the mean of the
1 For the interested reader, however, [4] provides a succinct introduction to the mathematical
subject.
2 Or, in the language of probability theory, each individual event represents a set of zero measure.](https://image.slidesharecdn.com/23428428-250509151846-2d979635/85/Dissipation-And-Control-In-Microscopic-Nonequilibrium-Systems-Steven-J-Large-35-320.jpg)
![2.1 Mathematical Preliminaries 17
distribution, which is also often referred to as the expectation value of X. Higher-
order moments encode more detailed information about the shape of the distribution,
with each successively higher-order n providing information about events that are
farther from the mean [2]. However, for n 1 there are alternative, more convenient
measures used to quantify the characteristics of the distribution. For instance, the
centered moments subtract the mean (first moment) to give moments relative to the
mean value,
δXn
=
x
(x − x)n
p(x)dx , (2.4)
where here, and in the remainder of the thesis, δX ≡ X−X indicates the difference
between a random variable and its mean value. Trivially δX = 0, and thus the first
nonzero centered moment is for n = 2. δX2 is the variance, which is often written
as σ2
x as it is equal to the square of the standard deviation σx [5]. More convenient
still are the cumulants κn of a distribution (see (2.7) below). The first two cumulants
are simply the mean (κ1) and variance (κ2), and, while higher-order cumulants can
be expressed as combinations of moments (2.3), no simple closed-form expression
is known to do so [6].
An alternative route to understanding properties of a probability distribution
comes through its Fourier transform, which is known as the characteristic function
(or moment-generating function)
φ(s) ≡ e−isx
=
x
e−isx
p(x)dx . (2.5)
The Fourier transform is used extensively in physics, particular when considering
periodic behavior, such as waves [7]. In the context of probability theory, the
characteristic function provides an often useful tool in solving problems, as well as a
route to more simply calculate the moments (2.3) and cumulants of a distribution. In
particular, the nth moment of a distribution can be calculated from the characteristic
function as
Xn
= i−n ∂nφ(s)
∂sn
s=0
. (2.6)
Furthermore, the characteristic function can be used to define, mathematically,
the cumulants through the cumulant-generating function g(s) ≡ − ln φ(s). The
cumulants κn can be determined from the cumulant-generating function, similar
to (2.6), as
κn ≡ i−n ∂ng(s)
∂sn
s=0
. (2.7)](https://image.slidesharecdn.com/23428428-250509151846-2d979635/85/Dissipation-And-Control-In-Microscopic-Nonequilibrium-Systems-Steven-J-Large-36-320.jpg)
![18 2 Theoretical Background
The cumulant-generating function has become central in the field of large deviation
theory, where a scaled cumulant-generating function is often used to determine the
large deviation rate function [8]. However, the cumulants also have fundamental
importance in standard treatments of probability theory. For instance, all cumulants
above κ2 (i.e., third order and higher) are zero for a Gaussian distribution, while no
analogous truncations exist for the moments [6].
For situations described by multiple random variables, such as the outcome of
several coin flips, the probability distribution of outcomes is multivariate. Here,
there are several other important concepts to understand. In particular, for two
random variables X and Y, the probability of a particular outcome is now given by
the joint distribution p(x, y), which is the probability that both X = x and Y = y.
The joint distribution obeys the normalization condition 1 =
x,y p(x, y)dxdy. The
univariate distribution p(x)—also called the marginal distribution in this context—
can be obtained by integrating—or marginalizing—over the y-coordinate as
p(x) =
y
p(x, y)dy . (2.8)
Alternatively, given we know the outcome of Y, the probability of X conditioned on
the outcome Y = y is known as the conditional distribution p(x|y).
We can relate all three distributions through the law of total probability as [5]
p(x, y) = p(x|y)p(y) = p(y|x)p(x) . (2.9)
Finally, if the random variables X and Y are independent of one another, then the
conditional and marginal distributions are equal (p(x|y) = p(x) and p(y|x) =
p(y)), and therefore the joint distribution can be expressed as a product of the
marginal distributions
p(x, y) = p(x)p(y) . (2.10)
2.2 Nonequilibrium Dynamics
To describe the state of a fluctuating system, such as the position of a diffusing
particle, we can gain tremendous insight by making use of the machinery of
probability theory. In particular, the state of a system at a time t is itself a
random variable X(t) that depends on time. For instance, if we observe the state
of the system at a series of times t0, t1, · · · , tn, so that increasing indices indicate
observations that are later in time (i j implies that ti tj ), and each successive
time is spaced with the increment ti+1 − ti = t, then the system trajectory is a
vector of random variables X(t) = (Xn, Xn−1, · · · , X0) ≡ Xt[0,n] . Here, Xi is the
state X at time ti, and the subscript t[0,n] is shorthand notation for the sequence
of times t0, t1, · · · , tn. This process is an example of a discrete-time stochastic](https://image.slidesharecdn.com/23428428-250509151846-2d979635/85/Dissipation-And-Control-In-Microscopic-Nonequilibrium-Systems-Steven-J-Large-37-320.jpg)
![2.2 Nonequilibrium Dynamics 19
process, because the random variable is observed at discrete time increments;
however, this can be turned into a continuous-time process by taking the limit of
t → 0.
The probability of observing a particular trajectory of the system is given by
the multivariate joint distribution p(x[0,n]). Furthermore, by viewing the trajectory
x[0,n] as a time series, the conditional distribution can be understood as a predictive
tool, quantifying the probability distribution of future states of the system, given a
previous history. For instance,
p(xn|x[0,n−1]) ≡
p(xn, x[0,n−1])
p(x[0,n−1])
(2.11)
defines the probability of observing the system in state xn given that the n previous
observations were x[0,n−1].
An assumption that greatly simplifies the analysis of stochastic systems is the
Markov assumption, which asserts that the state of the system at time tn only
depends on the most recent previous state, at tn−1, and is thus conditionally
independent of x[0,n−2]. More concretely, the Markov assumption states that
p(xn|xn−1, x[0,n−2]) = p(xn|xn−1) , (2.12)
and systems which obey this assumption are said to be Markovian. The single-
timestep conditional probabilities p(xi+1|xi) are often called transition probabili-
ties, as they represent the probability of observing the system in state xi+1 at time
ti+1 given that it was in state xi at time ti. While there are many other common
and useful assumptions that can be made on the interdependence of system states
along a stochastic trajectory, such as the Martingale property that has become central
to many formal treatments of stochastic processes and probability theory [9], the
Markov assumption is, by and large, the most ubiquitous in the modern treatment of
stochastic thermodynamics.
The primary utility of the Markov assumption is that it allows the decomposition
of stochastic trajectories into transition probabilities. In particular, for a system
which at time t0 is distributed over x states as pinit(x0), the trajectory probability
distribution over states after n time increments can be decomposed into a product of
transition probabilities as
p(x[0,n]) = p(xn|xn−1)p(xn−1|xn−2) · · · p(x1|x0)pinit(x0) . (2.13)
For a Markovian system, we can use (2.13) to write the trajectory probability
p(x2, x1, x0) = p(x2|x1)p(x1|x0)p(x0). By then marginalizing over x1 (2.8) we
obtain a tremendously useful identity in the study of stochastic processes called the
Chapman-Kolmogorov equation:
p(x2|x0) =
p(x2|x1)p(x1|x0)dx1 (2.14)](https://image.slidesharecdn.com/23428428-250509151846-2d979635/85/Dissipation-And-Control-In-Microscopic-Nonequilibrium-Systems-Steven-J-Large-38-320.jpg)
![20 2 Theoretical Background
for a continuous state space, or, for a discrete state space:
p(x2|x0) =
x1
p(x2|x1)p(x1|x0) . (2.15)
Under certain constraints on the limits of the transition probabilities as the time
increment t → 0, the Chapman-Kolmogorov equation (2.14) (or (2.15)) can be
written as a differential equation.3 Specifically, for the transition rate R(x|x, t) ≡
lim t→0
p(x,t+ t|x,t)
t , one can show that the differential Chapman-Kolmogorov
equation is [2]
∂t p(x, t|x
, t
) = −
i
∂xi
Ai(x, t)p(x, t|x
, t
)
+
1
2
i,j
∂2
xi ,xj
Bij (x, t)p(x, t|x
, t
)
+
R(x|x
, t)p(x
, t|x
, t
) − R(x|x
, t)p(x
, t|x
, t
)
dx . (2.16)
Here the state x is, in general, a vector, ∂t ≡ ∂/∂t is the partial derivative with
respect to time, and ∂xi ≡ ∂/∂xi is the partial derivative with respect to the ith
component of x. This form indicates three different types of motion that can arise
in stochastic dynamics. In particular, Ai(x, t) represents a drift term, quantifying
deterministic motion of the system, while Bji(x, t) is a positive-semidefinite
diffusion tensor, quantifying the impact of diffusive motion. The final term on the
RHS of (2.16) represents the impact of discrete jumps on the overall stochastic
dynamics of the system. In various limits, the differential Chapman-Komogorov
equation (2.16) reduces to common equations of motion for stochastic systems.
2.2.1 Master Equation
For systems in which the drift and diffusion coefficients are zero (Ai(x, t) =
Bij (x, t) = 0, for all i, j), the resulting differential equation is the master equation,
which describes the rate of change of the probability in a system where the source of
probability changes are solely due to jumps. While one can use the master equation
to describe the dynamics on continuous state spaces [2], for the purposes of this
thesis, we will use the master equation when discussing discrete-state systems. The
discrete master equation is often used to model the dynamics of chemical reaction
networks [10], or as a simple representation of a more complex problem, for instance
following coarse-graining [11].
3 See Gardiner [2] for a more in-depth discussion of these constraints, but simply put, the
constraints relate to the continuity of the underlying stochastic process in the continuous-time
limit.](https://image.slidesharecdn.com/23428428-250509151846-2d979635/85/Dissipation-And-Control-In-Microscopic-Nonequilibrium-Systems-Steven-J-Large-39-320.jpg)
![2.2 Nonequilibrium Dynamics 21
Thus, the discrete master equation quantifies the rate of change of the probability
distribution in a stochastic system due to jumps between discrete states. Mathemat-
ically, the discrete master equation is
dt pi =
j
Rij pj , (2.17)
where pi is the probability of the system being in state i at time t, dt ≡ d/dt
is shorthand for the time derivative, and Rij is the (i, j) element of the transition
rate matrix, quantifying the rate of transitions from state j → i per unit time. Rij
is shorthand for R(xi|xj ) (here assumed to be time-independent), and the indices
indicate that the state space is discrete. In order to conserve probability, the diagonal
elements of the transition rate matrix are constrained by
Rii = −
j=i
Rij . (2.18)
It can be shown, in several different ways, that in the long-time limit—and for
time-independent transition rates—solutions to the master equation converge to a
stationary distribution [6]. Furthermore, this distribution is unique if the transition
rate matrix is irreducible, which means that each state in the system is accessible—
via intermediate states—by any other state. Concretely, it is possible for a system
initially in state xi at t = 0 to be found in any other state of the system xj at a
later time. In what follows we will assume that there is a unique stationary solution,
and will refer to it in general as the steady-state distribution. Once the dynamics
have a thermodynamic interpretation, under certain circumstances the steady-state
distribution is equivalent to the Boltzmann equilibrium distribution (1.1).
At steady state, the probability distribution is unchanging: dt pi = 0. Therefore
the stationary distribution pss
i solves the equation
j
Rij pss
j = 0 . (2.19)
Thus the stationary distribution is the (unique) right eigenvector of the transition
rate matrix corresponding to the zero eigenvalue [6]. Thus for a time-independent
rate matrix, one only needs to solve for the eigenvectors of the transition rate matrix
to identify the steady-state distribution.
If the master equation (2.17) represents a thermodynamic system, the entries in
the transition rate matrix are further restricted to satisfy the local detailed balance
relationship
Rij
Rji
= exp(−β ωij ) , (2.20)](https://image.slidesharecdn.com/23428428-250509151846-2d979635/85/Dissipation-And-Control-In-Microscopic-Nonequilibrium-Systems-Steven-J-Large-40-320.jpg)
![22 2 Theoretical Background
where ωij = ωi − ωj is the difference in the thermodynamic potential of states
xi and xj . For instance, if the system is in contact with a thermal reservoir, the
thermodynamic potential of each state xi is its energy ωi = i, and thus asymmetries
in the transition rates are completely determined by the relative energies of the
states. The local detailed balance constraint is necessary in such systems so that
they remain consistent with the known predictions of thermodynamics. Specifically,
for a thermodynamic system the equilibrium distribution is known from the study
of statistical mechanics, and the rate definitions in (2.20) ensure that the stationary
distribution and the equilibrium distribution are equal.
To generate trajectories of a system obeying the master equation (2.17), we need
to know the transition kernel p(x, t+ t|x, t), which can be obtained by expanding
the time-dependent probability distribution from (2.17) to linear order in t and
using the master equation (2.17) to evaluate the derivatives [2]:
p(xi, t + t|xj , t) = δij 1 −
k
Rkj t + Rij t + O( t2
) , (2.21)
where δij = p(xi, t|xj , t). The first RHS term is the probability that during time
t the system remains at xj , while the second RHS term is the probability that
it transitioned to state xi from xj . Thus, for a system initially in state xj , after a
small time t has passed, we can calculate its state by drawing a new state (xi) at
random from p(xi, t+ t|xj , t). Numerically, this can be achieved by implementing
a kinetic Monte-Carlo algorithm.4 The details of our implementation are given in
App. A.
As an example, consider the underlying set of states xi ∈ Z, where Z is the set
of integers, subject to a harmonic potential centered at x0 = 0, so that the system
evolves in the potential i =
ktrap
2 x2
i where ktrap is the trap stiffness. Transition rates
of the general form
Rij = e− 1
2 β ij (2.22)
for constant kinetic bare rate and ij ≡ i − j satisfy local detailed
balance (2.20) and thus represent the stochastic dynamics of a physical system in
contact with a thermal reservoir.
Figure 2.1 shows a sample trajectory (left) of the system in the harmonic potential
for the first 50 discrete jumps, and the steady-state distribution pss
x (2.19) (right)
given by the eigenvector corresponding to the zero-eigenvalue of the transition rate
matrix. The trajectory consists of lag times, during which the system remains in a
fixed state, interspersed with instantaneous discrete jumps.
4 This is analogous, in this case, to the Gillespie algorithm used to propagate the chemical master
equation.](https://image.slidesharecdn.com/23428428-250509151846-2d979635/85/Dissipation-And-Control-In-Microscopic-Nonequilibrium-Systems-Steven-J-Large-41-320.jpg)
![2.2 Nonequilibrium Dynamics 23
0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5
Time t
−5.0
−2.5
0.0
2.5
5.0
State
x
0.0 0.2
px
Fig. 2.1 Discrete-state dynamics generated by the master equation in a harmonic potential.
(left) Sample of the first 50 steps of a discrete-state trajectory governed by the discrete master
equation (2.17) and subject to a harmonic confining potential. Each circle represents the start or
end of a discrete jump in the state. We set = 1 in the transition rates (2.22). (right) Empirical
distribution (circles) obtained from a long trajectory of the discrete dynamics compared to the
corresponding Boltzmann distribution (1.1) (bars) for a system in thermodynamic equilibrium. In
both subplots we take β = 1
2.2.2 Fokker-Planck Equation
For systems in which there are no discrete jumps (R(x|x, t) = 0 everywhere),
but nonzero drift and diffusion coefficients, the differential Chapman-Kolmogorov
equation (2.16) reduces to the Fokker-Plank equation which, in its general form, is
∂t p(x, t) = −
i
∂xi
Ai(x, t)p(x, t|x
, t
)
+
1
2
i,j
∂2
xi ,xj
Bij (x, t)p(x, t|x
, t
)
. (2.23)
For the duration of this thesis, we will be concerned with time- and space-
independent diffusion, so that Bij (x, t) = 2D (so that the factor of 1/2 in front of
the diffusive term is canceled), and further restrict our attention to the 1D Fokker-
Planck equation corresponding to an overdamped system, which is also known as
the Smoluchowski equation.5 Here, we consider the Smoluchowski equation for a
diffusing particle in a time-dependent potential E(x, t), which means that the drift
coefficient in (2.23) is replaced by (βD times) the force f (x, t) ≡ −∂xE(x, t):
A(x, t) = −βD∂xE(x, t).
5 To get an underdamped equation of motion, we need to also include the time evolution of the
distribution of velocities. The Kramers equation does just this for a 1D system, but we won’t
discuss it in detail here. For the interested reader, however, an excellent overview is given in Hannes
Risken’s book [12].](https://image.slidesharecdn.com/23428428-250509151846-2d979635/85/Dissipation-And-Control-In-Microscopic-Nonequilibrium-Systems-Steven-J-Large-42-320.jpg)
![24 2 Theoretical Background
Integrating the Fokker-Planck equation over an (arbitrary) initial distribution
p(x, t = 0) converts the conditional equation into an equation for the marginal
probability distribution p(x, t). Thus the 1D overdamped Fokker-Planck equation
is
∂t p(x, t) = βD ∂x [f (x, t)p(x, t)] + D ∂2
x p(x, t) . (2.24)
The stationary solution—if it exists—of the Fokker-Planck equation can be obtained
by setting ∂t p(x, t) = 0 and solving the RHS of (2.24).
In general, the full time-dependent distribution p(x, t) is rarely analytically
tractable, and must be obtained through numerical methods. However, when the
energy landscape is harmonic (E(x) =
ktrap
2 x2), the forces are linear, f (x) =
−∂xE(x) = −ktrapx, and the Smoluchowski equation becomes
∂t p(x, t) = −βDktrap [xp(x, t)] + D ∂2
x p(x, t) . (2.25)
Here—and, in fact, for any Fokker-Planck equation subject to a potential-derived
drift coefficient—the (equilibrium) steady-state solution can be found from the
Boltzmann distribution
peq(x) =
ktrap
2π
e−
βktrap
2 x2
, (2.26)
which here is Gaussian with mean xeq = 0 and variance δx2eq = (βktrap)−1.
Here, the angle brackets · · · eq indicate an average over the equilibrium distribu-
tion (2.26).
In this special case we can also find an analytical solution for the full time-
dependent distribution, for any initial conditions. For an initial distribution that is
localized at x = x so that p(x, t = 0) = δ(x − x), the time-dependent solution is
Gaussian at all times t 0,
p(x, t|x
, t
= 0) =
βktrap
2π(1 − e−2βDktrapt )
exp −
βktrap
2
(x − xe−βDktrapt )2
1 − e−2βDktrapt
,
(2.27)
with mean x(t) = xe−βDktrapt and variance δx2 = (1 − e−2βDktrapt )/(βktrap).
Asymptotically as t → ∞, the mean and variance approach the steady-state values.
We will make extensive use of this general form in Chap. 7.
Figure 2.2 shows the time-dependent probability distribution obtained by solving
the Smoluchowski equation (2.24) in a harmonic potential with βktrap = D = 1,
given an initial distribution p(x, t = 0) = δ(x−2). The distribution relaxes towards
the equilibrium distribution—indicated by a dashed red line—obtained from the
Boltzmann equation (2.26). The right panel shows the time-dependent average and
variance of the distribution, which converge towards their equilibrium values of
xeq = 0 and δx2eq = 1.](https://image.slidesharecdn.com/23428428-250509151846-2d979635/85/Dissipation-And-Control-In-Microscopic-Nonequilibrium-Systems-Steven-J-Large-43-320.jpg)
![2.2 Nonequilibrium Dynamics 25
−3 −2 −1 0 1 2 3
Position x
0.0
0.2
0.4
0.6
p(x,
t)
τ
1/4
1/2
1
2
4
πeq
0 5 10
Time t
−2
−1
0
1
Cumulant
x(t)
δx(t)2
Fig. 2.2 Time-dependent solution to the Smoluchowski equation for a diffusing particle in a
harmonic potential. (left) Time-dependent solution to the Smoluchowski equation for a Brownian
particle diffusing in a harmonic potential (2.27), given an initial distribution p(x, 0) = δ(x − 2).
Early times are indicated by light blue, while later times are indicated by progressively darker
shades of blue. The red dashed line is the Boltzmann equilibrium distribution towards which the
system is relaxing. (right) The values of the first two cumulants—mean (yellow) and variance
(green)—of the distribution p(x, t) as a function of time, showing that they asymptote to their
equilibrium values of x = 0 and δx2 = 1 as t → ∞
2.2.3 Langevin Equation
Similar to the use of the transition kernel of the master equation to generate
trajectories that represent the time evolution of single instances of the stochastic
process, the transition kernel of the Fokker-Planck equation gives rise to a stochastic
equation of motion for individual trajectories known as the Langevin equation. For
instance, one can show that for the Smoluchowski equation of a diffusing particle in
a potential E(x, t), the transition kernel is, to leading order in t [2],
p(x, t + t|x
, t) =
D
2π t
exp −
(x − x + βDf (x, t) t)2
2D t
, (2.28)
which is simply a Gaussian distribution with variance D t and mean x −
βDf (x, t) t, where f (x, t) = −∂xE(x, t) is the force experienced by the particle
at position x and time t. Thus, the resulting stochastic dynamics are those of
a system moving with a systematic drift of velocity βDf (x, t) with zero-mean
Gaussian fluctuations with variance D t.
Therefore, the update rule for a trajectory generated by this equation of motion
is
x(t + t) = x(t) + βDf (x, t) t +
√
2DN(0, t) , (2.29)](https://image.slidesharecdn.com/23428428-250509151846-2d979635/85/Dissipation-And-Control-In-Microscopic-Nonequilibrium-Systems-Steven-J-Large-44-320.jpg)
![26 2 Theoretical Background
where N(0, t) is a normal distribution with 0 mean and variance t. The
Gaussian-fluctuation term is known as a Wiener increment, and often indicated by
W. In the continuous-time limit ( t → 0), (2.29) becomes
dx = βDf (x, t)dt +
√
2D dW(t) , (2.30)
where dW(t) is the differential element of a continuous-time stochastic process
W(t), known as a Wiener process (also called a white-noise process as it has a
flat power spectral density over all frequencies) [2]. The Wiener process is defined
by a couple of important properties, namely it has zero mean (W(t) = 0),
unit variance (W(t)2 = 1), and its values at different times are independent,
W(t)W(t) = δ(t − t).
Alternatively, it is common to rewrite (2.30) as a differential equation, in terms
of derivatives as opposed to differentials:
ẋ = βDf (x, t) +
√
2D ξ(t) , (2.31)
where ẋ ≡ dx/dt, ξ(t) ≡ dW(t)/dt, and δξ(t)δξ(t) = δ(t − t). Throughout
this thesis, we will primarily use the derivative form (2.31), except when using the
differential form is more clear.
Equation (2.30) (or (2.31)) is an example of a stochastic differential equation
(SDE) because it contains a stochastic differential term (dW(t) or ξ(t)). As we
will see, this makes the subsequent analysis significantly different from standard
ordinary differential equations. Here, the sample paths generated by the Langevin
equation (2.30) are continuous everywhere, but differentiable nowhere, due to the
properties of the Wiener process [2].6
The trajectories generated by the SDE in (2.30) represent the same underlying
physical process as the Smoluchowski equation (2.24). In both cases we are
implicitly assuming that the system is overdamped. This means that there are no
inertial terms in the equations of motion, which is an oft-used approximation in
low-Reynolds number environments.
The Reynolds number ‘Re’ is a dimensionless quantity used in fluid mechanics
to classify the relative importance of inertial and viscous forces [13]. For small
values of the Reynolds number, viscous forces dominate, and inertial effects can
largely be ignored [14]. Approximations on a single bacterium predict that its
Reynolds number is ∼10−5 which implies that its dynamics are governed largely
6 In fact, this is an argument made by Jacobs in [3] for the use of differential notation for SDEs
in preference to writing them in terms of derivatives. The non-differentiability of sample paths
implies that the derivatives themselves don’t exist, and thus it is more mathematically consistent to
write the Langevin equation (or any SDE for that matter) in differential notation, such as in (2.30).](https://image.slidesharecdn.com/23428428-250509151846-2d979635/85/Dissipation-And-Control-In-Microscopic-Nonequilibrium-Systems-Steven-J-Large-45-320.jpg)
![2.3 Stochastic Thermodynamics 27
by viscous forces.7 Thus, the overdamped approximation plays a prominent role in
the microscopic physics of biological systems.
The overdamped equation of motion (2.30) is a particular limit of a more
general Langevin equation, where the fluctuations, which are assumed to impart
forces on the microscopic particle, appear in the equation of motion for the
acceleration. Specifically, the underdamped Langevin equation, which could be
derived, in principle, from the transition kernel for a Fokker-Planck equation of
an underdamped system–such as the Kramers equation–is
mẍ = −f (x, t) −
1
βD
ẋ +
2
β2D
ξ(t) . (2.32)
Here, the first term on the RHS represents the force applied to the system (either
by a potential, or some external source), the second term on the RHS represents
the frictional damping of velocities, and the final term on the RHS represents the
stochastic forces on the particle from the environment. To recover the overdamped
equation of motion (2.31) from the underdamped equation of motion, we simply set
ẍ = 0 and solve for ẋ.
2.3 Stochastic Thermodynamics
The preceding sections describe methods to generate the stochastic dynamics of
fluctuating systems, based solely on the mathematical concepts of probability
theory; however, these trajectories still lack a thermodynamic interpretation. This
interpretation is provided by the growing field of stochastic thermodynamics.
Stochastic thermodynamics is the study of thermodynamics at the microscopic
scale, where systems are small and fluctuations cannot be ignored. At its heart,
stochastic thermodynamics seeks to generalize the classical laws of thermody-
namics to fluctuating trajectories. As considered here, the systems of interest are
embedded within an aqueous environment, and can be pushed out of equilibrium in
a number of different ways. For instance, systems can be driven out of equilibrium
by altering their potential energy in a time-dependent manner through a control
parameter λ, which we discuss in more detail in Sect. 2.6. Here, we can study the
response of the system to a time-dependent control parameter λ(t), the spontaneous
relaxation towards equilibrium after an instantaneous change in the control param-
7 Specifically, the Reynolds number is defined mathematically as Re ≡ Ua/ν where U is the speed
of the object, a is a characteristic linear dimension, and ν is the kinematic viscosity of the fluid (for
water, ν ≈ 10−6 m2/s). For a typical bacterium (such as, for instance, E. coli) the characteristic
linear dimension is a ≈ 10−6 m and they travel at velocities of U ≈ 10−5 m/s and thus the
Reynolds number is Re ≈ 10−5 [14].](https://image.slidesharecdn.com/23428428-250509151846-2d979635/85/Dissipation-And-Control-In-Microscopic-Nonequilibrium-Systems-Steven-J-Large-46-320.jpg)
![28 2 Theoretical Background
eter value, or the properties of nonequilibrium steady states when the driving is
time-independent.
Alternatively, systems such as chemical reaction networks can be kept out of
equilibrium by chemostats that maintain the out-of-equilibrium concentrations of
chemical reactants. Here, the system is still driven out of equilibrium, even though
there is no active manipulation of the system through, for instance, a control
parameter.
Formally, the separation of system from surroundings in stochastic thermo-
dynamics involves identifying fast- and slow-relaxing degrees of freedom. The
slow-relaxing degrees of freedom constitute the system, while the fast-relaxing
degrees of freedom comprise the bath (or environment). This is one reason why
the study of microscopic biological systems has proved to be difficult: there is often
no obvious separation of timescales allowing one to separate, unambiguously, the
system from the environment.
At the microscopic scale, the mathematical forms of the laws of thermodynamics
are largely preserved, but a complete understanding of how exactly to attribute
increments of work, heat, and entropy to single trajectories is a nontrivial process.
Throughout this thesis we will adopt the convention that positive work and heat
correspond to energy flows into the system. Naturally, this gives rise to the
differential form of the first law
dE =d̄W +d̄Q , (2.33)
where dE represents a change in the internal energy of the system, and the inexact
differentialsd̄ indicate that the work and heat are path-dependent.
Given an overdamped Langevin equation describing a fluctuating particle, Seki-
moto was the first to suggest a thermodynamic interpretation [15]. In particular, how
can we quantify the heat and work associated with a single stochastic trajectory?
Conceptually, work is done on a system when an external agent inputs energy, and
heat represents the exchange of energy between the system and its environment.
Thus, in the absence of any external input, all energy changes are heat. For an
overdamped system fluctuating in a fixed potential energy landscape V (x, λ),
changes in energy are determined by the changes in potential dE = dV , which, by
using the chain rule, can be written in terms of a differential in the particle position
dV = (∂xV )dx = −d̄Q, quantifying the exchange of heat with the reservoir. Thus,
the heat along a stochastic trajectory is given by the integral expression [15, 16]
Q[x[0,τ]] = −
τ
0
f (x(t))ẋ dt , (2.34)
where here the notation Q[x[0,τ]] indicates that the heat is a functional of the
trajectory x during times t ∈ [0, τ]. We can use (2.34) to obtain an exact expression
for the heat in terms of the stochastic dynamics by substituting (2.31) for ẋ.
However, because ẋ is a stochastic differential equation, direct integration could
pose a problem, as the integral of the stochastic process has not yet been defined.](https://image.slidesharecdn.com/23428428-250509151846-2d979635/85/Dissipation-And-Control-In-Microscopic-Nonequilibrium-Systems-Steven-J-Large-47-320.jpg)
![2.3 Stochastic Thermodynamics 29
We omit discussion of the different definitions of stochastic integrals, but specify
here that the Sekimoto definitions of heat (2.34) (and work (2.35)) must use the
Stratonovich interpretation of the stochastic integral [15].
Alternatively, work is done on the system if the control parameter λ changes in
time. Thus, changes in energy due to work are dE = dV = (∂λV )dλ =d̄W, and the
work done along a trajectory is
W[x[0,τ]] =
τ
0
(∂λV (x(t))) λ̇ dt . (2.35)
Combining (2.35) with (2.34), the integrated first law is
E = V (x(τ), τ) − V (x(0), 0) = Q[x[0,τ]] + W[x[0,τ]] , (2.36)
on the level of an individual stochastic trajectory of the system.
Equation (2.36) provides a powerful conceptual separation of the work and heat
accumulation at the microscopic level: work accumulates when external influences
on the system results in changes of energy, while heat accumulates when the system
exchanges energy with the bath. The first law, which effectively is a statement of
energy conservation, links these two energy flows.
Identifying a valid form of the second law of thermodynamics requires a
consistent definition of entropy along a single trajectory. However, the second law
refers to the entropy of the universe, so we must identify the changes in entropy of
both the system and the environment. The entropy change in the environment during
a trajectory can be identified with the heat
Sres[x0,τ ] = −βQ[x[0,τ]], (2.37)
while the instantaneous entropy of the system at time t is
S(t) = − ln p(x(t), t) , (2.38)
where, here and throughout the thesis, we use natural units (kB = 1) for the entropy.
Here, the probability distribution p(x(t), t) is obtained by first solving, for instance,
the corresponding Fokker-Planck equation for the system dynamics. The second
law thus follows by measuring the total change in entropy of the universe over a
trajectory,
Stot[x[0,τ]] = Sres[x[0,τ]] + S(τ) − S(0) , (2.39)
where, the final two terms on the RHS capture the change in the entropy of the
system state over the time interval. Unlike the usual form of the second law, the
entropy change along a single trajectory can be negative [17]. For consistency with
the classical laws of thermodynamics, we require only that Stot[x0,τ ] ≥ 0
where · · · indicates an average over an ensemble of trajectories under the](https://image.slidesharecdn.com/23428428-250509151846-2d979635/85/Dissipation-And-Control-In-Microscopic-Nonequilibrium-Systems-Steven-J-Large-48-320.jpg)
![30 2 Theoretical Background
same driving. In (2.39), there is no requirement that the system starts and/or ends
in equilibrium, however to calculate the entropy from (2.38) we must know the
probability distribution p(x(t), t) over states.
The stochastic thermodynamic study of trajectory-level entropy led to a far more
general form of the second law through an entirely new class of mathematical
identities known collectively as fluctuation theorems.
2.4 Fluctuation Theorems
Historically, the identification of symmetries under time reversal of dynamical
systems began with a seminal paper by Gallavotti and Cohen in which they related
the change in Kolmogorov-Sinai entropy8 to the ratio of trajectory probabilities
under time-forward and time-reversed dynamics [19]. Subsequent study on the
probabilities of second-law-violating events in steady-state systems [20, 21] as well
as earlier research on the work integrated over driven trajectories by Bochkov and
Kuzovlev [22, 23] led to the well-known Jarzynski equality [24],
e−βW
= e−β F
, (2.40)
which relates the exponentiated trajectory work βW ≡ βW[x[0τ]|] averaged over
an ensemble of nonequilibrium trajectories (LHS), to the exponentiated change
in equilibrium free energy β F ≡ βF(τ) − βF(0) (RHS). In the Jarzynski
equality, the system is assumed to be driven through the manipulation of an external
control parameter λ—or, in Jarzynski’s words, a work parameter—and the average
· · · is taken over the ensemble of nonequilibrium responses of the system to the
particular driving protocol. Importantly, the Jarzynski equality holds regardless of
how far the system is driven out of thermodynamic equilibrium.
By making an appeal to Jensen’s inequality—an inequality pertaining to averages
over convex functions that is used ubiquitously in information theory [18]—the
Jarzynski equality can be reduced to the inequality,
W ≥ F (2.41)
or, alternatively
Wex ≥ 0 (2.42)
where here we define the excess work Wex ≡ W − F as the difference between
the work and the free energy difference. The lower bounds implied by the Jarzynski
8 The Kolmogorov-Sinai entropy is a definition of entropy, used often in dynamical systems theory,
that is calculated from the Lyapunov exponents in chaotic systems [18].](https://image.slidesharecdn.com/23428428-250509151846-2d979635/85/Dissipation-And-Control-In-Microscopic-Nonequilibrium-Systems-Steven-J-Large-49-320.jpg)
![2.4 Fluctuation Theorems 31
equality (2.41) (and (2.42)) can be seen as alternative statements of the second
law of thermodynamics, when the excess work is the only source of entropy
production [17].
The Jarzynski equality is an example of what is now called an integrated
fluctuation theorem (IFT), to contrast with a detailed fluctuation theorem (DFT).
While an IFT involves ensemble averages, a DFT relates the probabilities of
individual stochastic trajectories. The first DFT found was the Crooks fluctuation
theorem [25],
p[x|]
p̃[x̃| ˜
]
= e−βWex[x|]
, (2.43)
which relates the ratio of trajectory probabilities p[x|] for a system driven
through a time-dependent control protocol , to the probability of its time-reversed
trajectory p̃[x̃| ˜
] under the time-reversed protocol ˜
. Here the tilde indicates the
time-reversal operation, with x̃(t) = x(τ − t) and λ̃(t) ≡ λ(τ − t). The Crooks
theorem can be re-written in terms of the work distribution along forward and
reverse processes as p(Wex)/p̃(−Wex) = exp(βWex) and, in fact, can be used to
derive the Jarzynski equality. Thus, the Crooks theorem represents the DFT that,
upon integration, reproduces the Jarzynski IFT. In fact, the existence of a DFT
directly implies a corresponding IFT, and vice-versa [26].
Among the fluctuation theorems, the entropy production fluctuation theorem is
central, reducing to other fluctuation theorems, such as the Crooks IFT, in particular
settings. Proven in its current form by Seifert in 2005 [27], the entropy production
fluctuation theorem is
p( Stot)
p̃(− Stot)
= e Stot , (2.44)
relating the exponentiated change in entropy to the ratio of the probability of
observing a total entropy change Stot along a forward trajectory to the probability
of observing a corresponding decrease in entropy under the time-reversed process.
For a system driven by an external control parameter, the excess work (nondimen-
sionalized by β) and entropy production are equal, thus (2.44) reduces to the Crooks
fluctuation theorem (2.43). In fact many of the well-known fluctuation theorems
can be recovered from (2.44) by splitting the change in entropy into adiabatic
and nonadiabatic contributions, Stot = Sa + Sna, capturing the dissipation
due to external time-dependent driving Sna (such as by a time-dependent control
parameter) or due to nonequilibrium boundary conditions Sa (such as out-of-
equilibrium concentrations of chemical reactants), respectively [26, 28, 29].
Furthermore, by using Jensen’s inequality, (2.44) reduces to the familiar form
of the second law of thermodynamics, Stot ≥ 0. Thus, (2.44) represents a true
generalization of the second law of thermodynamics, which only became apparent
through the study of microscopic fluctuating systems. The exponential suppression
of ‘second-law violating’ trajectories makes the probability of observing such events](https://image.slidesharecdn.com/23428428-250509151846-2d979635/85/Dissipation-And-Control-In-Microscopic-Nonequilibrium-Systems-Steven-J-Large-50-320.jpg)
![32 2 Theoretical Background
in macroscopic systems vanishingly small [17]. Broadly speaking, the fluctuation
theorems have provided a far deeper understanding of the microscopic physics of
fluctuating systems, placing surprisingly general constraints on the fluctuations and
dissipation in thermodynamic systems, even far from equilibrium.
2.5 Entropy and Information Theory
From its inception in the 1940s by Claude Shannon at Bell Labs, the field of infor-
mation theory provided a comprehensive and powerful set of tools to describe the
mathematics of information. More recently, the field of nonequilibrium statistical
mechanics and stochastic systems have found several concepts from information
theory essential for understanding the physics of nonequilibrium systems. Here, we
briefly review some of the relevant results from information theory.
The idea of entropy is central to information theory where, unlike many classical
treatments in thermodynamics, it possesses a conceptually simple explanation.
The entropy is a measure of the amount of information required, on average, to
describe the state of a system. Low-entropy states (such as a long sequence of
heads in successive coin flips) are highly ordered, and thus require a relatively small
amount of information to specify, whereas high-entropy states are more disordered,
requiring more information to completely specify.9 Mathematically, the entropy of
a random variable X—or Shannon entropy as it is typically referred to—is
S(X) ≡ −
x
px ln px . (2.45)
When multiple random variables (X and Y) are involved, one can also define the
joint and conditional entropy respectively as
S(X, Y) = −
x,y
pxy ln pxy (2.46a)
S(X|Y) = −
y
py
x
px|y ln px|y . (2.46b)
The conditional entropy (2.46b) is useful in decomposing the joint entropy (2.46a)
via the chain rule for entropies [18]:
S(X, Y) = S(X) + S(Y|X) (2.47a)
= S(Y) + S(X|Y) . (2.47b)
9 This explanation of the entropy is clarified when the entropy is described in bits, by using base-2
logarithms. Here, the entropy is literally the average number of bits—yes or no questions—required
to completely specify the state of a system.](https://image.slidesharecdn.com/23428428-250509151846-2d979635/85/Dissipation-And-Control-In-Microscopic-Nonequilibrium-Systems-Steven-J-Large-51-320.jpg)
![2.5 Entropy and Information Theory 33
Central to the study of information theory is the measure of mutual information,
I(X; Y) ≡
x,y
pxy ln
pxy
pxpy
, (2.48)
which quantifies the information that is shared between two random variables. The
mutual information (2.48) is symmetric (I(X; Y) = I(Y; X)), nonnegative, and can
be used to relate the marginal entropy of a single random variable in a multivariable
system to its conditional entropy:
I(X; Y) = S(X) − S(X|Y) . (2.49)
This provides a conceptual interpretation of I(X; Y) as the reduction in uncertainty
(entropy) about a random variable X by observation of another random variable Y.
If X and Y are independent, then pxy = pxpy and I(X; Y) = 0 (2.48). Conversely,
if X and Y are correlated, then the non-negativity of mutual information requires
that S(X|Y) S(X).
Information theory also provides a number of so-called divergence measures
that quantify the differences between probability distributions. Such divergences
have become central to the study of nonequilibrium systems, in particular for lower
bounding statistical averages. The relative entropy (or Kullback-Leibler divergence)
is a common measure used in nonequilibrium statistical physics, and is defined for
probability distributions px and qx as
D (px||qx) ≡
x
px ln
px
qx
(2.50)
The relative entropy is nonnegative for any two probability distributions,
D(px||qx) ≥ 0—a property which is commonly used to prove the non-negativity
of physical quantities in nonequilibrium thermodynamics—but is asymmetric in its
arguments (D(px||qx) = D(qx||px) in general) [18]. However, one can symmetrize
the relative entropy in several different ways, such as taking the relative entropy
of each distribution to their arithmetic mean mx ≡ 1
2 (px + qx) to get the Jensen-
Shannon divergence [30]
JSD (px, qx) ≡
1
2
D (px||mx) +
1
2
D (qx||mx) . (2.51)
The divergence measures originally derived in the context of information theory
arise, in many forms, within the study of nonequilibrium systems. For instance, we
will see in Chap. 7 how the relative entropy is related to excess work in discretely
driven systems. In fact, the physical meanings of various divergence measures in the
context of trajectory ensembles in stochastic thermodynamics have been explored
in detail in [30], relating the relative entropy (2.50) to dissipation, Jensen-Shannon](https://image.slidesharecdn.com/23428428-250509151846-2d979635/85/Dissipation-And-Control-In-Microscopic-Nonequilibrium-Systems-Steven-J-Large-52-320.jpg)
![34 2 Theoretical Background
divergence (2.51) to time asymmetry, and providing physical interpretations of
various other divergence measures from information theory in the context of
thermodynamic trajectories.
When a system is controlled by a set of external parameters λ, the equilibrium
distribution πx(λ) is parameterized by the control parameters λ (here, assumed to
take on any of a continuous range of values). Thus, we can define a quantitative
measure of the difference between two equilibrium distributions by taking the
relative entropy D[πx(λ)||πx(λ)]. If the two distributions are close to one another,
so that λ ≈ λ + λ, then the Taylor expansion of the relative entropy is, to leading
order in λ,
D[πx(λ)||πx(λ
)] ≈
1
2
λi
λj
Iij (λ) . (2.52)
Here,
dIij (λ) ≡ ∂λi ln πx(λ) ∂λj ln πx(λ)λ (2.53)
is the (i, j)th component of the Fisher information matrix, and λi is the ith compo-
nent of λ [18]. Here, and throughout the remainder of the thesis we have employed
an Einstein summation notation, where repeated indices are implicitly summed over,
for instance, λi λj Iij (λ) ≡
i,j λi λj Iij (λ). The Fisher information plays
an important role in quantifying the dissipation in driven, nonequilibrium processes,
and will be encountered periodically throughout the remainder of this thesis.
2.6 Control in Microscopic Nonequilibrium Systems
Broadly speaking, control theory is the study of physical systems and how their
state can be controlled through interaction with the outside world. By identifying
particular goals, it is then possible to design ways to exercise control over a
system to achieve them [31]. For instance, a canonical example of a mechanical
controller is the centrifugal governor—an idea which was explored in detail by
James Clerk Maxwell in the 1800s [32]—where the motion of a centrifugal rotor,
coupled to the engine output, regulates engine speed by controlling the input valve.
Since its early applications, the scope of control theory has expanded greatly and
plays a major role in the modern practice of science, particularly in the context of
experimental physics, where many techniques rely fundamentally on the successful
implementation of control-theory-based feedback techniques [33].
Optimal control—one subfield of the broader subject of control theory—deals
with the implementation of control strategies that minimize some measure of
cost in a controlled system. Through much of this thesis, we will deal with the
utilization of optimal control to minimize the amount of work required to manipulate](https://image.slidesharecdn.com/23428428-250509151846-2d979635/85/Dissipation-And-Control-In-Microscopic-Nonequilibrium-Systems-Steven-J-Large-53-320.jpg)
![2.6 Control in Microscopic Nonequilibrium Systems 35
stochastic systems in a particular way.10 In particular, we will often consider
stochastic systems which fluctuate in the presence of a potential energy E(x, λ)
which is determined by a set of control parameters λ. The instantaneous value of
λ determines the equilibrium state π(x|λ) of the system through (1.1). A control
protocol ≡ λ[0,τ] is a time-dependent schedule for the control parameter λ,
driving the system from an initial state—often the equilibrium state at λ(t = 0)—to
a final state at λ(t = τ), where the protocol duration is denoted by τ.
Following from the Sekimoto definition (2.35), work is done on the system
(or extracted from the system) whenever the control parameter is changed. For
a particular time-dependent control protocol that transforms λi → λf over a
protocol duration τ, the mean work required is given by the functional
W = −
τ
0
fi(x, λ)(t)λ̇i
dt , (2.54)
where fi ≡ −∂λi E(x, λ) is the generalized force conjugate to control param-
eter λi, the angle brackets · · · (t) indicate an average over the instantaneous
(nonequilibrium) distribution of the system at time t during the protocol , and
the angle brackets · · · indicate an average over the (nonequilibrium) distribution
of system trajectories during the entire control protocol . Given that the initial
and final control parameter values are generally fixed,11 the equilibrium free energy
difference F ≡ F(λf) − F(λi) is constant for all protocols, and thus minimizing
the work is the same as minimizing the excess work Wex = W − F. For the
remainder of this thesis, we will focus on minimizing the excess work.
For a system bound by a harmonic potential (see Sect. 2.7.1), Schmiedl and
Seifert [36] found the exact optimal control protocols that minimize the average
work when the control parameter is either the position of the harmonic potential,
E(x, λ) =
ktrap
2 (x − λ)2, or the trap strength, E(x, λ) = λ
2 x2 [36]. Further, in
2011, Aurell, et al. expanded the set of exactly solvable minimum-work protocols
to a more general class of problems by finding a clever mapping of the stochastic
differential equations to a set of deterministic transport equations [37, 38]. While
this work provided a new means of analyzing such problems, aside from some
notable recent results regarding a finite-time Landauer limit [34, 35], the study of
analytically tractable solutions in optimal control of stochastic systems has seen
little progress since. Primarily, this is due to the intrinsic difficulty in solving the
general minimization problem in (2.54).
10 Actually, this represents a further subfield known as stochastic optimal control, as it pertains to
optimal control strategies in stochastic systems. However, in Chap. 6 we will refer to the control of
systems through stochastic driving protocols as ‘stochastic control’, and thus, to avoid confusion,
we will refer to stochastic optimal control in this context simply as optimal control.
11 Recent work on optimal bit erasure [34, 35] enforces a constraint on the final probability
distribution rather than requiring that the control parameter reaches a particular final value.](https://image.slidesharecdn.com/23428428-250509151846-2d979635/85/Dissipation-And-Control-In-Microscopic-Nonequilibrium-Systems-Steven-J-Large-54-320.jpg)
![O Jesus, I am heartily sorry for having hitherto loved Thee so little!
Why did I not love Thee sooner?
Would that I could redeem that time in which I did not love Thee!
Where will I find hearts that are going to help me to make up for the
glory that I should have given Thee up to this time?
Would that I loved Thee with that tender love with which Mary loved
Thee. The angels and saints loved Thee, my Jesus; let me also join
them in their love.
Offering.
I offer Thee my whole being; let it be unreservedly Thine. Thou hast
taken possession of my heart, Thou art its Master; my whole life
shall be used for Thy greater honor and glory. Wherever my
influence may reach, there will I carry the name of Jesus. Wherever
the word of God may be announced, I will glory in Thy name.
Thou, O Lord, art so full of love, and still Thou art so little loved! O
would that I could enkindle all hearts with the fire of Thy love! Thou,
O Lord, art the Master of all hearts; we belong to Thee because
Thou hast created us, and when we rebelled from Thee and became
slaves of Satan, Thou didst redeem us. What greater right have we
than to belong to Thee, to be subject to Thee, and to announce Thy
holy name to all, so that by our mutual example we may be
encouraged to live lives of sanctity?
O that all of us would have the love of the Infant Jesus in our souls,
and make earnest endeavors to [pg 064] spread Thy love among all
men! When, O Lord, will I be dissolved, to live only with Thee? To
Thee my sighs and desires go up day and night. When will I see Thy
blessed countenance? When my time comes to die, let me die in Thy
love, so that I may love Thee for all eternity. Thy will, O Jesus, be
done in all things, and let me delight to walk in Thy footsteps.](https://image.slidesharecdn.com/23428428-250509151846-2d979635/85/Dissipation-And-Control-In-Microscopic-Nonequilibrium-Systems-Steven-J-Large-56-320.jpg)
![Do with me what Thou wilt; I am willing to submit to joy or sorrow,
life or death, fortune or misfortune. It is all indifferent to me as long
as it is Thy will, only give me the grace to endure adversity with
resignation. It will be a joy to me to fulfil Thy holy will in all things.
Thanksgiving After Communion.
Adoration of Jesus in the Blessed Sacrament, according to St. Francis
de Sales.
How can I, O Lord Jesus Christ, praise Thee sufficiently for having
visited my soul in Thy infinite goodness, by Thy divinity and Thy
sacred humanity, with Thy body and blood? I give Thee, my Lord
and my God, a loving welcome, Thou Who art my Redeemer, my last
end, my consolation, my sweetest rest, my all. A thousand welcomes
to Thee, my dear Jesus, my good shepherd. In the abyss of my
nothingness I humbly adore Thee. I adore Thy sacred flesh and
blood, which Thou hast given me for my food, and the pledge of my
eternal union with Thee. I adore Thy sacred head crowned with
thorns; I adore Thy eyes that have shed so many tears for me; I
adore Thy mouth, with which Thou didst teach me eternal truths; I
adore Thy [pg 065] sacred countenance, which has been beaten for
me by cruel soldiers; Thy feet, that have been pierced by the nails
and fixed to the tree of the cross; Thy arms, which were
outstretched for me, for a loving embrace; Thy side which the lance
pierced, and from which blood and water issued forth, the witnesses
of my redemption; Thy Heart, which loved me even unto the death
of the cross. My dear Redeemer, I adore Thy sacred body, covered
with innumerable wounds which Thou didst suffer for me. I adore
Thy most holy soul, which was saddened unto death in the Garden
of Olives that I might reach eternal life.
My dear Jesus, with great love I embrace Thee, and I will remain
faithful to Thee until my last hour. Bless these feelings of a good
heart: grant that no storm of temptation may lead me again to be
unfaithful as I have repeatedly been heretofore.](https://image.slidesharecdn.com/23428428-250509151846-2d979635/85/Dissipation-And-Control-In-Microscopic-Nonequilibrium-Systems-Steven-J-Large-57-320.jpg)
![A Pious Salutation To The Infant Jesus.
Hail, holy and adorable Infant Jesus, source of mercy! Thou art the
life of the sinner, unfathomable ocean of divine sweetness, our hope,
our joy. To Thee, born in the dreary stable of Bethlehem, we cry,
poor children of Eve, to Thee we send forth our sighs, to Thee we
send up our lamentations in this valley of tears. Turn then, O
incarnate love, Thine eyes of mercy towards us, O blessed fruit of
Thy Virgin Mother. As Thou didst show Thyself full of kindness in the
stable of Bethlehem, so show unto us, after this miserable life, Thy
divine mercy, when Thou shalt come in the glory of Thy Father. O
sweet, O loving, O clement Infant Jesus! Amen.
[pg 066]
Pious Invocation To The Infant Jesus.
(Father Elias Avillon.)
Hail, Infant Jesus, Thou fountain of mercy, Thou life of the sinner
dead in sin, Thou unfathomable ocean of divine consolation, our
hope and our joy. Hail, to Thee, born in the desolate stable of
Bethlehem, do we cry, poor children of Eve. To Thee do we sigh and
bewail our misery from this valley of tears. Turn then, O incarnate
love of God, Thine eyes of mercy upon us, Thou blessed fruit of the
womb of the Virgin Mary. As Thou didst show poor humanity
unbounded mercy from the manger, so also continue Thy kindness
during our life, until after this miserable existence we arrive in the
glory of heaven, where we shall meet Thee, to praise and glorify
Thee for all eternity. O sweet, O pious, O merciful Infant Jesus!
Amen.
A Prayer To The Holy Name Of Jesus.
Eternal God, Father in heaven, daily do we pray that Thy sacred
name be sanctified, that Thy kingdom come and Thy will be done. It
is also Thy holy will that the name of Thy beloved Son should be
glorified. Thou hast given Him a name at which every knee should](https://image.slidesharecdn.com/23428428-250509151846-2d979635/85/Dissipation-And-Control-In-Microscopic-Nonequilibrium-Systems-Steven-J-Large-58-320.jpg)
![bend in heaven, on earth, and under the earth. I pray Thee with all
my heart that Thou give us the grace to pronounce that holy name
with the greatest devotion, and to serve the Lord with a good and
willing heart. Engrave this holy name on my heart, that I may never
forget it. Place it as a sign on my forehead, that I may ever be
faithful to it. O holy name of Jesus! my heart rejoices when I hear
thee mentioned. I feel great [pg 067] consolation as often as I hear
that holy name. Angels of heaven, who are gathered in ecstasy
about the throne of God, praise with us on earth the sacred name of
Jesus; sing its praises worthily, as we cannot sing it; invoke it for us
also, that when our feeble powers fail, thy holiness and zeal will
supply our deficiency. Thy holy name, O Jesus, be praised forever!
Adoration of the Child Jesus in the Twelve Mysteries of His Divine
Infancy.
Dear Jesus, divine Infant of incomparable beauty, of infinite
goodness, I adore Thee, because Thou art my Redeemer, and I love
Thee. I make a sacrifice to Thee of all my intellect, and of my heart,
and I thank Thee for having become a child for my sake. I adore
Thee in all the mysteries of Thy holy childhood, and pray Thee to let
me enter into the spirit of it. Give me the grace to honor Thy
childhood all my life, and that I may imitate the virtues which Thou
dost inculcate by Thy example.
I adore Thee, O God of purity, at the moment when the Holy Ghost
formed Thy sacred body in the bosom of the Virgin Mary, and beg of
Thee the grace to be always pure.
I adore Thee, O hidden God, hidden for nine months in the womb of
the Blessed Virgin, and I desire to honor Thee by my perfect life.
I adore Thee, O Child of grace, in Thy visitation to John the Baptist
in order to sanctify him. Visit also my soul, that I may live a saintly
life and remain faithful to the impulses of Thy grace.](https://image.slidesharecdn.com/23428428-250509151846-2d979635/85/Dissipation-And-Control-In-Microscopic-Nonequilibrium-Systems-Steven-J-Large-59-320.jpg)
![I adore Thee, O Child Jesus, at the moment of Thy birth, and I
desire, with Thy help, that no other [pg 068] fire burn in my heart
than that which Thou camest on this earth to kindle.
I adore Thee, O Spouse covered with wounds, in the painful mystery
of the circumcision. I conjure Thee by the sacred blood which Thou
didst then shed for the first time for our salvation, that Thou wilt
give me the virtue of meekness, with which I am to go through life,
for Thy glory and the edification of my neighbor.
I adore Thee, O Lord Jesus, with the Magi, who came from the
distant orient to adore Thee lying in the crib in the stable. Give me
the grace to be in earnest in the work of my salvation.
Great God, Author of all sanctity, I adore Thee in the Temple, where
Thou didst present Thyself according to the Law of Moses, to be
offered to Thy Father as His first-born Son. Give me grace to subject
myself to every law, even though it would imply that I am a sinner,
which I really am.
I adore Thee, humble Jesus, in Thy flight into Egypt, and beg of
Thee to give me the grace, by this holy mystery of Thy life, of
perfect humility of heart.
I adore Thee, O Child Jesus, in Thy poverty, which afflicted Thee in
Thy stay in Egypt. Give me the grace also to be poor in spirit, and
endure actual poverty with resignation.
I adore Thee, O Child Jesus, in Thy joyful and triumphal return from
Thy exile in Egypt to Nazareth; give me the grace to overcome the
enemies of my salvation.
I adore Thee, O obedient Child Jesus, in Thy exact compliance with
the commands of Mary and Joseph; give me the grace joyfully to
obey my superiors.
[pg 069]](https://image.slidesharecdn.com/23428428-250509151846-2d979635/85/Dissipation-And-Control-In-Microscopic-Nonequilibrium-Systems-Steven-J-Large-60-320.jpg)
![I adore Thee, O Child Jesus, in the midst of the doctors in the
Temple of Jerusalem, when Thou didst enlighten their minds on the
sacred prophecies of the Old Testament. Give me a Christian
simplicity, that I may believe all that Thou teachest me through the
holy Catholic Church.
[pg 070]](https://image.slidesharecdn.com/23428428-250509151846-2d979635/85/Dissipation-And-Control-In-Microscopic-Nonequilibrium-Systems-Steven-J-Large-61-320.jpg)
![the way, for save only in a few cases has a certain devotion been
assigned to a particular time. The devotion to the Holy Family is a
beautiful and instructive one; the Christian family should be built on
this great model. The Holy Family consisted of Jesus, Mary, and [pg
071] Joseph; the father, Mother, and Child. All other families are
made up of the same constituents.
Those who are actually in a family, or who intend to choose that
mode of life by which they may get to heaven, will love this devotion
and find instruction and consolation in it.
One of the greatest works of God in this world is the Holy Family at
Bethlehem and Nazareth. He sent down upon the earth His only
Son, Jesus Christ; prepared a most holy Mother for Him, Mary
immaculate, and selected for Him a foster-father and a protector. He
held them together in the most tender family ties of father, Mother,
and Child. He kept them in that relation until St. Joseph died a
blessed death, and the Lord Jesus went forth on His sacred mission
of teaching and redeeming mankind.
What a beautiful sight to us poor human beings! what a glory to God
was that Holy Family, dwelling in the humble abode of Nazareth! We
find here the model on which we shall reflect for this month of
February. We will consecrate this month to praising God with the
members of the Holy Family; we will study the ways by which they
were so pleasing to God, and we will draw from these considerations
many valuable lessons for our own conduct.
Our Holy Father, Pope Leo XIII., with an ever-watchful eye to the
necessities of our time, has seen the importance of a devotion to the
Holy Family, and has recommended it to the faithful, and with his
own authority established a sodality of the Holy Family; he has
invested it with many indulgences in order to encourage the faithful
in taking the Holy Family as their model. Here is, then, a practical
way to teach ourselves the way of salvation by the example of
others.](https://image.slidesharecdn.com/23428428-250509151846-2d979635/85/Dissipation-And-Control-In-Microscopic-Nonequilibrium-Systems-Steven-J-Large-63-320.jpg)
![It is therefore a most useful practice of piety to become members of
the sodality of the Holy Family; you will then place yourselves under
its special protection, and choose Jesus, Mary, and Joseph for your
particular advocates before God. Look upon the members of the
Holy Family as the most perfect models for imitation, whose
examples will teach you what to correct and what to avoid, what to
do for your temporal and eternal welfare, and that of your families.
[pg 072]](https://image.slidesharecdn.com/23428428-250509151846-2d979635/85/Dissipation-And-Control-In-Microscopic-Nonequilibrium-Systems-Steven-J-Large-64-320.jpg)
![Considerations and Prayers for Every Day.
First Day.
In an apostolic brief of June 14, 1892, the Holy Father, Pope Leo
XIII., demonstrates how the welfare of the family and of the State
depends chiefly on education, and that it is of the utmost
importance that a religious spirit be fostered in the Christian family.
From the first family, God so arranged the method and order of such
a life as to exhibit to the world a form of a divinely ordered
association, in which all human beings might behold a most
complete model of family life, and of all virtue and holiness. The
devotion to the Holy Family, a holy and a powerful institution before
God and man, has increased very much within a few years, and it is
worth our while to think of this on the first day of our monthly
devotion, and appreciate it as we ought.
Prayer.
O most loving Jesus, Who didst hallow by Thy surpassing virtues,
and the example of Thy home life, the household Thou didst choose
to live in whilst on earth, mercifully look down upon this family,
whose members, humbly prostrate before Thee, implore Thy
protection. Remember that we are Thine, bound and consecrated to
Thee by a special devotion. Protect us in Thy mercy, deliver us from
danger, help us in our necessities, and impart to us strength to
persevere always in the imitation of Thy Holy Family, so that, by
serving Thee and loving Thee faithfully during this mortal life, we
may at length give Thee eternal praise in heaven. [pg 073] O Mary,
dearest Mother, we implore thy assistance, knowing that thy divine](https://image.slidesharecdn.com/23428428-250509151846-2d979635/85/Dissipation-And-Control-In-Microscopic-Nonequilibrium-Systems-Steven-J-Large-65-320.jpg)
![Son will hearken to thy petitions; and do thou, most glorious
patriarch St. Joseph, help us with thy powerful patronage, and place
our petitions in Mary's hands that she may offer them to Jesus
Christ. Amen.
Second Day.
Within our own time the devotion to the Holy Family has grown
under the fostering care of the Supreme Pontiff, who has authorized
the establishment of associations throughout the world, by which
men and women, married and unmarried, are gathered into one fold
from the standpoint of the family. The Holy Father desires that all
the faithful of the Catholic Church should consider this association as
a consecration to the Christian life, and that they will feel that they
are bound to lead a holy life because their association is established
and legalized by God. For what is the Sacrament of Matrimony but
the legalizing of the family before God and man? Let us consider the
family as a special institution of God's providence for the
preservation of the world, and the propagating in it of sound
principles of learning and religion.
Prayer.
O most loving Jesus, etc., etc.
Third Day.
We are to consecrate ourselves to God under the union of a family.
That is the pretence with which we come before God, to claim His
kindness and mercy because we belong to a family. We are [pg 074]](https://image.slidesharecdn.com/23428428-250509151846-2d979635/85/Dissipation-And-Control-In-Microscopic-Nonequilibrium-Systems-Steven-J-Large-66-320.jpg)
![not isolated creatures, looking for our selfish ends, but we are a
union of individuals constituted under a certain authority, which
gives us a claim to the respect of God and man; for God has said,
“Where there are two or three gathered together in My name, there
am I in the midst of them.”—Matt. xviii. 20. The formula of
consecration of a Christian family has been given us by the
Sovereign Pontiff himself; rules, regulations, and by-laws have been
given to this society under the same authority of the Holy Father.
The whole Christian family should be so united among themselves,
that there be but one family under one head, set over all to rule
them by its supreme power. We belong to this Christian family of the
Church; let us then unite for the common good of all.
Prayer.
O most loving Jesus, etc., etc.
Fourth Day.
The scope of the pious association of the Holy Family is that all
Christian families be consecrated to the Holy Family of Nazareth,
placing it before themselves for veneration and imitation; offering up
every day before its image prayers in its honor, and practising in
their lives the sublime virtues which the Holy Family offered for
imitation to every grade of society. The rich will find a model before
them, the learned and highly educated will know exactly what to do
according to the dignity of their position, the working class,
especially, will find here the guidance and friendship needed in their
temptations and troubles. The Holy Family is a [pg 075] poor,
working, humble family, as poor as the poorest, as laborious as the
most hard-working, as humble as the most lowly.
Prayer.](https://image.slidesharecdn.com/23428428-250509151846-2d979635/85/Dissipation-And-Control-In-Microscopic-Nonequilibrium-Systems-Steven-J-Large-67-320.jpg)
![O most loving Jesus, etc., etc.
Fifth Day.
The picture of the Holy Family should be in every household; it is a
perpetual reminder, placed in tangible form, of our dear Lord, our
blessed Lady, and our friend, St. Joseph, who were the members of
the Holy Family. We often are interested in the pictures of great
persons, and take delight in representations of angels and holy
people. What is the secret of this inclination? Why, we can see those
whom we wish to imitate, and grow to know their good and holy
lives through their pictures. The Holy Father has approved one
special picture which is to be the emblem of this association: Mary
and Joseph, holding the youth Jesus between them by the hand.
Jesus is not an infant, for this picture is to bring to mind the fact
that His parents had to suffer care and anxiety in order to bring Him
to this stage of boyhood, for which noble duty they were fitted by
special providence and by special faithfulness.
Prayer.
O most loving Jesus, etc., etc.
Sixth Day.
“We have good hopes,” says the Holy Father, concluding his
encyclical letter, “that all [pg 076] to whom the salvation of souls is
committed, especially the bishops, will make themselves partners
and sharers of our zeal in promoting this pious association. For those
who recognize and deplore with us the change and corruption of](https://image.slidesharecdn.com/23428428-250509151846-2d979635/85/Dissipation-And-Control-In-Microscopic-Nonequilibrium-Systems-Steven-J-Large-68-320.jpg)
![Christian morals, the extinction of the love of religion and piety in
families, and the passion for earthly goods, enkindled beyond
measure, will desire exceedingly to apply suitable remedies for so
many grievous evils. Since nothing can be more salutary and
efficacious for Christian families than the example of the Holy Family,
let care be taken that as many families as possible, especially those
of the working classes, against which insidious forces are more
strongly exercised, enroll themselves in this association. Let the
association be on its guard, lest it swerve from its purpose, or
change its spirit—rather let the practices of piety and the prayers
which have been determined on to be preserved intact. May Jesus,
Mary, and Joseph, thus besought within the family circle, be
graciously present! May they foster charity, regulate morals, incite all
that imitate them to virtue, and alleviate and render more bearable
the hardships which oppress mankind!”
Prayer.
O most loving Jesus, etc., etc.
Seventh Day.
A plenary indulgence after a sincere confession and a worthy
communion, and praying for the intentions of His Holiness, may be
gained by the members of the association on the following days:
First, on the day of their entrance into the association, [pg 077] after
they have recited the act of consecration. Second, on the day of a
general meeting, when all go to communion in a body, and renew
their promises. Third, on the feasts of the Nativity (Christmas), the
Circumcision, Epiphany, Easter and Ascension. Fourth, on the feasts
of the Blessed Virgin: Immaculate Conception, Nativity,
Annunciation, Purification, Assumption. Fifth, also on the following
days: feast of St. Joseph in March, Patronage of St. Joseph, Third](https://image.slidesharecdn.com/23428428-250509151846-2d979635/85/Dissipation-And-Control-In-Microscopic-Nonequilibrium-Systems-Steven-J-Large-69-320.jpg)
![Sunday after Easter, Espousal of the Blessed Virgin on the twenty-
third of January. Sixth, on the feast of the Holy Family, and Seventh,
at the hour of death.
Prayer.
O most loving Jesus, etc., etc.
Eighth Day.
Partial indulgences may be gained when visiting a church where the
association is established, provided the members pray for the
prosperity of Christendom, and for the intention of the Holy Father.
Seven years and seven quarantines may be gained on the feasts of
the Visitation, Presentation, and the Patronage of the Blessed Virgin.
The same indulgences may be gained by the family in the reunion in
prayer among themselves, if they pray before a picture of the Holy
Family. The same also, whenever the members attend a public
meeting of the association. Three hundred days' indulgence is
granted as often as a member of the association recites, before a
picture of the Holy Family, the prayer “O most loving Jesus,” etc.,
etc. The members [pg 078] may gain an indulgence of two hundred
days when they make the salutation: “Jesus, Mary, and Joseph,
enlighten us, aid us, save us. Amen.” One hundred days for every
new member that is brought to the association, and sixty days for
every good work done in honor of the Holy Family. All these
indulgences are applicable to the souls in purgatory.
Prayer.
O most loving Jesus, etc., etc.](https://image.slidesharecdn.com/23428428-250509151846-2d979635/85/Dissipation-And-Control-In-Microscopic-Nonequilibrium-Systems-Steven-J-Large-70-320.jpg)
![Ninth Day.
The Catholic Christian has the true faith which comes to him from
Jesus Christ, the Founder of the true Church. He ought, then, to
show by his conduct that his faith has made him better than so
many others, who have not had the graces and advantages which
came to him. In other words, it is not enough to believe the truths
that God has revealed; it is not enough to belong to the true Church
by the internal adhesion of the mind; it is indispensably necessary to
manifest our faith in exterior works. Our faith should so have
penetrated our whole being, that the profession of religion should
show itself in all our actions. Faith without works is dead, and at the
Last Judgment the almighty Judge will demand of us an account of
all our actions, and then will He render to every one the merited
reward or punishment. Let it be our aim in life to fill the days of our
stay on earth with many good actions, the outcome of our faith, so
that when we stand before the throne of God, we may have many
glorious deeds to our credit.
[pg 079]
Prayer.
O most loving Jesus, etc., etc.
Tenth Day.
A Christian considers his faith as a gift of heaven, a priceless
treasure far surpassing any earthly wealth, because it raises man to
a true knowledge of God and secures for him his eternal salvation.
He rejects with horror the maxims of our modern infidels, who say,
“One religion is as good as another”; “Hell is only a bugbear”; “The
faith of the heart is enough for salvation,” and many others of the](https://image.slidesharecdn.com/23428428-250509151846-2d979635/85/Dissipation-And-Control-In-Microscopic-Nonequilibrium-Systems-Steven-J-Large-71-320.jpg)
![same nature. He rejects them because he knows that God is one,
that the truth that comes from God is one, and that therefore faith
must be one, and religion one. He knows that religion cannot be
framed according to the whims of man, but only can come from the
authority of God in His holy revelation. He will not associate with
those who hold the above false doctrines; he will be an enemy of
bad books, which teach errors of faith and which drag those who
read them into the mire of immorality, and he will caution his friends
against them.
Prayer.
O most loving Jesus, etc., etc.
Eleventh Day.
The Catholic Church was founded by Jesus Christ as a perfect
society, with authority to make laws, with power to punish the guilty,
and to expel rebellious subjects from her midst. This power was [pg
080] given to the Church by Christ when He said, “And whatsoever
thou shalt bind upon earth, it shall be bound also in heaven.” It is
clear that Christ intended His Church to be our guide in all our
actions, as an authority to teach us what the revelation is concerning
our future state. We should, therefore, be obedient and faithful
children of the Church. We should be grateful to God, Who in His
mercy has established certain fountains of grace, which are found in
the Church, and are guarded by her. These fountains of grace are
the sacraments, which point out the holy states of life, and the true
manner of pleasing God. We should use the Sacraments of Penance
and Holy Eucharist for the remedy of our faults, and the strength of
our weakness.
Prayer.](https://image.slidesharecdn.com/23428428-250509151846-2d979635/85/Dissipation-And-Control-In-Microscopic-Nonequilibrium-Systems-Steven-J-Large-72-320.jpg)
![O most loving Jesus, etc., etc.
Twelfth Day.
Confession is the correction of our faults, and if we have sinned let
us remember we have an advocate in heaven, to Whom we wish to
return in the sincerity of our hearts. The Holy Eucharist is the body
and blood of Christ and communion the partaking of it.
We should not, therefore, be deterred from frequenting these
sacraments by human considerations, or by the mockery of the
people of this world. We should have these words of Christ deeply
engraven in our hearts: “Except you eat the flesh of the Son of man,
and drink His blood, you shall not have life in you.”—John vi. 54. A
devout Catholic is [pg 081] easily distinguished from the crowd of
careless ones, when we see him humbly and frequently going to
confession and to holy communion.
Prayer.
O most loving Jesus, etc., etc.
Thirteenth Day.
Be also reverent and devout in the house of God; not brought there
by vain curiosity, or by fashion, but by unfeigned piety, rendering to
God an external tribute of dependence and adoration. Look upon
priests as the ambassadors of God, treat them with respect, listen to
their teaching, and put it into practice. Reverence the bishops as
divinely constituted guardians and teachers in the Church: especially](https://image.slidesharecdn.com/23428428-250509151846-2d979635/85/Dissipation-And-Control-In-Microscopic-Nonequilibrium-Systems-Steven-J-Large-73-320.jpg)
![the see of St. Peter, the Vicar of Christ, the Roman Pontiff, the Father
and teacher, in whom is intrusted the plentitude of power to rule the
whole Catholic family. Reverence the infallible authority of the Pope,
which guides you in matters of faith, in form of worship, and
morality.
Accept with docility and obedience the decisions of the Holy See,
and conform to them your opinions and thoughts. Do not follow the
changeable and novel opinions of our infidel age.
Prayer.
O most loving Jesus, etc., etc.
Fourteenth Day.
The true Christian must not only profess the faith, but also the laws
of Christ. He is [pg 082] anxious to observe them exactly and to
observe them all, knowing that he is guilty of damnation who
violates the law in one important point. This law bids us to love all
men as fellow creatures, to love our relatives, our country, but above
all, and before all, we must love God, the Author of our being, the
great loving Father of heaven. The honor of God is the very first duty
of man, who as a rational being knows God and His infinite
goodness; we wish to serve Him as His subjects, render Him the
homage due to His immensity, a worship which our infinite littleness
renders to God. Never profane the word of God yourself, and
prevent curses and oaths in others as far as possible. By acts of
praise and benediction let us repair the offences against God when
we cannot prevent them. At least pray for those who use the name
of God in vain, and thus endeavor to ward off from them the eternal
punishment due to that wicked practice.](https://image.slidesharecdn.com/23428428-250509151846-2d979635/85/Dissipation-And-Control-In-Microscopic-Nonequilibrium-Systems-Steven-J-Large-74-320.jpg)
![Prayer.
O most loving Jesus, etc., etc.
Fifteenth Day.
Contribute your share towards the glorification of the name of God,
by observing the Sundays and festivals of the Church. God has
expressly reserved these days to Himself, and has pointed them out
by the authority of the Church. In the Old and in the New Law, God
has had days of rest and of religious practices; and for the
observances of these He has promised publicly that there should be
many rewards. Every good, God-fearing man will give a [pg 083]
just tribute of respect to God, because God wills it, and because he
is looking for some benefit from God. Abstain then from servile
works on those days, no matter what temporal gains may be
expected, and be careful that others, too, keep holy the Sabbath of
the Lord, particularly those who are intrusted to your care and
command. Do not have work done on Sunday, and allow none to be
done about your premises.
Prayer.
O most loving Jesus, etc., etc.
Sixteenth Day.
Respect your parents, superiors, and masters, and all those who
hold positions of trust towards you. They hold the authority of God,
and he who despises them despises God Himself. Honor and respect](https://image.slidesharecdn.com/23428428-250509151846-2d979635/85/Dissipation-And-Control-In-Microscopic-Nonequilibrium-Systems-Steven-J-Large-75-320.jpg)
![those superiors as representatives of God, and obey them in all
things that are not against His law; and so we come to that great
second commandment, which is like unto the first: “Thou shalt love
thy neighbor as thyself.” We should do unto others as we would wish
that others should do unto us; that is, we should look upon one
another as children of one great family, of which God is the heavenly
Father. “By this shall all men know that you are My disciples, if you
have love one for another,” not showing this love through politeness
only, but through a real, downright feeling of interest in others, and
without a selfish regard of our own interests. Try to perform spiritual
and corporal works of mercy for your neighbor.
[pg 084]
Prayer.
O most loving Jesus, etc., etc.
Seventeenth Day.
In all your intercourse with others respect their persons and
property; do not look for unjust gains; be faithful to your bargains
and contracts; never look to your own selfish interests solely. Never
speak ill of anybody, nor circulate detractions, nor reveal secrets and
defects that might lessen the esteem in which any one is held;
excuse the faults of others, and find some excuse for the intention
with which even an evil action is committed. We are all temples of
the Holy Ghost, sanctified and ennobled by the blood of Christ. We
are the dwellings of the Holy Trinity, called to a heavenly inheritance.
Do not desecrate this sacred temple by impurity; guard against
impure thoughts and immodest desires, flee from dangerous
occasions. Avoid foolishly pursuing the luxuries and vanities of the
world, improper company, and bad conversation. Do not enter
theatres or places of amusement where your morals are](https://image.slidesharecdn.com/23428428-250509151846-2d979635/85/Dissipation-And-Control-In-Microscopic-Nonequilibrium-Systems-Steven-J-Large-76-320.jpg)
![endangered, and from which you carry nothing but pictures of
immoral objects. Arm yourself most effectually against the approach
of evil by the powerful shield of prayer, and walk in the presence of
God.
Prayer.
O most loving Jesus, etc., etc.
Eighteenth Day.
So far we have considered the law of God practically interpreted in
our every-day life; let us [pg 085] go still further and endeavor to
learn the laws of the Church, for the Lord has said: “He that heareth
you, heareth Me; and he that despiseth you despiseth Me,” and “If
he will not hear the Church let him be to thee as the heathen and
publican.” On Sundays and holydays of obligation, we ought to hear
Mass. We should observe the feasts of the Church and the restriction
from flesh meat on Fridays and other days of abstinence. Remember
that these little acts of mortification are a great benefit to us, since
the Lord has commanded, “That we should bring forth fruits worthy
of penance.” Make your Easter duty, for the Church has laid down
the law that you should fulfil those duties, not from routine or
human respect, but with the knowledge of the needs of your soul.
Prayer.
O most loving Jesus, etc., etc.
Nineteenth Day.](https://image.slidesharecdn.com/23428428-250509151846-2d979635/85/Dissipation-And-Control-In-Microscopic-Nonequilibrium-Systems-Steven-J-Large-77-320.jpg)
![What an exalted opinion we should have of Christian piety! It
inspires the Christian man and woman with lofty ideas, and prepares
them for noble undertakings. These lessons of piety should be
planted early in the hearts of the young, that they may take root and
grow up to a magnificent fruit of mature virtue. Serve God as a
loving and dutiful child, cherish a great devotion to the Blessed
Virgin, and have recourse to her in all your wants, being sure that all
your petitions will meet with a ready and hearty response. Never
forget these three truths, which should be the main considerations
of [pg 086] the true Christian on every occasion: First, That sin is
the only evil which should be feared; Second, The grace of God is
the real good for which we should strive with all our heart; Third,
The salvation of the soul is the all-important business of our lives,
for which we were created, and which should be looked after with
that care which it deserves.
Prayer.
O most loving Jesus, etc., etc.
Twentieth Day.
The husband, as head of the family, owes to his wife fidelity, love,
and support. Fidelity is that constancy of affection which he has
promised at his marriage, and which must be preserved inviolable
until death; it means that purity of soul and body which will not
permit itself to be degraded by impurity and adultery. Conjugal
fidelity is a great and holy duty, in which matrimony is held sacred.
There you find peace, happiness, and the blessing of almighty God.
Our Lord was very stringent upon this point, for He says:
“Whosoever shall look on a woman to lust after her, hath already
committed adultery with her in his heart.”—Matt. v. 28. Whosoever
then commits adultery transgresses a most important divine](https://image.slidesharecdn.com/23428428-250509151846-2d979635/85/Dissipation-And-Control-In-Microscopic-Nonequilibrium-Systems-Steven-J-Large-78-320.jpg)
![commandment. In the Old Law this crime was punished by stoning
the guilty person to death, and in the primitive Church by a severe
public penance of many years' duration. What fidelity does not the
husband demand of the wife! With the same strictness is he obliged
to be faithful to her whom he has chosen as his life companion.
[pg 087]
Prayer.
O most loving Jesus, etc., etc.
Twenty-first Day.
The second duty of the husband and wife is conjugal love. The word
conjugal means joined together, because husband and wife are
united to bear the same burden. The Apostle says: “Husbands, love
your wives, as Christ also loved the Church, and delivered Himself up
for it.”—Ephes. v. 25. Husband and wife are individuals whom God
has joined in inseparable companionship. The greatest bond
between mankind, and the sweetest one, is conjugal love, of which
we are thinking on this day consecrated to the Holy Family. Keep this
great duty before your eyes and never forget it, for it is easily
destroyed. It is from that love, too, that should spring your children,
who are to grow up to take your place in the Church and the State.
These children you are to bring up in the fear and love of God,
faithful to the Church and their fatherland. A tremendous
responsibility!
Prayer.
O most loving Jesus, etc., etc.](https://image.slidesharecdn.com/23428428-250509151846-2d979635/85/Dissipation-And-Control-In-Microscopic-Nonequilibrium-Systems-Steven-J-Large-79-320.jpg)
![Twenty-second Day.
The husband is the main worker in the family, so that his duty is to
provide for his family by his industry and economy. He must look for
employment [pg 088] and strive to keep it, so as to have a never-
failing source of income, by which his people may live in comparative
comfort. There are husbands who will allow their wives and children
to work, while they themselves live idle lives, which is the fashion of
the untutored Indian. Not only must the husband work, but he must
live in economy, and not throw his money away foolishly in gaming
or in drunkenness. The most frequent cause of a husband and
father's failure to provide for his family is drunkenness. Drunkenness
causes woe, sin, sorrow and shame. Drunkenness besots the mind,
and makes of an intelligent being a brute in his passions, and a fool
in his actions; it extinguishes the spirit of God in him, all sentiments
of religion are lost, the Church of God is despised and disgraced.
Drunkenness does as much harm as the greatest vices, bringing ruin
with it whenever indulged in.
Prayer.
O most loving Jesus, etc., etc.
Twenty-third Day.
The husband has to share the care of the children and he should
look after the instruction of the child. Children are a great treasure,
worth more than all the wealth of the world. The Lord said of them
to the apostles: “Suffer the little children to come unto Me, and
forbid them not: for of such is the kingdom of God.” And why is the
Lord so anxious for the welfare of the innocent child? Because it is a
weak human being, unable to help itself, [pg 089] destined for](https://image.slidesharecdn.com/23428428-250509151846-2d979635/85/Dissipation-And-Control-In-Microscopic-Nonequilibrium-Systems-Steven-J-Large-80-320.jpg)
![heaven, redeemed by the blood of Jesus, and a temple of the Holy
Ghost, more fitted to be such on account of the purity of its soul.
Now the care of this child is given to the father and the mother,
under whose care it is to grow up a true Christian, an exemplary
member of the Church of God; to live on until it has fulfilled its days
and the duties of its state of life, when, like yourself, having come to
the fulness of maturity, it is gathered in to render an account of its
life work to almighty God. The success of the child's life depends
chiefly on the manner in which its parents fulfil their duty.
Prayer.
O most loving Jesus, etc., etc.
Twenty-fourth Day.
In what special duties are you to instruct your children? First of all,
let the young children learn early to pray, make them think of God,
speak to them of His love for mankind, teach them to adore Him,
because He has created them, to thank Him for all His benefits
which flow to them so abundantly, to ask Him with confidence for all
the graces that they need. Correct the children for their faults: Lying,
stealing, cursing, stubbornness, disobedience, fighting, and cruelty.
Have an eye very early on their morals, for little children learn to do
wicked things, by which they lose the love and grace of God. Be not
a tyrant, but a sensible, religious father or mother, and see to it that
the children are free from these vices. Give them no bad example,
[pg 090] especially by using profanity, or by getting intoxicated. All
this presupposes perpetual vigilance; remember you will have to
render a strict account of these things before God.
Prayer.](https://image.slidesharecdn.com/23428428-250509151846-2d979635/85/Dissipation-And-Control-In-Microscopic-Nonequilibrium-Systems-Steven-J-Large-81-320.jpg)
![O most loving Jesus, etc., etc.
Twenty-fifth Day.
Having considered the husband's duties, we must now look at the
duty of a wife. The Scriptures say of the wife: “A good wife is a good
portion; she shall be given in the portion of them that fear God, to a
man for his good deeds.”—Eccles. xxvi. 3. There is nothing in the
whole world more precious than a good wife. “A wise woman
buildeth her house: but the foolish will pull down with her hands that
also which is built.”—Prov. xiv. 1. A wife must love her husband, and
she owes him the most scrupulous fidelity; if the first duty of the
husband is to love his wife, so also is there a corresponding duty to
return that love. She must be patient with him when he comes home
murmuring against his fate; she must make the home agreeable to
him by cleanliness and cheerfulness. She must bear the burdens of
this life with her husband, and encourage him, that he may not be
despondent. The wife must be sober, not given to scolding and fault-
finding. “Let women be subject to their husbands, as to the Lord;
because the husband is the head of the wife, as Christ is the head of
the Church.”—Ephes. v. 22.
[pg 091]
Prayer.
O most loving Jesus, etc., etc.
Twenty-sixth Day.](https://image.slidesharecdn.com/23428428-250509151846-2d979635/85/Dissipation-And-Control-In-Microscopic-Nonequilibrium-Systems-Steven-J-Large-82-320.jpg)
![The conscientious observance of the marriage vows is to be a
supreme law to the wife. Purity must be the virtue principally looked
to in marriage, according to the laws of the Sacrament of
Matrimony; the wife's motto must be that of Susanna of old, who
said: “I will rather die than sin before God.” The wife owes her
husband a compliance at least to his wishes; not exactly an abject
obedience, but that the husband and wife consult with each other,
and that she comply in all lawful and sensible things. This subjection
is founded on the Scriptures. God Himself said to Eve, “Thou shalt
be under thy husband's power, and he shall have dominion over
thee” (Gen. iii. 16), and St. Paul declares, “Let women be subject to
their husbands as to the Lord.”—Ephes. v. 22. The loving, true, and
obedient wife exerts an unbounded influence for good over her
husband. She will make him great in the eyes of men, she will make
him respectable and presentable in society; in short, she will make
the married life a truly happy one from beginning to end.
Prayer.
O most loving Jesus, etc., etc.
Twenty-seventh Day.
The wife has a sublime calling to be a mother. What is more
beautiful than motherhood? [pg 092] what is more useful to the
new-born humanity than the mother? All the world recognizes her
dignity, and respects her. A Christian mother will consider the child a
gift from God, which though given to her, still belongs to God; hers
is the first care of the newly-born infant, her care and love will not
relax for all time to come. She is always the mother. Have the child
presented for Baptism at the very earliest moment; if it be in danger
of death see that it receives private baptism. Then start out in
patience and kindness to rear it, giving it a secular but above all a](https://image.slidesharecdn.com/23428428-250509151846-2d979635/85/Dissipation-And-Control-In-Microscopic-Nonequilibrium-Systems-Steven-J-Large-83-320.jpg)
Dissipation And Control In Microscopic Nonequilibrium Systems Steven J Large
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- 5. SpringerTheses Recognizing Outstanding Ph.D. Research Steven J. Large Dissipation and Control in Microscopic Nonequilibrium Systems
- 6. Springer Theses Recognizing Outstanding Ph.D. Research
- 7. Aims and Scope The series “Springer Theses” brings together a selection of the very best Ph.D. theses from around the world and across the physical sciences. Nominated and endorsed by two recognized specialists, each published volume has been selected for its scientific excellence and the high impact of its contents for the pertinent field of research. For greater accessibility to non-specialists, the published versions include an extended introduction, as well as a foreword by the student’s supervisor explaining the special relevance of the work for the field. As a whole, the series will provide a valuable resource both for newcomers to the research fields described, and for other scientists seeking detailed background information on special questions. Finally, it provides an accredited documentation of the valuable contributions made by today’s younger generation of scientists. Theses may be nominated for publication in this series by heads of department at internationally leading universities or institutes and should fulfill all of the following criteria • They must be written in good English. • The topic should fall within the confines of Chemistry, Physics, Earth Sciences, Engineering and related interdisciplinary fields such as Materials, Nanoscience, Chemical Engineering, Complex Systems and Biophysics. • The work reported in the thesis must represent a significant scientific advance. • If the thesis includes previously published material, permission to reproduce this must be gained from the respective copyright holder (a maximum 30% of the thesis should be a verbatim reproduction from the author’s previous publications). • They must have been examined and passed during the 12 months prior to nomination. • Each thesis should include a foreword by the supervisor outlining the significance of its content. • The theses should have a clearly defined structure including an introduction accessible to new PhD students and scientists not expert in the relevant field. Indexed by zbMATH. More information about this series at http://www.springer.com/series/8790
- 8. Steven J. Large Dissipation and Control in Microscopic Nonequilibrium Systems Doctoral Thesis accepted by Simon Fraser University, BC, Canada
- 9. Steven J. Large Viewpoint Investment Partners Calgary, AB, Canada ISSN 2190-5053 ISSN 2190-5061 (electronic) Springer Theses ISBN 978-3-030-85824-7 ISBN 978-3-030-85825-4 (eBook) https://doi.org/10.1007/978-3-030-85825-4 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
- 10. – To the memory of Thomas George Smith
- 11. Supervisor’s Foreword Encompassing several independent research thrusts, Dr. Steven Large’s thesis tackles a wide range of important open questions in nonequilibrium statistical mechanics, a subfield of physics that promises insights into the design principles of molecular-scale biological systems. Steve’s thesis work establishes a multifaceted extension of the deterministic control framework that has been the workhorse of the field for the past 20 years to systems that are strongly fluctuating and autonomous, thereby facilitating the application of stochastic thermodynamics ideas to understand molecular machines in nanotechnology and in living things. This thesis contains several distinct types of contributions: purely theoretical derivations, numerical simulations to systematically characterize model systems, experimental design, and analysis of experimental data. In his examination of the control of strongly fluctuating microscopic systems, Steve derived simple, tractable, and surprisingly general conclusions (that upon reflection are intuitive). He subsequently generalized this work to autonomous settings, establishing connections to the chemical kinetics literature. As part of this project, Steve uncovered a conceptual blind spot in recent stochastic ther- modynamics research, establishing the rigorous generalization of fundamental thermodynamics concepts to autonomous complex systems. On the applied side, Steve put these theoretical developments to work by designing, analyzing, and interpreting the first experimental demonstration (by collaborators Sara Tafoya and Carlos Bustamante, UC Berkeley) of the utility of this generally applicable method (that some naturally evolved machines show evidence of using) for designing energetically efficient control in biomolecules: drive slowly where friction is large and rapidly where friction is small. He analyzed raw data from equilibrium single-molecule experiments on small DNA molecules and designed subsequent nonequilibrium experiments demonstrating that protocols optimized using this framework systematically and statistically significantly reduced energy loss compared to naive protocols, across a wide range of experimental unfolding speeds and wildly different DNA molecules. Taken as a whole, these theoretical advances and experimental demonstrations give a scale on which to evaluate the energetic efficiency of molecular machines, vii
- 12. viii Supervisor’s Foreword guidelines for designing effective synthetic machines, and a perspective on the engineering principles that govern effective microscopic energy transmission far from equilibrium. Burnaby, Canada Prof. David A. Sivak June 2021
- 13. Acknowledgment First and foremost, I would like to thank my supervisor David Sivak for his support and guidance over the past 5 years. You granted me independence to follow my interests and immerse myself in the truest essence of academic research. The experience I have gained by working with you has been essential in strengthening my capabilities, and has been vital to my continued development as a scientist. Throughout my graduate education I have had the great opportunity of working closely with a number of collaborators, from whom I have benefited greatly. I would like to thank Carlos Bustamante, Shixin Liu, and particularly Sara Tafoya for their patience and perseverance working on the DNA hairpin project. I would also like to thank Raphaël Chetrite and Jannik Ehrich for their collaboration on various projects throughout the past few years, I greatly enjoyed working with you both. I would also like to thank the guidance of my supervisory committee, John Bechhoefer and Eldon Emberly. Your unique approaches to scientific problems and continued feedback through the years has helped to broaden the scope of my thinking. The content of this thesis has been improved and clarified significantly from your encouragement and suggestions. I greatly appreciate the financial support I have received throughout my graduate studies, including NSERC CGS-M and CGS-D3 fellowships, as well as the Billy Jones and Howard Malm graduate awards in physics, made possible through the continued generosity of SFU donors. I would also like to thank my parents, Andrew and Sandra, for believing in me through the past 5 years and beyond. Ever since I left to start my undergraduate degree in 2011 at the University of Guelph, I could always depend on your encouragement and unwavering support. I also would like to thank my sister Jenny, her husband Ian, and son Nate, for always being there for me. Through the ups and downs of graduate school, friends and colleagues make the worst of times better and the best of times great. I would like to acknowledge the entire cast of Sivak group members over the past 5 years, with a special mention of Aidan Brown and Alexandra Kasper, who helped immensely in the early years of grad school. I also benefited greatly from many colleagues and friends in the broader physics graduate student body, with a particular thanks owing to Chapin ix
- 14. x Acknowledgment Korosec, Konstantin Lehmann, and Benny Jäger. I also want to thank the friends who have been along for the whole ride, Jordan Elvedahl and Scott Matthews, such lasting friendships burn a mark on us all, your role in keeping me tethered to reality through the last decade cannot be understated. Finally, I would like to thank my future wife Maegan Kelleway. Your support and encouragement throughout graduate school have been indispensable, ultimately, you—above all others—have helped me keep a clear head through this whole process. The past 5 years especially have brought many changes to our life, and this work wouldn’t have been possible without you. This closes one chapter on our life and begins another, one day at a time.
- 15. The general struggle for existence is not a struggle for raw materials—these for organisms are air, soil and water, all abundantly available—nor for energy which exists in plenty in the sun and any hot body in the form of heat, but rather a struggle for entropy, which becomes available through the transition of energy from the hot sun to the cold earth. – Ludwig Boltzmann1 1 L. Boltzmann, The second law of thermodynamics, in Theoretical Physics and Philosophical Problems, vol. 5, ed. by B. McGuinness (Springer, 1974).
- 16. Contents 1 Introduction ................................................................. 1 1.1 Molecular Machines.................................................. 3 1.1.1 Kinesin ...................................................... 4 1.1.2 ATP Synthase ............................................... 5 1.2 Nonequilibrium Statistical Physics .................................. 6 1.3 Overview of This Thesis ............................................. 9 1.4 Contributions to This Thesis ......................................... 9 References.................................................................... 10 2 Theoretical Background ................................................... 15 2.1 Mathematical Preliminaries.......................................... 15 2.1.1 Random Variables, Probabilities, and Characteristic Functions.................................... 15 2.2 Nonequilibrium Dynamics........................................... 18 2.2.1 Master Equation ............................................ 20 2.2.2 Fokker-Planck Equation .................................... 23 2.2.3 Langevin Equation.......................................... 25 2.3 Stochastic Thermodynamics ......................................... 27 2.4 Fluctuation Theorems ................................................ 30 2.5 Entropy and Information Theory .................................... 32 2.6 Control in Microscopic Nonequilibrium Systems .................. 34 2.6.1 Linear-Response Theory ................................... 36 2.6.2 Generalized Friction Tensor................................ 38 2.6.3 Minimal-Work Control Protocols .......................... 40 2.7 Model Systems ....................................................... 42 2.7.1 Harmonic Trap.............................................. 42 2.7.2 Periodic Potential ........................................... 43 2.7.3 Fast-Switching Potential ................................... 44 References.................................................................... 46 xiii
- 17. xiv Contents Part I Experimental Tests of Nonequilibrium Theory 3 DNA Hairpins I: Equilibrium ............................................. 51 3.1 Introduction........................................................... 51 3.2 Experimental Setup .................................................. 53 3.3 Equilibrium Sampling................................................ 55 3.3.1 Estimating the Generalized Friction Coefficient .......... 56 3.3.2 Designing Protocols ........................................ 58 References.................................................................... 61 4 DNA Hairpins II: Nonequilibrium ....................................... 63 4.1 Unfolding/Refolding Force identification ........................... 65 4.2 Excess Work Measurements ......................................... 68 4.2.1 Excess Power in Designed and Naive Protocols .......... 70 4.2.2 Cycle Work and Dissipation in Designed and Naive Protocols ............................................. 72 4.3 Protocol Work Ratios ................................................ 72 4.3.1 Excess Work Ratio for Variable Bin Widths .............. 74 4.3.2 Protocol-Work Ratio for DNA Hairpins ................... 75 References.................................................................... 76 5 DNA Hairpins III: Conclusions ........................................... 79 5.1 Alternative Hairpin Sequence........................................ 79 5.2 Mean-Variance Trade-Offs for Excess Work........................ 83 5.3 Alternative Buffer Conditions ....................................... 84 5.4 Discussion ............................................................ 86 References.................................................................... 87 Part II Dissipation in Nonequilibrium Systems Through the Lens of Control Theory 6 Stochastic Control .......................................................... 91 6.1 Introduction........................................................... 91 6.2 Revisiting Linear Response .......................................... 92 6.3 Protocol Ensembles .................................................. 94 6.3.1 Expansion of the Excess Power ............................ 95 6.3.2 Lower Bound on Excess Work ............................. 100 6.4 Model Ensembles .................................................... 101 6.4.1 Periodic-Potential Ensemble ............................... 102 6.4.2 Stochastically Driven Protocols............................ 107 6.5 Discussion ............................................................ 109 References.................................................................... 110 7 Optimal Discrete Control .................................................. 113 7.1 Introduction........................................................... 113 7.2 Background ........................................................... 114 7.3 Infinite-Time Work ................................................... 115
- 18. Contents xv 7.4 Nonequilibrium Excess Work ....................................... 117 7.4.1 Nonequilibrium Excess Work: Linear Response for Time-Dependent Protocols ............................. 119 7.5 Minimum-Work Protocols ........................................... 121 7.5.1 Minimum-Work Protocols for a Single Control Parameter ................................................... 123 7.6 Harmonic Trap ....................................................... 125 7.6.1 Infinite-Time Limit ......................................... 125 7.6.2 General Solution: Finite-Time Work ...................... 126 7.7 Periodic Potential..................................................... 128 7.8 Discussion ............................................................ 132 References.................................................................... 133 8 On Dissipation Bounds..................................................... 135 8.1 Introduction........................................................... 135 8.2 Discrete Stochastic Protocols ........................................ 136 8.3 A Cost for Control ................................................... 139 8.4 Harmonic System .................................................... 140 8.4.1 Timescale-Separated Limit................................. 142 8.4.2 Nonequilibrium Excess Work .............................. 142 8.4.3 General Dissipation Bound................................. 145 8.5 Discussion ............................................................ 146 References.................................................................... 148 Part III The Nonequilibrium Physics of Autonomous Machines 9 Free Energy Transduction ................................................. 151 9.1 Introduction........................................................... 151 9.2 Strongly Coupled Multi-component Systems....................... 152 9.2.1 Entropy Production ......................................... 153 9.2.2 Excess Work ................................................ 155 9.3 Classes of Upstream Dynamics...................................... 156 9.3.1 External Control Parameter ................................ 156 9.3.2 Thermodynamically Complete System .................... 157 9.4 Model System ........................................................ 159 9.4.1 Excess Power Does Not Equal Entropy Production ...... 160 9.4.2 Excess Power Can Become Negative ...................... 161 9.4.3 Entropy Production in Thermodynamically Complete or Incomplete Systems .......................... 163 9.5 Discussion ............................................................ 163 References.................................................................... 164 10 Hidden Excess Power....................................................... 167 10.1 Introduction........................................................... 167 10.2 Coarse-Grained Representations of Mechanochemical Systems... 169 10.3 Hidden Excess Work in Molecular Machines....................... 171
- 19. xvi Contents 10.3.1 TSS Excess Work........................................... 172 10.3.2 Nonequilibrium Excess Work .............................. 174 10.4 Model Systems ....................................................... 175 10.4.1 Linear-Transport Motor .................................... 175 10.4.2 Rotary Motor................................................ 176 10.5 Discussion ............................................................ 178 References.................................................................... 179 11 Conclusions and Outlook .................................................. 183 11.1 Outlook ............................................................... 187 11.2 Final Remark ......................................................... 189 References.................................................................... 190 A Code and Data .............................................................. 193 A.1 Master Equation ...................................................... 193 A.1.1 Trajectory Simulation ...................................... 194 A.2 Langevin Equation ................................................... 195 A.2.1 Underdamped Dynamics ................................... 195 A.2.2 Overdamped Dynamics .................................... 196 A.3 Coupled Discrete and Continuous Dynamics ....................... 197 B DNA Hairpins ............................................................... 199 B.1 Folding Forces........................................................ 199 B.2 Alternative Excess Work Measures.................................. 200 C Stochastic Control .......................................................... 203 C.1 Generalization of Lower Dissipation Bound ........................ 203 C.2 Disagreement Between Theoretical Predictions and Numerical Results .................................................... 205 C.3 Equivalence of Ensembles ........................................... 206 D Optimal Discrete Control .................................................. 209 D.1 Expansion of the Relative Entropy .................................. 209 D.2 Harmonic Trap: Exact Result ........................................ 211 E On Dissipation Bounds..................................................... 215 E.1 Generalized Friction for Gamma-Distributed Dwell Times ........ 215 E.2 Generalized Friction for Gamma-Distributed Dwell Times: Harmonic-Trap ............................................... 216 E.3 Average Step Number for Uniform Jump Rates .................... 216 F Free Energy Transduction ................................................. 221 F.1 Detailed Derivation of Transduced Additional Free Energy Rate .......................................................... 221 F.2 At Steady State, Excess Power Equals Heat Flow .................. 222
- 20. Contents xvii G Hidden Excess Power....................................................... 225 G.1 Expansion of the TSS Work ......................................... 225 G.2 Nonequilibrium Excess Work in Autonomous Systems............ 230 G.3 Simulation Details: Linear-Transport Motor ........................ 234 References.................................................................... 235
- 21. Chapter 1 Introduction The classical theory of thermodynamics deals with phenomenological relationships between flows of energy in macroscopic systems. While originally motivated by a need to understand and optimize the performance of steam engines in the 1800s [1], the scope of thermodynamics has expanded through the past two centuries, finding its place as a mathematical framework that pervades the branches of science and industry. At its heart, the theory of classical thermodynamics captures the mathematical relationships between physical quantities, such as heat and work, subject to a set of underlying assumptions. So long as these assumptions are valid, the classical theory of thermodynamics can be encompassed by four axiomatic rules known as the laws of thermodynamics [2]. Despite the widespread successes of classical thermodynamics, there are a number of shortcomings that limit its utility. Central to the focus of this thesis is the assumption of systems being in a state of equilibrium. The restriction of thermodynamic analysis to equilibrium precludes the treatment of many interesting situations, such as biological systems, which are manifestly out of equilibrium. Furthermore, through the subsequent development of a statistical theory of thermo- dynamics, known as statistical mechanics, it became clear that the laws of classical thermodynamics pertain to ensemble-average quantities. For macroscopic systems, an appeal to the central limit theorem makes clear that for N ∼ 1023 interacting molecules, the fluctuations in energy and energy flows can be assumed to be vanishingly small (so long as the system is not sufficiently close to a critical point). However, for small systems—often on the scale of micro- or nano-meters—thermal fluctuations can no longer be ignored, and have significant implications for the overall physical description of the system of interest. Furthermore, ever since Clausius, in 1865, first articulated the law of increase in entropy [3], what has come to be known as the second law has posed a deep, unresolved question: how can microscopic equations of motion, that are symmetric under time reversal, give rise to macroscopic behavior that is not? © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 S. J. Large, Dissipation and Control in Microscopic Nonequilibrium Systems, Springer Theses, https://doi.org/10.1007/978-3-030-85825-4_1 1
- 22. 2 1 Introduction Over the past 25 years, the development of stochastic thermodynamics has ushered in a new era in thermodynamics, largely overcoming these limitations, and deepening our understanding of the second law as a statistical relation (Sect. 2.3). At its heart, stochastic thermodynamics seeks to understand how the laws of thermo- dynamics manifest in microscopic strongly fluctuating systems that are potentially far from equilibrium. The theoretical framework of stochastic thermodynamics provides a consistent method of assigning physical quantities–such as work, heat, and entropy–to fluctuating systems in contact with thermodynamic reservoirs, even when those systems are far from equilibrium. These physical quantities can be identified along a single trajectory or at the level of probability distributions, thus permitting a diverse set of methods to understand the physics of thermodynamic systems across all scales. Given a set of state energies, the equilibrium ensemble of a system can be determined without any knowledge of its dynamics. Thus, the entire theory of equilibrium statistical mechanics can be built without the need to model the microscopic dynamics. In nonequilibrium systems, however, the same is not true. In general, the state of a particular nonequilibrium system depends not only on its present conditions, but also on its history [4]. In addition to deriving a set of consistent laws of stochastic thermodynamics, the study of fluctuating systems led to the development of an entirely new class of results, known collectively as fluctuation theorems [5–11] (Sect. 2.4). These mathematical identities place stringent constraints on the fluctuations in stochastic systems, even far from thermodynamic equilibrium, and can be viewed as general- izations of the second law of thermodynamics. In particular, the entropy production fluctuation theorem can be used to derive the second law, and provides a much deeper understanding of its physical origins. In fact, all fluctuation theorems–even those not relating to entropy–give rise to second-law-like inequalities [12, 13]. This realization has led to a significant improvement in our understanding of the physical origins of irreversibility in nonequilibrium systems. For instance, the generalized Jarzynski equality [14] has shown how incorporating information— in the form of feedback control—into the thermodynamics of fluctuating systems can lead to sub-zero bounds on entropy production, a result which has been experimentally verified [15]. The incorporation of information into the theory of thermodynamics has effectively ‘exorcised’ the long-standing thought experiment known as Maxwell’s demon [16]: by treating information as a physical quantity— which was suggested by Landauer in 1961 [17]—the familiar form of the second law is restored. In tandem with the theoretical developments of stochastic thermodynamics, a new set of experimental techniques were developed to directly probe and perturb microscopic systems [18]. Early experiments were concerned with measuring the fluctuations in microscopic systems, and primarily aimed at verifying the fluctuation theorems. Examples include measurements of heat and work fluctuations in the tip of an AFM [19], the rotational angle of a torsion pendulum [20], and in an electrical resistor [21]. Furthermore, the development—and subsequent refinement—of opti-
- 23. 1.1 Molecular Machines 3 cal tweezers for single-molecule force spectroscopy allowed the direct verification of theoretical predictions in biological systems, such as a DNA or RNA hairpin [22– 24]. Biological systems have served as a central focus for applying stochastic thermodynamics. Ultimately, this comes as a result of the nonequilibrium nature of microscopic biological systems. The essential importance of nonequilibrium physics in our understanding of biology is summarized effectively by Erwin Schrödinger in What is Life? where he famously equated the ‘decay into thermody- namic equilibrium’ with death [25]. More recently, a great deal of effort has focused on our understanding of molecular machines, which are a class of nanoscale objects that operate out of equilibrium to perform useful tasks within biological cells. These molecular machines manage to remain out of thermodynamic equilibrium by siphoning free energy off of nonequilibrium environmental conditions, such as out-of-equilibrium concentrations of chemical reactants and products. In fact, the understanding of molecular machines has served as a central motivation for the continued development of stochastic thermodynamics as a whole [26, 27]. Stochastic thermodynamics can help in furthering our understanding of the physics of molecular machines, in particular, the fundamental physical constraints such nonequilibrium systems face in vivo, and how these constraints affect the limits of performance in molecular machines. Thus, there is a need to define a performance metric in such systems that quantifies the loss of capacity in the system. The entropy production, or dissipation, can serve such a role: entropy production represents a fundamental loss of system capacity to perform useful work. Low-entropy states have a higher capacity to perform work than high-entropy states; for instance a fully extended polymer chain has the capacity to exert forces during its compaction. Many recent significant advances, such as the thermodynamic uncertainty relation [28, 29] and its generalizations [30–33], have been motivated as providing such fundamental functional limitations on the performance of molecular machines out of equilibrium. 1.1 Molecular Machines At the sub-cellular scale, biological systems exhibit a strikingly high degree of organization which is inconsistent with an equilibrium state [26]. This organization in maintained, in large part, through the concerted effort of a host of molecular machines [34]. These nanoscale machines consume energy, typically in the form of high-energy chemical bonds, to perform useful functions in the cell [27]. Physically, molecular machines are made up of several interacting soft-matter components, which experience large thermal fluctuations. Over the past several decades, molecular machines have been a subject of intense focus within the biophysics community, and have been studied using a host of modern experimental techniques, such as optical tweezers [35–38], or electrorotation [15, 39–41], to probe and perturb individual molecular machines. Efforts to better understand the physics of molecular machines promise to deepen our understanding of the fundamental
- 24. 4 1 Introduction operational constraints facing evolved molecular machines, but also promise a range of practical benefits, such as the design and synthesis of de novo molecular machines, perhaps accelerating their use in next-generation nanomedicine [42, 43]. 1.1.1 Kinesin A canonical example of a molecular transport motor is kinesin [44]. The first of many variants, kinesin-1, was discovered in 1985 in observations of the mobility of organelles in the extruded cytoplasm from the giant axon of a squid [45–47]. Soon after, this ‘axoplasmic motility’ was understood to be due to the effect of ‘a novel force-generating molecule’ which has since come to be known as kinesin-1 [48]. Many distinct variations of the kinesin motor were found soon after by identifying a common genetic motif in the genome of Drosophila melanogaster [49], and subsequently found to occur, in one form or another, in virtually all forms of eukaryotic cells [50]. Kinesin is a molecular motor that is responsible for the transport of cellular cargoes throughout individual cells. Morphologically, kinesin consists of two motor head domains, which are connected through a long stalk to two cargo binding domains. By making use of the high-energy phosphate bond in the cellular energy currency ATP, kinesin moves by its head domains processively walking along a microtubule, typically in a ‘+’-end directed fashion, although certain kinesin motors can, in fact, travel in the opposite direction [51]. In vivo, the cell ensures directionality by maintaining an out-of-equilibrium concentration of ATP and ADP molecules [44, 50]. Through the use of modern experimental techniques, kinesin has been studied in great detail, informing much of our modern understanding of the mechanics of nanoscale machines. For instance, early studies with optical trapping allowed direct measurement of the stall force—the maximum opposing force under which the kinesin can still move in the forward direction—of ∼6–8 pN [35–38]. When this stall force is multiplied by the kinesin step size of ∼8 nm, the resulting work done by the kinesin motor is ∼48–64 pN· nm, which, when compared with the free energy ∼80 pN·nm liberated by hydrolysis of a single ATP, indicates an efficiency of up to ∼80% [52]. Additionally, single-particle fluorescence tracking has measured the in vivo speeds of kinesin motors directly, clocking in at a maximum of ∼1 µm/s, or 125 (8-nm) steps/s [53]. In addition to providing an interesting biophysical model for understanding the physics of molecular machines, kinesins are suspected to play a central role in pathophysiology of many diseases. In particular, their role in transport implicates them in many neuronal disorders, such as amyotrophic lateral sclerosis (ALS) and Alzheimer’s, where neurons’ extremely polarized cell morphologies make proces- sive transport motors—such as kinesin—vital to their healthy function [54, 55].
- 25. 1.1 Molecular Machines 5 1.1.2 ATP Synthase With ATP serving as the primary chemical fuel source for many molecular machines (such as kinesin), the production of ATP is essential to the operation of biological cells. The molecular machine ATP synthase is responsible, in large part, for the production and maintenance of cellular ATP stores. As with kinesin, several variations of ATP synthase exist in different organisms, but its presence is ubiquitous amongst both eukaryotic and prokaryotic cells [56, 57]. In eukaryotic cells, ATP synthase motors are often located in the membranes of organelles, whereas in prokaryotic cells they are found in the cell membrane. In fact, the internalization of ATP synthase—and corresponding increase in cellular energy capacity—during the emergence of eukaryotic cells in the evolutionary past is thought to have been a central factor in the subsequent formation of higher-order structures, such as complex multicellular organisms [58]. While there are several different forms of ATP synthase, the most well-studied is FoF1 ATP synthase, which is found in eukaryotic mitochondria and chloroplasts as well as prokaryotic cell membranes. In contrast, variants such as the VoV1 ATPase inhabit organelle membranes other than mitochondria, such as endosomes and lysosomes [59, 60], while AoA1 ATP synthase appears in the membrane of extremophilic archaea. VoV1 ATPase operates in the opposite direction of FoF1, using ATP as an energy source to acidify the interior of organelles, and contains a biochemical control mechanism, whereby the depletion of cellular glucose causes dissociation of the Vo and V1 components, inactivating the complex as an ATPase [60, 61]. Even within the class of FoF1 ATP synthase motors, however, there is significant diversity across organisms. For instance, the number of protein subunits in the Fo component can vary between organisms, and can have a significant impact on its overall energetics [62]. The remarkable efficiencies achieved by ATP synthase make it a fascinating example of energy transduction in out-of-equilibrium biological systems. The ATP synthase motor provides a straightforward example of a molecular machine that converts energy stored in out-of-equilibrium chemical concentrations ([H+] difference across the membrane) into an essential cellular resource (ATP). This has made ATP synthase a canonical system in which to model and understand the fundamental physics of molecular machines and the inherent trade-offs in the functional capabilities of biomolecular machines [63–66]. Throughout this thesis, the simple models we use to test theoretical models are motivated by the physics of molecular machines, such as kinesin and ATP synthase. These represent a minute fraction of the diversity of molecular machines that exist within biological cells, performing a wide array of functions, and largely responsible for maintaining the nonequilibrium conditions necessary for life. Fundamentally, these nanoscale engines operate in the presence of strong thermal fluctuations and out of thermodynamic equilibrium, and thus to understand their physics we need to make use of nonequilibrium theories of statistical physics and stochastic processes.
- 26. 6 1 Introduction 1.2 Nonequilibrium Statistical Physics For a physical system in contact with a thermal reservoir, at equilibrium the microstates x are distributed in accordance with the Boltzmann distribution π(x|λ) = e−βE(x|λ)+βF(λ) , (1.1) where β ≡ (kBT )−1 is the inverse temperature of the reservoir, E(x|λ) is the energy of microstate x given a set of external parameters λ—such as pressure or volume—and F(λ) is the equilibrium free energy of the system. The equilibrium free energy is related to the partition function by F(λ) ≡ −kBT ln Z, where Z ≡ exp [−βE(x|λ)] dx normalizes the equilibrium distribution. Thus, to determine the equilibrium distribution for any system of interest, we only need to know the energies of each of the microstates. In contrast, determining the nonequilibrium distribution of a system requires knowledge of its dynamics. Furthermore, aside from special cases, like a nonequi- librium steady state, determining the instantaneous nonequilibrium distribution over microstates also requires knowledge of the previous history of the system. For systems that we are generally interested in, such as a polymer in solution, there are a very large number of individual molecules, N ≈ 1023 (counting all of the waters) that can have non-negligible interactions with the system. The explicit modeling of all degrees of freedom in such a system is analytically intractable, and even outside of the realm of numerical calculations. Fortunately, it is often the case that a relatively small number of these molecules represent our system of interest. Thus, we seek a method of obtaining equations of motion for the system of interest that do not require explicitly accounting for all the molecules that make up the environment. Mathematically, such reduced descriptions of the system dynamics can be justified by an assumption of weak interactions between the system and its surroundings, or a separation of timescales between the system and environment. Weak interactions between the system and its environment are a primary assumption behind, for instance, Zwanzig’s projection operator method [4], where the Liouville equation for the entire system is projected onto a low-dimensional space that captures the dynamics of the system of interest, and the effective interactions between the system and environment are captured through an additional fluctuating force. Conversely, the assumption of a separation of time scales enforces a typical axiom of irreversible thermodynamics: thermodynamic reservoirs always remain at equilibrium [67]. In either case, the same equations of motion are the result, which transform the typical differential equations of classical mechanics to stochastic differential equations, with fluctuating components that capture the effective interactions between a system and its surroundings. A canonical example for the equation of motion of a stochastic system is the diffusive motion of a small object, such as a pollen grain, in water. This phenomenon was first documented by the botanist Robert Brown in 1827 while observing the motion of pollen grains in water [68], however it wasn’t until the early twentieth
- 27. 1.2 Nonequilibrium Statistical Physics 7 century when Einstein [69]—and, independently, Smoluchowski [70]—provided a clear and consistent interpretation of the seemingly random dynamics. Einstein proposed that the random motion of the pollen grain was due to its repeated interaction with the water molecules surrounding it. Furthermore, because the motion of all of the individual water molecules was exceedingly complex, the motion of the pollen grain could only be understood statistically. As such, Einstein’s treatment of what is now called Brownian motion is often seen as the first treatment of a physical phenomenon using stochastic modeling [71]. These two main points led Einstein to the famous diffusion equation, quantifying the temporal evolution of the probability of observing the position of the pollen grain at a particular point in space ∂p(x, t|x0, t0) ∂t = D∇2 p(x, t|x0, t0) , (1.2) where p(x, t|x0, t0) is the probability density of observing the pollen grain at position x at time t given it was at x0 at time t0, ∇2 is the Laplacian operator, and D is the diffusion coefficient. The diffusion coefficient D is related to the rate at which the probability distribution spreads out in time. Specifically, the mean- squared displacement (MSD) of a diffusive particle obeys the relationship δx2 (t) = 2NdimDt , (1.3) where the angle brackets · · · indicate an average over an ensemble of pollen grains, all initialized at the same position at time t = 0, δx ≡ x − x is the deviation of x from its average position, and Ndim is the number of dimensions in which the particle is diffusing. For a pollen grain diffusing on the surface of water, Ndim = 2.1 Not long after Einstein’s discovery of the diffusion equation, Paul Langevin, a French physicist, presented an alternative treatment of the process of diffusion using stochastic processes. Specifically, he claimed that the equation of motion of an individual pollen grain, as per Newtonian mechanics, is given by the differential equation d2x dt2 = −γ dx dt + F(t) (1.4) 1 There are, however, some subtleties involved in diffusion of a particle at an air-water interface. Specifically, the diffusion coefficient predicted by the Einstein relation (1.7) may be modified due to the interaction of the particle with the interface. For instance, an increase in drag forces due to surface tension [72], the Marangoni effect and capillary forces [73], and even the elastic response of the interface to fluctuation-induced surface-wave formation [74] can have significant impact on the value of the diffusion coefficient.
- 28. 8 1 Introduction where the first term on the RHS represents a frictional damping force Ffric(t) = −γ v(t), with γ the viscous damping coefficient for the pollen grain in the fluid, and F(t) is a random variable that represents a fluctuating force, arising from the frequent impacts of water molecules with the pollen grain. While it would take another 40 years to construct a mathematically rigorous derivation of the form of the fluctuating force, we will simply state two properties of the fluctuating force that turn out to be true: F(t) = 0 (1.5a) δF(t)δF(t ) = 2kBT γ δ(t − t ) . (1.5b) The first of these properties (1.5a) implies that the fluctuating force does not, on average, impart any net force on the pollen grain, while the second (1.5b) suggests that the random forces at different times are uncorrelated. The constant of proportionality in (1.5b) is fixed by the equipartition theorem [4]. Using (1.5a) and (1.5b), we can write the Langevin equation in its more familiar form d2x dt = −γ dx dt + 2kBT γ ξ(t) , (1.6) where now ξ(t) is a zero-mean ξ(t) = 0, delta-correlated white noise process with ξ(t)ξ(t) = δ(t − t). Finally, we can connect the phenomenological constants γ from the Langevin equation (1.6) and D from the diffusion equation (1.2) through the Einstein relation D = kBT γ . (1.7) Fundamentally, the diffusion equation (1.2) and Langevin equation (1.6) repre- sent two different approaches to solving the same problem: what is the equation of motion for a fluctuating particle in contact with a thermal reservoir? In each case, the influence of the thermal reservoir comes into the mathematical description of the system through a transport coefficient (γ or D). The diffusion equation solves this problem by calculating the full probability distribution function p(x, t) as the solution of the partial differential equation in (1.2), whereas the Langevin equation provides a description in terms of a stochastic differential equation, where the solution is a fluctuating trajectory consistent with the statistical properties of the dynamics. Figure 1.1 shows a comparison between trajectory-level and probability distribu- tion descriptions of a stochastic system. For n = 1000 Langevin trajectories (left), the empirical distribution over positions x at specific times match the solution to the corresponding (right) diffusion equation (1.2).
- 29. 1.4 Contributions to This Thesis 9 0 1 2 3 4 5 Time t −10 −5 0 5 10 Position x 0.0 0.2 0.4 P(x|t) Fig. 1.1 Trajectory-level and distribution-level simulations of diffusive motion give alterna- tive descriptions of the same physical process. (left) A sample of n = 1000 individual trajectories of a Langevin simulation of Brownian motion compared to the (right) time-dependent probability distribution obtained by solving the diffusion equation (solid curves). Circles indicate the empirical distribution of positions obtained from the ensemble of Langevin trajectories at the same times (indicated by the red vertical bars on the left), showing agreement 1.3 Overview of This Thesis The remainder of this thesis consists of three parts, discussing research that broadly falls into three categories: experimental tests of nonequilibrium theory (Part I), dis- sipation through the lens of control theory (Part II), and the nonequilibrium physics of autonomous machines (Part III). The content for Part I presents and elaborates on results published in [24], Part II discusses the theoretical work presented in [75, 76] and elaborates upon previous discussions in [75] on lower bounds on dissipation (Chap. 8). Finally, in Part III we discuss the implications of control-theoretical models of nonequilibrium physics in understanding the microscopic physics of molecular machines. Specifically, we investigate the relationship between excess work and entropy production in strongly coupled nonequilibrium system from [77], and derive a near-equilibrium phenomenological method of quantifying the energy flows between the components of such systems presented in [78]. 1.4 Contributions to This Thesis Several chapters in this thesis contain results from collaborative efforts. The content in Part I (Chaps. 3, 4, and 5) is drawn from material published in [24] regarding experiments performed in collaboration with Sara Tafoya, Shixin Liu, and Carlos Bustamante at the University of California, Berkeley. In this project, Sara Tafoya performed the experiments and gathered the raw data, and all parties
- 30. 10 1 Introduction in the collaboration participated in conceptualization of the project. I performed the data curation and formal analysis of the resulting data, designed experiments, and wrote the necessary software for the project, including both the data analysis and visualization. In summarizing the results and discussing how to present them, all parties in the collaboration took part. Furthermore, Raphaël Chetrite helped with the formal analysis and mathematical details of our derivations in Chap. 6 (which are published in [75]), particularly for the exact solutions found in Sect. 6.4.1. Jannik Ehrich helped with conceptualization of the project discussed in Chap. 9 and published in [77], performed some of the formal analysis, and helped in writing up the results. References 1. S. Carnot, Réflexions sur la puissance motrice du feu et sur les machines propres a développer cette puissance (Bachelier Libraire, 1824) 2. D.V. Schroeder, An Introduction to Thermal Physics (Pearson, 1999) 3. R. Clausius, Ueber verschiedene für die Anwendung bequeme Formen der Hauptgleichungen der mechanischen Wärmetheorie. Annalen der physik und chemie 23, 125 (1865) 4. R. Zwanzig, Nonequilibrium Statistical Mechanics. Oxford University Press, 125 (2001) 5. C. Jarzynski, Nonequilibrium equality for free energy differences. Phys. Rev. Lett. 78, 2690 (1997) 6. G.E. Crooks, Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences. Phys. Rev. E 60, 2721 (1999) 7. G. Gallavotti, E.G.D. Cohen, Dynamical ensembles in nonequilibrium statistical mechanics. Phys. Rev. Lett. 74, 2694 (1995) 8. D.J. Evans, E.G.D. Cohen, G.P. Morriss, Probability of second law violations in shearing steady states. Phys. Rev. Lett. 71, 2401 (1993) 9. D.J. Evans, D.J. Searles, Equilibrium microstates which generate second law violating steady states. Phys. Rev. E 50, 1645 (1994) 10. T. Hatano, S.-I. Sasa, Steady-state thermodynamics of Langevin systems. Phys. Rev. Lett. 86, 3463 (2001) 11. U. Seifert, Entropy production along a stochastic trajectory and an integral fluctuation theorem. Phys. Rev. Lett. 95, 040602 (2005) 12. M. Esposito, R. Kawai, K. Lindenberg, C. Van den Broeck, Efficiency at maximum power of low-dissipation Carnot engines. Phys. Rev. Lett. 105, 150603 (2010) 13. J.M. Horowitz, H. Sandberg, Second-law-like inequalities with information and their interpre- tations. New J. Phys. 16, 125007 (2014) 14. T. Sagawa, M. Ueda, Generalized Jarzynski equality under nonequilibrium feedback control. Phys. Rev. Lett. 104, 090602 (2010) 15. S. Toyabe, T. Sagawa, M. Ueda, E. Muneyuki, M. Sano, Experimental demonstration of information-to-energy conversion and validation of the generalized Jarzynski equality. Nat. Phys 6, 988 (2010) 16. K. Maruyama, F. Nori, V. Vedral, Colloquium: the physics of Maxwell’s demon and informa- tion. Rev. Mod. Phys. 81, 1–17 (2009) 17. R. Landauer, Irreversibility and heat generation in the computing process. IBM J. Res. Dev. 5, 183 (1961) 18. S. Ciliberto, Experiments in stochastic thermodynamics: short history and perspectives. Phys. Rev. X 7, 021051 (2017)
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- 34. Chapter 2 Theoretical Background 2.1 Mathematical Preliminaries We now develop the mathematical framework behind the stochastic equations of motion governing microscopic, fluctuating systems. In particular we will derive the three primary methods of describing stochastic dynamics: the master equation, the Langevin equation, and the Fokker-Planck equation. Each of these descriptions of stochastic dynamics is used throughout the remainder of the thesis. The master equation is used in Chap. 9, the Langevin equation is used in Chaps. 6 and 10, and the Fokker-Planck equation is used in Chap. 7. 2.1.1 Random Variables, Probability Distributions and Characteristic Functions The fundamental mathematical underpinning of stochastic thermodynamics and nonequilibrium physics is probability theory. As a mathematical theory originally axiomatized by Andrey Kolmogorov in the 1930s [1], probability theory has become central to a number of fields of science and industry, from the fluctuating proteins within a biological cell to the statistics of returns in an investment portfolio [2, 3]. At its core, probability theory deals with the mathematics of uncertainty, quantifying the likelihoods of random events. The outcome of a random event X, such as a coin toss, is referred to as a random variable because its outcome is uncertain. While we do not know what the exact outcome of the event will be, we know what the possible outcomes x may be, and can assign relative likelihoods to them. For instance, in tossing a fair coin, the possible outcomes are heads or tails, and both are equally likely. The function that quantifies the relative likelihoods for different outcomes of the random variable X is called the probability px, where here px is the likelihood that the outcome of the © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 S. J. Large, Dissipation and Control in Microscopic Nonequilibrium Systems, Springer Theses, https://doi.org/10.1007/978-3-030-85825-4_2 15
- 35. 16 2 Theoretical Background random event will be X = x. While a full treatise on the theory of probability is beyond the scope of this introduction,1 a central axiom of probability theory is that the probability distribution is normalized, x px = 1 , (2.1) where the summation is taken over all possible outcomes of the random event. As defined in (2.1) the probability distribution is applicable to discrete random variables, and often referred to as the probability mass function. However, when the outcome of a random variable can be any value in a continuous domain, we instead use the probability density function p(x), which satisfies the alternative normalization condition x p(x)dx = 1 . (2.2) Intuitively, the need for a generalized definition of the probability in continuous spaces derives from the fact that there is no way of assigning the probability of a single outcome X = x when x has a continuous domain, as each individual outcome has zero probability.2 Instead, we must refer to the probability that the outcome x lies within a range of possible values p(a x b) = b a p(x)dx [3]. For the majority of this thesis, we will work with the probability density function, as most model systems we look at are continuous, but will distinguish between the two distributions with the notation used above: in probability mass functions the outcome appears as a subscript px, while for probability density functions the outcome appears as an argument p(x). While a mathematically complete description of a random variable requires specification of the probability distribution, we can describe particular properties of the random variable through its moments. The nth moment of the distribution is Xn = x xn p(x)dx , (2.3) where here the angle brackets · · · indicate an average over the distribution of x. Here, the distribution being averaged over is unambiguous, however in cases where it is not, the averaging distribution will be made explicitly clear from the notation, often with a subscript or superscript. From (2.3), the zeroth moment (n = 0) of any distribution simply gives the normalization condition (2.2), and the first moment (n = 1) is the mean of the 1 For the interested reader, however, [4] provides a succinct introduction to the mathematical subject. 2 Or, in the language of probability theory, each individual event represents a set of zero measure.
- 36. 2.1 Mathematical Preliminaries 17 distribution, which is also often referred to as the expectation value of X. Higher- order moments encode more detailed information about the shape of the distribution, with each successively higher-order n providing information about events that are farther from the mean [2]. However, for n 1 there are alternative, more convenient measures used to quantify the characteristics of the distribution. For instance, the centered moments subtract the mean (first moment) to give moments relative to the mean value, δXn = x (x − x)n p(x)dx , (2.4) where here, and in the remainder of the thesis, δX ≡ X−X indicates the difference between a random variable and its mean value. Trivially δX = 0, and thus the first nonzero centered moment is for n = 2. δX2 is the variance, which is often written as σ2 x as it is equal to the square of the standard deviation σx [5]. More convenient still are the cumulants κn of a distribution (see (2.7) below). The first two cumulants are simply the mean (κ1) and variance (κ2), and, while higher-order cumulants can be expressed as combinations of moments (2.3), no simple closed-form expression is known to do so [6]. An alternative route to understanding properties of a probability distribution comes through its Fourier transform, which is known as the characteristic function (or moment-generating function) φ(s) ≡ e−isx = x e−isx p(x)dx . (2.5) The Fourier transform is used extensively in physics, particular when considering periodic behavior, such as waves [7]. In the context of probability theory, the characteristic function provides an often useful tool in solving problems, as well as a route to more simply calculate the moments (2.3) and cumulants of a distribution. In particular, the nth moment of a distribution can be calculated from the characteristic function as Xn = i−n ∂nφ(s) ∂sn s=0 . (2.6) Furthermore, the characteristic function can be used to define, mathematically, the cumulants through the cumulant-generating function g(s) ≡ − ln φ(s). The cumulants κn can be determined from the cumulant-generating function, similar to (2.6), as κn ≡ i−n ∂ng(s) ∂sn s=0 . (2.7)
- 37. 18 2 Theoretical Background The cumulant-generating function has become central in the field of large deviation theory, where a scaled cumulant-generating function is often used to determine the large deviation rate function [8]. However, the cumulants also have fundamental importance in standard treatments of probability theory. For instance, all cumulants above κ2 (i.e., third order and higher) are zero for a Gaussian distribution, while no analogous truncations exist for the moments [6]. For situations described by multiple random variables, such as the outcome of several coin flips, the probability distribution of outcomes is multivariate. Here, there are several other important concepts to understand. In particular, for two random variables X and Y, the probability of a particular outcome is now given by the joint distribution p(x, y), which is the probability that both X = x and Y = y. The joint distribution obeys the normalization condition 1 = x,y p(x, y)dxdy. The univariate distribution p(x)—also called the marginal distribution in this context— can be obtained by integrating—or marginalizing—over the y-coordinate as p(x) = y p(x, y)dy . (2.8) Alternatively, given we know the outcome of Y, the probability of X conditioned on the outcome Y = y is known as the conditional distribution p(x|y). We can relate all three distributions through the law of total probability as [5] p(x, y) = p(x|y)p(y) = p(y|x)p(x) . (2.9) Finally, if the random variables X and Y are independent of one another, then the conditional and marginal distributions are equal (p(x|y) = p(x) and p(y|x) = p(y)), and therefore the joint distribution can be expressed as a product of the marginal distributions p(x, y) = p(x)p(y) . (2.10) 2.2 Nonequilibrium Dynamics To describe the state of a fluctuating system, such as the position of a diffusing particle, we can gain tremendous insight by making use of the machinery of probability theory. In particular, the state of a system at a time t is itself a random variable X(t) that depends on time. For instance, if we observe the state of the system at a series of times t0, t1, · · · , tn, so that increasing indices indicate observations that are later in time (i j implies that ti tj ), and each successive time is spaced with the increment ti+1 − ti = t, then the system trajectory is a vector of random variables X(t) = (Xn, Xn−1, · · · , X0) ≡ Xt[0,n] . Here, Xi is the state X at time ti, and the subscript t[0,n] is shorthand notation for the sequence of times t0, t1, · · · , tn. This process is an example of a discrete-time stochastic
- 38. 2.2 Nonequilibrium Dynamics 19 process, because the random variable is observed at discrete time increments; however, this can be turned into a continuous-time process by taking the limit of t → 0. The probability of observing a particular trajectory of the system is given by the multivariate joint distribution p(x[0,n]). Furthermore, by viewing the trajectory x[0,n] as a time series, the conditional distribution can be understood as a predictive tool, quantifying the probability distribution of future states of the system, given a previous history. For instance, p(xn|x[0,n−1]) ≡ p(xn, x[0,n−1]) p(x[0,n−1]) (2.11) defines the probability of observing the system in state xn given that the n previous observations were x[0,n−1]. An assumption that greatly simplifies the analysis of stochastic systems is the Markov assumption, which asserts that the state of the system at time tn only depends on the most recent previous state, at tn−1, and is thus conditionally independent of x[0,n−2]. More concretely, the Markov assumption states that p(xn|xn−1, x[0,n−2]) = p(xn|xn−1) , (2.12) and systems which obey this assumption are said to be Markovian. The single- timestep conditional probabilities p(xi+1|xi) are often called transition probabili- ties, as they represent the probability of observing the system in state xi+1 at time ti+1 given that it was in state xi at time ti. While there are many other common and useful assumptions that can be made on the interdependence of system states along a stochastic trajectory, such as the Martingale property that has become central to many formal treatments of stochastic processes and probability theory [9], the Markov assumption is, by and large, the most ubiquitous in the modern treatment of stochastic thermodynamics. The primary utility of the Markov assumption is that it allows the decomposition of stochastic trajectories into transition probabilities. In particular, for a system which at time t0 is distributed over x states as pinit(x0), the trajectory probability distribution over states after n time increments can be decomposed into a product of transition probabilities as p(x[0,n]) = p(xn|xn−1)p(xn−1|xn−2) · · · p(x1|x0)pinit(x0) . (2.13) For a Markovian system, we can use (2.13) to write the trajectory probability p(x2, x1, x0) = p(x2|x1)p(x1|x0)p(x0). By then marginalizing over x1 (2.8) we obtain a tremendously useful identity in the study of stochastic processes called the Chapman-Kolmogorov equation: p(x2|x0) = p(x2|x1)p(x1|x0)dx1 (2.14)
- 39. 20 2 Theoretical Background for a continuous state space, or, for a discrete state space: p(x2|x0) = x1 p(x2|x1)p(x1|x0) . (2.15) Under certain constraints on the limits of the transition probabilities as the time increment t → 0, the Chapman-Kolmogorov equation (2.14) (or (2.15)) can be written as a differential equation.3 Specifically, for the transition rate R(x|x, t) ≡ lim t→0 p(x,t+ t|x,t) t , one can show that the differential Chapman-Kolmogorov equation is [2] ∂t p(x, t|x , t ) = − i ∂xi Ai(x, t)p(x, t|x , t ) + 1 2 i,j ∂2 xi ,xj Bij (x, t)p(x, t|x , t ) + R(x|x , t)p(x , t|x , t ) − R(x|x , t)p(x , t|x , t ) dx . (2.16) Here the state x is, in general, a vector, ∂t ≡ ∂/∂t is the partial derivative with respect to time, and ∂xi ≡ ∂/∂xi is the partial derivative with respect to the ith component of x. This form indicates three different types of motion that can arise in stochastic dynamics. In particular, Ai(x, t) represents a drift term, quantifying deterministic motion of the system, while Bji(x, t) is a positive-semidefinite diffusion tensor, quantifying the impact of diffusive motion. The final term on the RHS of (2.16) represents the impact of discrete jumps on the overall stochastic dynamics of the system. In various limits, the differential Chapman-Komogorov equation (2.16) reduces to common equations of motion for stochastic systems. 2.2.1 Master Equation For systems in which the drift and diffusion coefficients are zero (Ai(x, t) = Bij (x, t) = 0, for all i, j), the resulting differential equation is the master equation, which describes the rate of change of the probability in a system where the source of probability changes are solely due to jumps. While one can use the master equation to describe the dynamics on continuous state spaces [2], for the purposes of this thesis, we will use the master equation when discussing discrete-state systems. The discrete master equation is often used to model the dynamics of chemical reaction networks [10], or as a simple representation of a more complex problem, for instance following coarse-graining [11]. 3 See Gardiner [2] for a more in-depth discussion of these constraints, but simply put, the constraints relate to the continuity of the underlying stochastic process in the continuous-time limit.
- 40. 2.2 Nonequilibrium Dynamics 21 Thus, the discrete master equation quantifies the rate of change of the probability distribution in a stochastic system due to jumps between discrete states. Mathemat- ically, the discrete master equation is dt pi = j Rij pj , (2.17) where pi is the probability of the system being in state i at time t, dt ≡ d/dt is shorthand for the time derivative, and Rij is the (i, j) element of the transition rate matrix, quantifying the rate of transitions from state j → i per unit time. Rij is shorthand for R(xi|xj ) (here assumed to be time-independent), and the indices indicate that the state space is discrete. In order to conserve probability, the diagonal elements of the transition rate matrix are constrained by Rii = − j=i Rij . (2.18) It can be shown, in several different ways, that in the long-time limit—and for time-independent transition rates—solutions to the master equation converge to a stationary distribution [6]. Furthermore, this distribution is unique if the transition rate matrix is irreducible, which means that each state in the system is accessible— via intermediate states—by any other state. Concretely, it is possible for a system initially in state xi at t = 0 to be found in any other state of the system xj at a later time. In what follows we will assume that there is a unique stationary solution, and will refer to it in general as the steady-state distribution. Once the dynamics have a thermodynamic interpretation, under certain circumstances the steady-state distribution is equivalent to the Boltzmann equilibrium distribution (1.1). At steady state, the probability distribution is unchanging: dt pi = 0. Therefore the stationary distribution pss i solves the equation j Rij pss j = 0 . (2.19) Thus the stationary distribution is the (unique) right eigenvector of the transition rate matrix corresponding to the zero eigenvalue [6]. Thus for a time-independent rate matrix, one only needs to solve for the eigenvectors of the transition rate matrix to identify the steady-state distribution. If the master equation (2.17) represents a thermodynamic system, the entries in the transition rate matrix are further restricted to satisfy the local detailed balance relationship Rij Rji = exp(−β ωij ) , (2.20)
- 41. 22 2 Theoretical Background where ωij = ωi − ωj is the difference in the thermodynamic potential of states xi and xj . For instance, if the system is in contact with a thermal reservoir, the thermodynamic potential of each state xi is its energy ωi = i, and thus asymmetries in the transition rates are completely determined by the relative energies of the states. The local detailed balance constraint is necessary in such systems so that they remain consistent with the known predictions of thermodynamics. Specifically, for a thermodynamic system the equilibrium distribution is known from the study of statistical mechanics, and the rate definitions in (2.20) ensure that the stationary distribution and the equilibrium distribution are equal. To generate trajectories of a system obeying the master equation (2.17), we need to know the transition kernel p(x, t+ t|x, t), which can be obtained by expanding the time-dependent probability distribution from (2.17) to linear order in t and using the master equation (2.17) to evaluate the derivatives [2]: p(xi, t + t|xj , t) = δij 1 − k Rkj t + Rij t + O( t2 ) , (2.21) where δij = p(xi, t|xj , t). The first RHS term is the probability that during time t the system remains at xj , while the second RHS term is the probability that it transitioned to state xi from xj . Thus, for a system initially in state xj , after a small time t has passed, we can calculate its state by drawing a new state (xi) at random from p(xi, t+ t|xj , t). Numerically, this can be achieved by implementing a kinetic Monte-Carlo algorithm.4 The details of our implementation are given in App. A. As an example, consider the underlying set of states xi ∈ Z, where Z is the set of integers, subject to a harmonic potential centered at x0 = 0, so that the system evolves in the potential i = ktrap 2 x2 i where ktrap is the trap stiffness. Transition rates of the general form Rij = e− 1 2 β ij (2.22) for constant kinetic bare rate and ij ≡ i − j satisfy local detailed balance (2.20) and thus represent the stochastic dynamics of a physical system in contact with a thermal reservoir. Figure 2.1 shows a sample trajectory (left) of the system in the harmonic potential for the first 50 discrete jumps, and the steady-state distribution pss x (2.19) (right) given by the eigenvector corresponding to the zero-eigenvalue of the transition rate matrix. The trajectory consists of lag times, during which the system remains in a fixed state, interspersed with instantaneous discrete jumps. 4 This is analogous, in this case, to the Gillespie algorithm used to propagate the chemical master equation.
- 42. 2.2 Nonequilibrium Dynamics 23 0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 Time t −5.0 −2.5 0.0 2.5 5.0 State x 0.0 0.2 px Fig. 2.1 Discrete-state dynamics generated by the master equation in a harmonic potential. (left) Sample of the first 50 steps of a discrete-state trajectory governed by the discrete master equation (2.17) and subject to a harmonic confining potential. Each circle represents the start or end of a discrete jump in the state. We set = 1 in the transition rates (2.22). (right) Empirical distribution (circles) obtained from a long trajectory of the discrete dynamics compared to the corresponding Boltzmann distribution (1.1) (bars) for a system in thermodynamic equilibrium. In both subplots we take β = 1 2.2.2 Fokker-Planck Equation For systems in which there are no discrete jumps (R(x|x, t) = 0 everywhere), but nonzero drift and diffusion coefficients, the differential Chapman-Kolmogorov equation (2.16) reduces to the Fokker-Plank equation which, in its general form, is ∂t p(x, t) = − i ∂xi Ai(x, t)p(x, t|x , t ) + 1 2 i,j ∂2 xi ,xj Bij (x, t)p(x, t|x , t ) . (2.23) For the duration of this thesis, we will be concerned with time- and space- independent diffusion, so that Bij (x, t) = 2D (so that the factor of 1/2 in front of the diffusive term is canceled), and further restrict our attention to the 1D Fokker- Planck equation corresponding to an overdamped system, which is also known as the Smoluchowski equation.5 Here, we consider the Smoluchowski equation for a diffusing particle in a time-dependent potential E(x, t), which means that the drift coefficient in (2.23) is replaced by (βD times) the force f (x, t) ≡ −∂xE(x, t): A(x, t) = −βD∂xE(x, t). 5 To get an underdamped equation of motion, we need to also include the time evolution of the distribution of velocities. The Kramers equation does just this for a 1D system, but we won’t discuss it in detail here. For the interested reader, however, an excellent overview is given in Hannes Risken’s book [12].
- 43. 24 2 Theoretical Background Integrating the Fokker-Planck equation over an (arbitrary) initial distribution p(x, t = 0) converts the conditional equation into an equation for the marginal probability distribution p(x, t). Thus the 1D overdamped Fokker-Planck equation is ∂t p(x, t) = βD ∂x [f (x, t)p(x, t)] + D ∂2 x p(x, t) . (2.24) The stationary solution—if it exists—of the Fokker-Planck equation can be obtained by setting ∂t p(x, t) = 0 and solving the RHS of (2.24). In general, the full time-dependent distribution p(x, t) is rarely analytically tractable, and must be obtained through numerical methods. However, when the energy landscape is harmonic (E(x) = ktrap 2 x2), the forces are linear, f (x) = −∂xE(x) = −ktrapx, and the Smoluchowski equation becomes ∂t p(x, t) = −βDktrap [xp(x, t)] + D ∂2 x p(x, t) . (2.25) Here—and, in fact, for any Fokker-Planck equation subject to a potential-derived drift coefficient—the (equilibrium) steady-state solution can be found from the Boltzmann distribution peq(x) = ktrap 2π e− βktrap 2 x2 , (2.26) which here is Gaussian with mean xeq = 0 and variance δx2eq = (βktrap)−1. Here, the angle brackets · · · eq indicate an average over the equilibrium distribu- tion (2.26). In this special case we can also find an analytical solution for the full time- dependent distribution, for any initial conditions. For an initial distribution that is localized at x = x so that p(x, t = 0) = δ(x − x), the time-dependent solution is Gaussian at all times t 0, p(x, t|x , t = 0) = βktrap 2π(1 − e−2βDktrapt ) exp − βktrap 2 (x − xe−βDktrapt )2 1 − e−2βDktrapt , (2.27) with mean x(t) = xe−βDktrapt and variance δx2 = (1 − e−2βDktrapt )/(βktrap). Asymptotically as t → ∞, the mean and variance approach the steady-state values. We will make extensive use of this general form in Chap. 7. Figure 2.2 shows the time-dependent probability distribution obtained by solving the Smoluchowski equation (2.24) in a harmonic potential with βktrap = D = 1, given an initial distribution p(x, t = 0) = δ(x−2). The distribution relaxes towards the equilibrium distribution—indicated by a dashed red line—obtained from the Boltzmann equation (2.26). The right panel shows the time-dependent average and variance of the distribution, which converge towards their equilibrium values of xeq = 0 and δx2eq = 1.
- 44. 2.2 Nonequilibrium Dynamics 25 −3 −2 −1 0 1 2 3 Position x 0.0 0.2 0.4 0.6 p(x, t) τ 1/4 1/2 1 2 4 πeq 0 5 10 Time t −2 −1 0 1 Cumulant x(t) δx(t)2 Fig. 2.2 Time-dependent solution to the Smoluchowski equation for a diffusing particle in a harmonic potential. (left) Time-dependent solution to the Smoluchowski equation for a Brownian particle diffusing in a harmonic potential (2.27), given an initial distribution p(x, 0) = δ(x − 2). Early times are indicated by light blue, while later times are indicated by progressively darker shades of blue. The red dashed line is the Boltzmann equilibrium distribution towards which the system is relaxing. (right) The values of the first two cumulants—mean (yellow) and variance (green)—of the distribution p(x, t) as a function of time, showing that they asymptote to their equilibrium values of x = 0 and δx2 = 1 as t → ∞ 2.2.3 Langevin Equation Similar to the use of the transition kernel of the master equation to generate trajectories that represent the time evolution of single instances of the stochastic process, the transition kernel of the Fokker-Planck equation gives rise to a stochastic equation of motion for individual trajectories known as the Langevin equation. For instance, one can show that for the Smoluchowski equation of a diffusing particle in a potential E(x, t), the transition kernel is, to leading order in t [2], p(x, t + t|x , t) = D 2π t exp − (x − x + βDf (x, t) t)2 2D t , (2.28) which is simply a Gaussian distribution with variance D t and mean x − βDf (x, t) t, where f (x, t) = −∂xE(x, t) is the force experienced by the particle at position x and time t. Thus, the resulting stochastic dynamics are those of a system moving with a systematic drift of velocity βDf (x, t) with zero-mean Gaussian fluctuations with variance D t. Therefore, the update rule for a trajectory generated by this equation of motion is x(t + t) = x(t) + βDf (x, t) t + √ 2DN(0, t) , (2.29)
- 45. 26 2 Theoretical Background where N(0, t) is a normal distribution with 0 mean and variance t. The Gaussian-fluctuation term is known as a Wiener increment, and often indicated by W. In the continuous-time limit ( t → 0), (2.29) becomes dx = βDf (x, t)dt + √ 2D dW(t) , (2.30) where dW(t) is the differential element of a continuous-time stochastic process W(t), known as a Wiener process (also called a white-noise process as it has a flat power spectral density over all frequencies) [2]. The Wiener process is defined by a couple of important properties, namely it has zero mean (W(t) = 0), unit variance (W(t)2 = 1), and its values at different times are independent, W(t)W(t) = δ(t − t). Alternatively, it is common to rewrite (2.30) as a differential equation, in terms of derivatives as opposed to differentials: ẋ = βDf (x, t) + √ 2D ξ(t) , (2.31) where ẋ ≡ dx/dt, ξ(t) ≡ dW(t)/dt, and δξ(t)δξ(t) = δ(t − t). Throughout this thesis, we will primarily use the derivative form (2.31), except when using the differential form is more clear. Equation (2.30) (or (2.31)) is an example of a stochastic differential equation (SDE) because it contains a stochastic differential term (dW(t) or ξ(t)). As we will see, this makes the subsequent analysis significantly different from standard ordinary differential equations. Here, the sample paths generated by the Langevin equation (2.30) are continuous everywhere, but differentiable nowhere, due to the properties of the Wiener process [2].6 The trajectories generated by the SDE in (2.30) represent the same underlying physical process as the Smoluchowski equation (2.24). In both cases we are implicitly assuming that the system is overdamped. This means that there are no inertial terms in the equations of motion, which is an oft-used approximation in low-Reynolds number environments. The Reynolds number ‘Re’ is a dimensionless quantity used in fluid mechanics to classify the relative importance of inertial and viscous forces [13]. For small values of the Reynolds number, viscous forces dominate, and inertial effects can largely be ignored [14]. Approximations on a single bacterium predict that its Reynolds number is ∼10−5 which implies that its dynamics are governed largely 6 In fact, this is an argument made by Jacobs in [3] for the use of differential notation for SDEs in preference to writing them in terms of derivatives. The non-differentiability of sample paths implies that the derivatives themselves don’t exist, and thus it is more mathematically consistent to write the Langevin equation (or any SDE for that matter) in differential notation, such as in (2.30).
- 46. 2.3 Stochastic Thermodynamics 27 by viscous forces.7 Thus, the overdamped approximation plays a prominent role in the microscopic physics of biological systems. The overdamped equation of motion (2.30) is a particular limit of a more general Langevin equation, where the fluctuations, which are assumed to impart forces on the microscopic particle, appear in the equation of motion for the acceleration. Specifically, the underdamped Langevin equation, which could be derived, in principle, from the transition kernel for a Fokker-Planck equation of an underdamped system–such as the Kramers equation–is mẍ = −f (x, t) − 1 βD ẋ + 2 β2D ξ(t) . (2.32) Here, the first term on the RHS represents the force applied to the system (either by a potential, or some external source), the second term on the RHS represents the frictional damping of velocities, and the final term on the RHS represents the stochastic forces on the particle from the environment. To recover the overdamped equation of motion (2.31) from the underdamped equation of motion, we simply set ẍ = 0 and solve for ẋ. 2.3 Stochastic Thermodynamics The preceding sections describe methods to generate the stochastic dynamics of fluctuating systems, based solely on the mathematical concepts of probability theory; however, these trajectories still lack a thermodynamic interpretation. This interpretation is provided by the growing field of stochastic thermodynamics. Stochastic thermodynamics is the study of thermodynamics at the microscopic scale, where systems are small and fluctuations cannot be ignored. At its heart, stochastic thermodynamics seeks to generalize the classical laws of thermody- namics to fluctuating trajectories. As considered here, the systems of interest are embedded within an aqueous environment, and can be pushed out of equilibrium in a number of different ways. For instance, systems can be driven out of equilibrium by altering their potential energy in a time-dependent manner through a control parameter λ, which we discuss in more detail in Sect. 2.6. Here, we can study the response of the system to a time-dependent control parameter λ(t), the spontaneous relaxation towards equilibrium after an instantaneous change in the control param- 7 Specifically, the Reynolds number is defined mathematically as Re ≡ Ua/ν where U is the speed of the object, a is a characteristic linear dimension, and ν is the kinematic viscosity of the fluid (for water, ν ≈ 10−6 m2/s). For a typical bacterium (such as, for instance, E. coli) the characteristic linear dimension is a ≈ 10−6 m and they travel at velocities of U ≈ 10−5 m/s and thus the Reynolds number is Re ≈ 10−5 [14].
- 47. 28 2 Theoretical Background eter value, or the properties of nonequilibrium steady states when the driving is time-independent. Alternatively, systems such as chemical reaction networks can be kept out of equilibrium by chemostats that maintain the out-of-equilibrium concentrations of chemical reactants. Here, the system is still driven out of equilibrium, even though there is no active manipulation of the system through, for instance, a control parameter. Formally, the separation of system from surroundings in stochastic thermo- dynamics involves identifying fast- and slow-relaxing degrees of freedom. The slow-relaxing degrees of freedom constitute the system, while the fast-relaxing degrees of freedom comprise the bath (or environment). This is one reason why the study of microscopic biological systems has proved to be difficult: there is often no obvious separation of timescales allowing one to separate, unambiguously, the system from the environment. At the microscopic scale, the mathematical forms of the laws of thermodynamics are largely preserved, but a complete understanding of how exactly to attribute increments of work, heat, and entropy to single trajectories is a nontrivial process. Throughout this thesis we will adopt the convention that positive work and heat correspond to energy flows into the system. Naturally, this gives rise to the differential form of the first law dE =d̄W +d̄Q , (2.33) where dE represents a change in the internal energy of the system, and the inexact differentialsd̄ indicate that the work and heat are path-dependent. Given an overdamped Langevin equation describing a fluctuating particle, Seki- moto was the first to suggest a thermodynamic interpretation [15]. In particular, how can we quantify the heat and work associated with a single stochastic trajectory? Conceptually, work is done on a system when an external agent inputs energy, and heat represents the exchange of energy between the system and its environment. Thus, in the absence of any external input, all energy changes are heat. For an overdamped system fluctuating in a fixed potential energy landscape V (x, λ), changes in energy are determined by the changes in potential dE = dV , which, by using the chain rule, can be written in terms of a differential in the particle position dV = (∂xV )dx = −d̄Q, quantifying the exchange of heat with the reservoir. Thus, the heat along a stochastic trajectory is given by the integral expression [15, 16] Q[x[0,τ]] = − τ 0 f (x(t))ẋ dt , (2.34) where here the notation Q[x[0,τ]] indicates that the heat is a functional of the trajectory x during times t ∈ [0, τ]. We can use (2.34) to obtain an exact expression for the heat in terms of the stochastic dynamics by substituting (2.31) for ẋ. However, because ẋ is a stochastic differential equation, direct integration could pose a problem, as the integral of the stochastic process has not yet been defined.
- 48. 2.3 Stochastic Thermodynamics 29 We omit discussion of the different definitions of stochastic integrals, but specify here that the Sekimoto definitions of heat (2.34) (and work (2.35)) must use the Stratonovich interpretation of the stochastic integral [15]. Alternatively, work is done on the system if the control parameter λ changes in time. Thus, changes in energy due to work are dE = dV = (∂λV )dλ =d̄W, and the work done along a trajectory is W[x[0,τ]] = τ 0 (∂λV (x(t))) λ̇ dt . (2.35) Combining (2.35) with (2.34), the integrated first law is E = V (x(τ), τ) − V (x(0), 0) = Q[x[0,τ]] + W[x[0,τ]] , (2.36) on the level of an individual stochastic trajectory of the system. Equation (2.36) provides a powerful conceptual separation of the work and heat accumulation at the microscopic level: work accumulates when external influences on the system results in changes of energy, while heat accumulates when the system exchanges energy with the bath. The first law, which effectively is a statement of energy conservation, links these two energy flows. Identifying a valid form of the second law of thermodynamics requires a consistent definition of entropy along a single trajectory. However, the second law refers to the entropy of the universe, so we must identify the changes in entropy of both the system and the environment. The entropy change in the environment during a trajectory can be identified with the heat Sres[x0,τ ] = −βQ[x[0,τ]], (2.37) while the instantaneous entropy of the system at time t is S(t) = − ln p(x(t), t) , (2.38) where, here and throughout the thesis, we use natural units (kB = 1) for the entropy. Here, the probability distribution p(x(t), t) is obtained by first solving, for instance, the corresponding Fokker-Planck equation for the system dynamics. The second law thus follows by measuring the total change in entropy of the universe over a trajectory, Stot[x[0,τ]] = Sres[x[0,τ]] + S(τ) − S(0) , (2.39) where, the final two terms on the RHS capture the change in the entropy of the system state over the time interval. Unlike the usual form of the second law, the entropy change along a single trajectory can be negative [17]. For consistency with the classical laws of thermodynamics, we require only that Stot[x0,τ ] ≥ 0 where · · · indicates an average over an ensemble of trajectories under the
- 49. 30 2 Theoretical Background same driving. In (2.39), there is no requirement that the system starts and/or ends in equilibrium, however to calculate the entropy from (2.38) we must know the probability distribution p(x(t), t) over states. The stochastic thermodynamic study of trajectory-level entropy led to a far more general form of the second law through an entirely new class of mathematical identities known collectively as fluctuation theorems. 2.4 Fluctuation Theorems Historically, the identification of symmetries under time reversal of dynamical systems began with a seminal paper by Gallavotti and Cohen in which they related the change in Kolmogorov-Sinai entropy8 to the ratio of trajectory probabilities under time-forward and time-reversed dynamics [19]. Subsequent study on the probabilities of second-law-violating events in steady-state systems [20, 21] as well as earlier research on the work integrated over driven trajectories by Bochkov and Kuzovlev [22, 23] led to the well-known Jarzynski equality [24], e−βW = e−β F , (2.40) which relates the exponentiated trajectory work βW ≡ βW[x[0τ]|] averaged over an ensemble of nonequilibrium trajectories (LHS), to the exponentiated change in equilibrium free energy β F ≡ βF(τ) − βF(0) (RHS). In the Jarzynski equality, the system is assumed to be driven through the manipulation of an external control parameter λ—or, in Jarzynski’s words, a work parameter—and the average · · · is taken over the ensemble of nonequilibrium responses of the system to the particular driving protocol. Importantly, the Jarzynski equality holds regardless of how far the system is driven out of thermodynamic equilibrium. By making an appeal to Jensen’s inequality—an inequality pertaining to averages over convex functions that is used ubiquitously in information theory [18]—the Jarzynski equality can be reduced to the inequality, W ≥ F (2.41) or, alternatively Wex ≥ 0 (2.42) where here we define the excess work Wex ≡ W − F as the difference between the work and the free energy difference. The lower bounds implied by the Jarzynski 8 The Kolmogorov-Sinai entropy is a definition of entropy, used often in dynamical systems theory, that is calculated from the Lyapunov exponents in chaotic systems [18].
- 50. 2.4 Fluctuation Theorems 31 equality (2.41) (and (2.42)) can be seen as alternative statements of the second law of thermodynamics, when the excess work is the only source of entropy production [17]. The Jarzynski equality is an example of what is now called an integrated fluctuation theorem (IFT), to contrast with a detailed fluctuation theorem (DFT). While an IFT involves ensemble averages, a DFT relates the probabilities of individual stochastic trajectories. The first DFT found was the Crooks fluctuation theorem [25], p[x|] p̃[x̃| ˜ ] = e−βWex[x|] , (2.43) which relates the ratio of trajectory probabilities p[x|] for a system driven through a time-dependent control protocol , to the probability of its time-reversed trajectory p̃[x̃| ˜ ] under the time-reversed protocol ˜ . Here the tilde indicates the time-reversal operation, with x̃(t) = x(τ − t) and λ̃(t) ≡ λ(τ − t). The Crooks theorem can be re-written in terms of the work distribution along forward and reverse processes as p(Wex)/p̃(−Wex) = exp(βWex) and, in fact, can be used to derive the Jarzynski equality. Thus, the Crooks theorem represents the DFT that, upon integration, reproduces the Jarzynski IFT. In fact, the existence of a DFT directly implies a corresponding IFT, and vice-versa [26]. Among the fluctuation theorems, the entropy production fluctuation theorem is central, reducing to other fluctuation theorems, such as the Crooks IFT, in particular settings. Proven in its current form by Seifert in 2005 [27], the entropy production fluctuation theorem is p( Stot) p̃(− Stot) = e Stot , (2.44) relating the exponentiated change in entropy to the ratio of the probability of observing a total entropy change Stot along a forward trajectory to the probability of observing a corresponding decrease in entropy under the time-reversed process. For a system driven by an external control parameter, the excess work (nondimen- sionalized by β) and entropy production are equal, thus (2.44) reduces to the Crooks fluctuation theorem (2.43). In fact many of the well-known fluctuation theorems can be recovered from (2.44) by splitting the change in entropy into adiabatic and nonadiabatic contributions, Stot = Sa + Sna, capturing the dissipation due to external time-dependent driving Sna (such as by a time-dependent control parameter) or due to nonequilibrium boundary conditions Sa (such as out-of- equilibrium concentrations of chemical reactants), respectively [26, 28, 29]. Furthermore, by using Jensen’s inequality, (2.44) reduces to the familiar form of the second law of thermodynamics, Stot ≥ 0. Thus, (2.44) represents a true generalization of the second law of thermodynamics, which only became apparent through the study of microscopic fluctuating systems. The exponential suppression of ‘second-law violating’ trajectories makes the probability of observing such events
- 51. 32 2 Theoretical Background in macroscopic systems vanishingly small [17]. Broadly speaking, the fluctuation theorems have provided a far deeper understanding of the microscopic physics of fluctuating systems, placing surprisingly general constraints on the fluctuations and dissipation in thermodynamic systems, even far from equilibrium. 2.5 Entropy and Information Theory From its inception in the 1940s by Claude Shannon at Bell Labs, the field of infor- mation theory provided a comprehensive and powerful set of tools to describe the mathematics of information. More recently, the field of nonequilibrium statistical mechanics and stochastic systems have found several concepts from information theory essential for understanding the physics of nonequilibrium systems. Here, we briefly review some of the relevant results from information theory. The idea of entropy is central to information theory where, unlike many classical treatments in thermodynamics, it possesses a conceptually simple explanation. The entropy is a measure of the amount of information required, on average, to describe the state of a system. Low-entropy states (such as a long sequence of heads in successive coin flips) are highly ordered, and thus require a relatively small amount of information to specify, whereas high-entropy states are more disordered, requiring more information to completely specify.9 Mathematically, the entropy of a random variable X—or Shannon entropy as it is typically referred to—is S(X) ≡ − x px ln px . (2.45) When multiple random variables (X and Y) are involved, one can also define the joint and conditional entropy respectively as S(X, Y) = − x,y pxy ln pxy (2.46a) S(X|Y) = − y py x px|y ln px|y . (2.46b) The conditional entropy (2.46b) is useful in decomposing the joint entropy (2.46a) via the chain rule for entropies [18]: S(X, Y) = S(X) + S(Y|X) (2.47a) = S(Y) + S(X|Y) . (2.47b) 9 This explanation of the entropy is clarified when the entropy is described in bits, by using base-2 logarithms. Here, the entropy is literally the average number of bits—yes or no questions—required to completely specify the state of a system.
- 52. 2.5 Entropy and Information Theory 33 Central to the study of information theory is the measure of mutual information, I(X; Y) ≡ x,y pxy ln pxy pxpy , (2.48) which quantifies the information that is shared between two random variables. The mutual information (2.48) is symmetric (I(X; Y) = I(Y; X)), nonnegative, and can be used to relate the marginal entropy of a single random variable in a multivariable system to its conditional entropy: I(X; Y) = S(X) − S(X|Y) . (2.49) This provides a conceptual interpretation of I(X; Y) as the reduction in uncertainty (entropy) about a random variable X by observation of another random variable Y. If X and Y are independent, then pxy = pxpy and I(X; Y) = 0 (2.48). Conversely, if X and Y are correlated, then the non-negativity of mutual information requires that S(X|Y) S(X). Information theory also provides a number of so-called divergence measures that quantify the differences between probability distributions. Such divergences have become central to the study of nonequilibrium systems, in particular for lower bounding statistical averages. The relative entropy (or Kullback-Leibler divergence) is a common measure used in nonequilibrium statistical physics, and is defined for probability distributions px and qx as D (px||qx) ≡ x px ln px qx (2.50) The relative entropy is nonnegative for any two probability distributions, D(px||qx) ≥ 0—a property which is commonly used to prove the non-negativity of physical quantities in nonequilibrium thermodynamics—but is asymmetric in its arguments (D(px||qx) = D(qx||px) in general) [18]. However, one can symmetrize the relative entropy in several different ways, such as taking the relative entropy of each distribution to their arithmetic mean mx ≡ 1 2 (px + qx) to get the Jensen- Shannon divergence [30] JSD (px, qx) ≡ 1 2 D (px||mx) + 1 2 D (qx||mx) . (2.51) The divergence measures originally derived in the context of information theory arise, in many forms, within the study of nonequilibrium systems. For instance, we will see in Chap. 7 how the relative entropy is related to excess work in discretely driven systems. In fact, the physical meanings of various divergence measures in the context of trajectory ensembles in stochastic thermodynamics have been explored in detail in [30], relating the relative entropy (2.50) to dissipation, Jensen-Shannon
- 53. 34 2 Theoretical Background divergence (2.51) to time asymmetry, and providing physical interpretations of various other divergence measures from information theory in the context of thermodynamic trajectories. When a system is controlled by a set of external parameters λ, the equilibrium distribution πx(λ) is parameterized by the control parameters λ (here, assumed to take on any of a continuous range of values). Thus, we can define a quantitative measure of the difference between two equilibrium distributions by taking the relative entropy D[πx(λ)||πx(λ)]. If the two distributions are close to one another, so that λ ≈ λ + λ, then the Taylor expansion of the relative entropy is, to leading order in λ, D[πx(λ)||πx(λ )] ≈ 1 2 λi λj Iij (λ) . (2.52) Here, dIij (λ) ≡ ∂λi ln πx(λ) ∂λj ln πx(λ)λ (2.53) is the (i, j)th component of the Fisher information matrix, and λi is the ith compo- nent of λ [18]. Here, and throughout the remainder of the thesis we have employed an Einstein summation notation, where repeated indices are implicitly summed over, for instance, λi λj Iij (λ) ≡ i,j λi λj Iij (λ). The Fisher information plays an important role in quantifying the dissipation in driven, nonequilibrium processes, and will be encountered periodically throughout the remainder of this thesis. 2.6 Control in Microscopic Nonequilibrium Systems Broadly speaking, control theory is the study of physical systems and how their state can be controlled through interaction with the outside world. By identifying particular goals, it is then possible to design ways to exercise control over a system to achieve them [31]. For instance, a canonical example of a mechanical controller is the centrifugal governor—an idea which was explored in detail by James Clerk Maxwell in the 1800s [32]—where the motion of a centrifugal rotor, coupled to the engine output, regulates engine speed by controlling the input valve. Since its early applications, the scope of control theory has expanded greatly and plays a major role in the modern practice of science, particularly in the context of experimental physics, where many techniques rely fundamentally on the successful implementation of control-theory-based feedback techniques [33]. Optimal control—one subfield of the broader subject of control theory—deals with the implementation of control strategies that minimize some measure of cost in a controlled system. Through much of this thesis, we will deal with the utilization of optimal control to minimize the amount of work required to manipulate
- 54. 2.6 Control in Microscopic Nonequilibrium Systems 35 stochastic systems in a particular way.10 In particular, we will often consider stochastic systems which fluctuate in the presence of a potential energy E(x, λ) which is determined by a set of control parameters λ. The instantaneous value of λ determines the equilibrium state π(x|λ) of the system through (1.1). A control protocol ≡ λ[0,τ] is a time-dependent schedule for the control parameter λ, driving the system from an initial state—often the equilibrium state at λ(t = 0)—to a final state at λ(t = τ), where the protocol duration is denoted by τ. Following from the Sekimoto definition (2.35), work is done on the system (or extracted from the system) whenever the control parameter is changed. For a particular time-dependent control protocol that transforms λi → λf over a protocol duration τ, the mean work required is given by the functional W = − τ 0 fi(x, λ)(t)λ̇i dt , (2.54) where fi ≡ −∂λi E(x, λ) is the generalized force conjugate to control param- eter λi, the angle brackets · · · (t) indicate an average over the instantaneous (nonequilibrium) distribution of the system at time t during the protocol , and the angle brackets · · · indicate an average over the (nonequilibrium) distribution of system trajectories during the entire control protocol . Given that the initial and final control parameter values are generally fixed,11 the equilibrium free energy difference F ≡ F(λf) − F(λi) is constant for all protocols, and thus minimizing the work is the same as minimizing the excess work Wex = W − F. For the remainder of this thesis, we will focus on minimizing the excess work. For a system bound by a harmonic potential (see Sect. 2.7.1), Schmiedl and Seifert [36] found the exact optimal control protocols that minimize the average work when the control parameter is either the position of the harmonic potential, E(x, λ) = ktrap 2 (x − λ)2, or the trap strength, E(x, λ) = λ 2 x2 [36]. Further, in 2011, Aurell, et al. expanded the set of exactly solvable minimum-work protocols to a more general class of problems by finding a clever mapping of the stochastic differential equations to a set of deterministic transport equations [37, 38]. While this work provided a new means of analyzing such problems, aside from some notable recent results regarding a finite-time Landauer limit [34, 35], the study of analytically tractable solutions in optimal control of stochastic systems has seen little progress since. Primarily, this is due to the intrinsic difficulty in solving the general minimization problem in (2.54). 10 Actually, this represents a further subfield known as stochastic optimal control, as it pertains to optimal control strategies in stochastic systems. However, in Chap. 6 we will refer to the control of systems through stochastic driving protocols as ‘stochastic control’, and thus, to avoid confusion, we will refer to stochastic optimal control in this context simply as optimal control. 11 Recent work on optimal bit erasure [34, 35] enforces a constraint on the final probability distribution rather than requiring that the control parameter reaches a particular final value.
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- 56. O Jesus, I am heartily sorry for having hitherto loved Thee so little! Why did I not love Thee sooner? Would that I could redeem that time in which I did not love Thee! Where will I find hearts that are going to help me to make up for the glory that I should have given Thee up to this time? Would that I loved Thee with that tender love with which Mary loved Thee. The angels and saints loved Thee, my Jesus; let me also join them in their love. Offering. I offer Thee my whole being; let it be unreservedly Thine. Thou hast taken possession of my heart, Thou art its Master; my whole life shall be used for Thy greater honor and glory. Wherever my influence may reach, there will I carry the name of Jesus. Wherever the word of God may be announced, I will glory in Thy name. Thou, O Lord, art so full of love, and still Thou art so little loved! O would that I could enkindle all hearts with the fire of Thy love! Thou, O Lord, art the Master of all hearts; we belong to Thee because Thou hast created us, and when we rebelled from Thee and became slaves of Satan, Thou didst redeem us. What greater right have we than to belong to Thee, to be subject to Thee, and to announce Thy holy name to all, so that by our mutual example we may be encouraged to live lives of sanctity? O that all of us would have the love of the Infant Jesus in our souls, and make earnest endeavors to [pg 064] spread Thy love among all men! When, O Lord, will I be dissolved, to live only with Thee? To Thee my sighs and desires go up day and night. When will I see Thy blessed countenance? When my time comes to die, let me die in Thy love, so that I may love Thee for all eternity. Thy will, O Jesus, be done in all things, and let me delight to walk in Thy footsteps.
- 57. Do with me what Thou wilt; I am willing to submit to joy or sorrow, life or death, fortune or misfortune. It is all indifferent to me as long as it is Thy will, only give me the grace to endure adversity with resignation. It will be a joy to me to fulfil Thy holy will in all things. Thanksgiving After Communion. Adoration of Jesus in the Blessed Sacrament, according to St. Francis de Sales. How can I, O Lord Jesus Christ, praise Thee sufficiently for having visited my soul in Thy infinite goodness, by Thy divinity and Thy sacred humanity, with Thy body and blood? I give Thee, my Lord and my God, a loving welcome, Thou Who art my Redeemer, my last end, my consolation, my sweetest rest, my all. A thousand welcomes to Thee, my dear Jesus, my good shepherd. In the abyss of my nothingness I humbly adore Thee. I adore Thy sacred flesh and blood, which Thou hast given me for my food, and the pledge of my eternal union with Thee. I adore Thy sacred head crowned with thorns; I adore Thy eyes that have shed so many tears for me; I adore Thy mouth, with which Thou didst teach me eternal truths; I adore Thy [pg 065] sacred countenance, which has been beaten for me by cruel soldiers; Thy feet, that have been pierced by the nails and fixed to the tree of the cross; Thy arms, which were outstretched for me, for a loving embrace; Thy side which the lance pierced, and from which blood and water issued forth, the witnesses of my redemption; Thy Heart, which loved me even unto the death of the cross. My dear Redeemer, I adore Thy sacred body, covered with innumerable wounds which Thou didst suffer for me. I adore Thy most holy soul, which was saddened unto death in the Garden of Olives that I might reach eternal life. My dear Jesus, with great love I embrace Thee, and I will remain faithful to Thee until my last hour. Bless these feelings of a good heart: grant that no storm of temptation may lead me again to be unfaithful as I have repeatedly been heretofore.
- 58. A Pious Salutation To The Infant Jesus. Hail, holy and adorable Infant Jesus, source of mercy! Thou art the life of the sinner, unfathomable ocean of divine sweetness, our hope, our joy. To Thee, born in the dreary stable of Bethlehem, we cry, poor children of Eve, to Thee we send forth our sighs, to Thee we send up our lamentations in this valley of tears. Turn then, O incarnate love, Thine eyes of mercy towards us, O blessed fruit of Thy Virgin Mother. As Thou didst show Thyself full of kindness in the stable of Bethlehem, so show unto us, after this miserable life, Thy divine mercy, when Thou shalt come in the glory of Thy Father. O sweet, O loving, O clement Infant Jesus! Amen. [pg 066] Pious Invocation To The Infant Jesus. (Father Elias Avillon.) Hail, Infant Jesus, Thou fountain of mercy, Thou life of the sinner dead in sin, Thou unfathomable ocean of divine consolation, our hope and our joy. Hail, to Thee, born in the desolate stable of Bethlehem, do we cry, poor children of Eve. To Thee do we sigh and bewail our misery from this valley of tears. Turn then, O incarnate love of God, Thine eyes of mercy upon us, Thou blessed fruit of the womb of the Virgin Mary. As Thou didst show poor humanity unbounded mercy from the manger, so also continue Thy kindness during our life, until after this miserable existence we arrive in the glory of heaven, where we shall meet Thee, to praise and glorify Thee for all eternity. O sweet, O pious, O merciful Infant Jesus! Amen. A Prayer To The Holy Name Of Jesus. Eternal God, Father in heaven, daily do we pray that Thy sacred name be sanctified, that Thy kingdom come and Thy will be done. It is also Thy holy will that the name of Thy beloved Son should be glorified. Thou hast given Him a name at which every knee should
- 59. bend in heaven, on earth, and under the earth. I pray Thee with all my heart that Thou give us the grace to pronounce that holy name with the greatest devotion, and to serve the Lord with a good and willing heart. Engrave this holy name on my heart, that I may never forget it. Place it as a sign on my forehead, that I may ever be faithful to it. O holy name of Jesus! my heart rejoices when I hear thee mentioned. I feel great [pg 067] consolation as often as I hear that holy name. Angels of heaven, who are gathered in ecstasy about the throne of God, praise with us on earth the sacred name of Jesus; sing its praises worthily, as we cannot sing it; invoke it for us also, that when our feeble powers fail, thy holiness and zeal will supply our deficiency. Thy holy name, O Jesus, be praised forever! Adoration of the Child Jesus in the Twelve Mysteries of His Divine Infancy. Dear Jesus, divine Infant of incomparable beauty, of infinite goodness, I adore Thee, because Thou art my Redeemer, and I love Thee. I make a sacrifice to Thee of all my intellect, and of my heart, and I thank Thee for having become a child for my sake. I adore Thee in all the mysteries of Thy holy childhood, and pray Thee to let me enter into the spirit of it. Give me the grace to honor Thy childhood all my life, and that I may imitate the virtues which Thou dost inculcate by Thy example. I adore Thee, O God of purity, at the moment when the Holy Ghost formed Thy sacred body in the bosom of the Virgin Mary, and beg of Thee the grace to be always pure. I adore Thee, O hidden God, hidden for nine months in the womb of the Blessed Virgin, and I desire to honor Thee by my perfect life. I adore Thee, O Child of grace, in Thy visitation to John the Baptist in order to sanctify him. Visit also my soul, that I may live a saintly life and remain faithful to the impulses of Thy grace.
- 60. I adore Thee, O Child Jesus, at the moment of Thy birth, and I desire, with Thy help, that no other [pg 068] fire burn in my heart than that which Thou camest on this earth to kindle. I adore Thee, O Spouse covered with wounds, in the painful mystery of the circumcision. I conjure Thee by the sacred blood which Thou didst then shed for the first time for our salvation, that Thou wilt give me the virtue of meekness, with which I am to go through life, for Thy glory and the edification of my neighbor. I adore Thee, O Lord Jesus, with the Magi, who came from the distant orient to adore Thee lying in the crib in the stable. Give me the grace to be in earnest in the work of my salvation. Great God, Author of all sanctity, I adore Thee in the Temple, where Thou didst present Thyself according to the Law of Moses, to be offered to Thy Father as His first-born Son. Give me grace to subject myself to every law, even though it would imply that I am a sinner, which I really am. I adore Thee, humble Jesus, in Thy flight into Egypt, and beg of Thee to give me the grace, by this holy mystery of Thy life, of perfect humility of heart. I adore Thee, O Child Jesus, in Thy poverty, which afflicted Thee in Thy stay in Egypt. Give me the grace also to be poor in spirit, and endure actual poverty with resignation. I adore Thee, O Child Jesus, in Thy joyful and triumphal return from Thy exile in Egypt to Nazareth; give me the grace to overcome the enemies of my salvation. I adore Thee, O obedient Child Jesus, in Thy exact compliance with the commands of Mary and Joseph; give me the grace joyfully to obey my superiors. [pg 069]
- 61. I adore Thee, O Child Jesus, in the midst of the doctors in the Temple of Jerusalem, when Thou didst enlighten their minds on the sacred prophecies of the Old Testament. Give me a Christian simplicity, that I may believe all that Thou teachest me through the holy Catholic Church. [pg 070]
- 62. February. The Holy Family. What a beautiful sight is the Holy Family dwelling on earth! God could not create anything more beautiful than this Holy Family; He reached the limit of possibilities in creation when He had accomplished this. It is indeed a great example to us, and a model which we ought to hold before our eyes during this month, in order that we may study the foundations of what a holy family ought to be; what a life that should be, where the holiest are bound together in ties of most affectionate friendship and relationship. Our Holy Father Leo XIII. has seen the great importance of the Christian family: the family established by the laws of the Church and the country, where marriages are legalized and solemnized; where continence and modesty reign; where children, under kind and parental subjection, are brought up to be good men and women, good for religion and for society. When poverty strikes such a family there is patience and forbearance shown; when sickness and death come, there is resignation to the will of God. In the Christian home there is surpassing peace, and not that crazy restlessness that looks for satisfaction in grasping at possessions, never satisfied day nor night. I do not think that this devotion has been appointed for this month by any authority, except the idea that after having adored the divine infancy during the month of January, the devotion to the Holy Family may follow as a natural sequence. Should any one desire to make this devotion during another month, it would certainly not be out of
- 63. the way, for save only in a few cases has a certain devotion been assigned to a particular time. The devotion to the Holy Family is a beautiful and instructive one; the Christian family should be built on this great model. The Holy Family consisted of Jesus, Mary, and [pg 071] Joseph; the father, Mother, and Child. All other families are made up of the same constituents. Those who are actually in a family, or who intend to choose that mode of life by which they may get to heaven, will love this devotion and find instruction and consolation in it. One of the greatest works of God in this world is the Holy Family at Bethlehem and Nazareth. He sent down upon the earth His only Son, Jesus Christ; prepared a most holy Mother for Him, Mary immaculate, and selected for Him a foster-father and a protector. He held them together in the most tender family ties of father, Mother, and Child. He kept them in that relation until St. Joseph died a blessed death, and the Lord Jesus went forth on His sacred mission of teaching and redeeming mankind. What a beautiful sight to us poor human beings! what a glory to God was that Holy Family, dwelling in the humble abode of Nazareth! We find here the model on which we shall reflect for this month of February. We will consecrate this month to praising God with the members of the Holy Family; we will study the ways by which they were so pleasing to God, and we will draw from these considerations many valuable lessons for our own conduct. Our Holy Father, Pope Leo XIII., with an ever-watchful eye to the necessities of our time, has seen the importance of a devotion to the Holy Family, and has recommended it to the faithful, and with his own authority established a sodality of the Holy Family; he has invested it with many indulgences in order to encourage the faithful in taking the Holy Family as their model. Here is, then, a practical way to teach ourselves the way of salvation by the example of others.
- 64. It is therefore a most useful practice of piety to become members of the sodality of the Holy Family; you will then place yourselves under its special protection, and choose Jesus, Mary, and Joseph for your particular advocates before God. Look upon the members of the Holy Family as the most perfect models for imitation, whose examples will teach you what to correct and what to avoid, what to do for your temporal and eternal welfare, and that of your families. [pg 072]
- 65. Considerations and Prayers for Every Day. First Day. In an apostolic brief of June 14, 1892, the Holy Father, Pope Leo XIII., demonstrates how the welfare of the family and of the State depends chiefly on education, and that it is of the utmost importance that a religious spirit be fostered in the Christian family. From the first family, God so arranged the method and order of such a life as to exhibit to the world a form of a divinely ordered association, in which all human beings might behold a most complete model of family life, and of all virtue and holiness. The devotion to the Holy Family, a holy and a powerful institution before God and man, has increased very much within a few years, and it is worth our while to think of this on the first day of our monthly devotion, and appreciate it as we ought. Prayer. O most loving Jesus, Who didst hallow by Thy surpassing virtues, and the example of Thy home life, the household Thou didst choose to live in whilst on earth, mercifully look down upon this family, whose members, humbly prostrate before Thee, implore Thy protection. Remember that we are Thine, bound and consecrated to Thee by a special devotion. Protect us in Thy mercy, deliver us from danger, help us in our necessities, and impart to us strength to persevere always in the imitation of Thy Holy Family, so that, by serving Thee and loving Thee faithfully during this mortal life, we may at length give Thee eternal praise in heaven. [pg 073] O Mary, dearest Mother, we implore thy assistance, knowing that thy divine
- 66. Son will hearken to thy petitions; and do thou, most glorious patriarch St. Joseph, help us with thy powerful patronage, and place our petitions in Mary's hands that she may offer them to Jesus Christ. Amen. Second Day. Within our own time the devotion to the Holy Family has grown under the fostering care of the Supreme Pontiff, who has authorized the establishment of associations throughout the world, by which men and women, married and unmarried, are gathered into one fold from the standpoint of the family. The Holy Father desires that all the faithful of the Catholic Church should consider this association as a consecration to the Christian life, and that they will feel that they are bound to lead a holy life because their association is established and legalized by God. For what is the Sacrament of Matrimony but the legalizing of the family before God and man? Let us consider the family as a special institution of God's providence for the preservation of the world, and the propagating in it of sound principles of learning and religion. Prayer. O most loving Jesus, etc., etc. Third Day. We are to consecrate ourselves to God under the union of a family. That is the pretence with which we come before God, to claim His kindness and mercy because we belong to a family. We are [pg 074]
- 67. not isolated creatures, looking for our selfish ends, but we are a union of individuals constituted under a certain authority, which gives us a claim to the respect of God and man; for God has said, “Where there are two or three gathered together in My name, there am I in the midst of them.”—Matt. xviii. 20. The formula of consecration of a Christian family has been given us by the Sovereign Pontiff himself; rules, regulations, and by-laws have been given to this society under the same authority of the Holy Father. The whole Christian family should be so united among themselves, that there be but one family under one head, set over all to rule them by its supreme power. We belong to this Christian family of the Church; let us then unite for the common good of all. Prayer. O most loving Jesus, etc., etc. Fourth Day. The scope of the pious association of the Holy Family is that all Christian families be consecrated to the Holy Family of Nazareth, placing it before themselves for veneration and imitation; offering up every day before its image prayers in its honor, and practising in their lives the sublime virtues which the Holy Family offered for imitation to every grade of society. The rich will find a model before them, the learned and highly educated will know exactly what to do according to the dignity of their position, the working class, especially, will find here the guidance and friendship needed in their temptations and troubles. The Holy Family is a [pg 075] poor, working, humble family, as poor as the poorest, as laborious as the most hard-working, as humble as the most lowly. Prayer.
- 68. O most loving Jesus, etc., etc. Fifth Day. The picture of the Holy Family should be in every household; it is a perpetual reminder, placed in tangible form, of our dear Lord, our blessed Lady, and our friend, St. Joseph, who were the members of the Holy Family. We often are interested in the pictures of great persons, and take delight in representations of angels and holy people. What is the secret of this inclination? Why, we can see those whom we wish to imitate, and grow to know their good and holy lives through their pictures. The Holy Father has approved one special picture which is to be the emblem of this association: Mary and Joseph, holding the youth Jesus between them by the hand. Jesus is not an infant, for this picture is to bring to mind the fact that His parents had to suffer care and anxiety in order to bring Him to this stage of boyhood, for which noble duty they were fitted by special providence and by special faithfulness. Prayer. O most loving Jesus, etc., etc. Sixth Day. “We have good hopes,” says the Holy Father, concluding his encyclical letter, “that all [pg 076] to whom the salvation of souls is committed, especially the bishops, will make themselves partners and sharers of our zeal in promoting this pious association. For those who recognize and deplore with us the change and corruption of
- 69. Christian morals, the extinction of the love of religion and piety in families, and the passion for earthly goods, enkindled beyond measure, will desire exceedingly to apply suitable remedies for so many grievous evils. Since nothing can be more salutary and efficacious for Christian families than the example of the Holy Family, let care be taken that as many families as possible, especially those of the working classes, against which insidious forces are more strongly exercised, enroll themselves in this association. Let the association be on its guard, lest it swerve from its purpose, or change its spirit—rather let the practices of piety and the prayers which have been determined on to be preserved intact. May Jesus, Mary, and Joseph, thus besought within the family circle, be graciously present! May they foster charity, regulate morals, incite all that imitate them to virtue, and alleviate and render more bearable the hardships which oppress mankind!” Prayer. O most loving Jesus, etc., etc. Seventh Day. A plenary indulgence after a sincere confession and a worthy communion, and praying for the intentions of His Holiness, may be gained by the members of the association on the following days: First, on the day of their entrance into the association, [pg 077] after they have recited the act of consecration. Second, on the day of a general meeting, when all go to communion in a body, and renew their promises. Third, on the feasts of the Nativity (Christmas), the Circumcision, Epiphany, Easter and Ascension. Fourth, on the feasts of the Blessed Virgin: Immaculate Conception, Nativity, Annunciation, Purification, Assumption. Fifth, also on the following days: feast of St. Joseph in March, Patronage of St. Joseph, Third
- 70. Sunday after Easter, Espousal of the Blessed Virgin on the twenty- third of January. Sixth, on the feast of the Holy Family, and Seventh, at the hour of death. Prayer. O most loving Jesus, etc., etc. Eighth Day. Partial indulgences may be gained when visiting a church where the association is established, provided the members pray for the prosperity of Christendom, and for the intention of the Holy Father. Seven years and seven quarantines may be gained on the feasts of the Visitation, Presentation, and the Patronage of the Blessed Virgin. The same indulgences may be gained by the family in the reunion in prayer among themselves, if they pray before a picture of the Holy Family. The same also, whenever the members attend a public meeting of the association. Three hundred days' indulgence is granted as often as a member of the association recites, before a picture of the Holy Family, the prayer “O most loving Jesus,” etc., etc. The members [pg 078] may gain an indulgence of two hundred days when they make the salutation: “Jesus, Mary, and Joseph, enlighten us, aid us, save us. Amen.” One hundred days for every new member that is brought to the association, and sixty days for every good work done in honor of the Holy Family. All these indulgences are applicable to the souls in purgatory. Prayer. O most loving Jesus, etc., etc.
- 71. Ninth Day. The Catholic Christian has the true faith which comes to him from Jesus Christ, the Founder of the true Church. He ought, then, to show by his conduct that his faith has made him better than so many others, who have not had the graces and advantages which came to him. In other words, it is not enough to believe the truths that God has revealed; it is not enough to belong to the true Church by the internal adhesion of the mind; it is indispensably necessary to manifest our faith in exterior works. Our faith should so have penetrated our whole being, that the profession of religion should show itself in all our actions. Faith without works is dead, and at the Last Judgment the almighty Judge will demand of us an account of all our actions, and then will He render to every one the merited reward or punishment. Let it be our aim in life to fill the days of our stay on earth with many good actions, the outcome of our faith, so that when we stand before the throne of God, we may have many glorious deeds to our credit. [pg 079] Prayer. O most loving Jesus, etc., etc. Tenth Day. A Christian considers his faith as a gift of heaven, a priceless treasure far surpassing any earthly wealth, because it raises man to a true knowledge of God and secures for him his eternal salvation. He rejects with horror the maxims of our modern infidels, who say, “One religion is as good as another”; “Hell is only a bugbear”; “The faith of the heart is enough for salvation,” and many others of the
- 72. same nature. He rejects them because he knows that God is one, that the truth that comes from God is one, and that therefore faith must be one, and religion one. He knows that religion cannot be framed according to the whims of man, but only can come from the authority of God in His holy revelation. He will not associate with those who hold the above false doctrines; he will be an enemy of bad books, which teach errors of faith and which drag those who read them into the mire of immorality, and he will caution his friends against them. Prayer. O most loving Jesus, etc., etc. Eleventh Day. The Catholic Church was founded by Jesus Christ as a perfect society, with authority to make laws, with power to punish the guilty, and to expel rebellious subjects from her midst. This power was [pg 080] given to the Church by Christ when He said, “And whatsoever thou shalt bind upon earth, it shall be bound also in heaven.” It is clear that Christ intended His Church to be our guide in all our actions, as an authority to teach us what the revelation is concerning our future state. We should, therefore, be obedient and faithful children of the Church. We should be grateful to God, Who in His mercy has established certain fountains of grace, which are found in the Church, and are guarded by her. These fountains of grace are the sacraments, which point out the holy states of life, and the true manner of pleasing God. We should use the Sacraments of Penance and Holy Eucharist for the remedy of our faults, and the strength of our weakness. Prayer.
- 73. O most loving Jesus, etc., etc. Twelfth Day. Confession is the correction of our faults, and if we have sinned let us remember we have an advocate in heaven, to Whom we wish to return in the sincerity of our hearts. The Holy Eucharist is the body and blood of Christ and communion the partaking of it. We should not, therefore, be deterred from frequenting these sacraments by human considerations, or by the mockery of the people of this world. We should have these words of Christ deeply engraven in our hearts: “Except you eat the flesh of the Son of man, and drink His blood, you shall not have life in you.”—John vi. 54. A devout Catholic is [pg 081] easily distinguished from the crowd of careless ones, when we see him humbly and frequently going to confession and to holy communion. Prayer. O most loving Jesus, etc., etc. Thirteenth Day. Be also reverent and devout in the house of God; not brought there by vain curiosity, or by fashion, but by unfeigned piety, rendering to God an external tribute of dependence and adoration. Look upon priests as the ambassadors of God, treat them with respect, listen to their teaching, and put it into practice. Reverence the bishops as divinely constituted guardians and teachers in the Church: especially
- 74. the see of St. Peter, the Vicar of Christ, the Roman Pontiff, the Father and teacher, in whom is intrusted the plentitude of power to rule the whole Catholic family. Reverence the infallible authority of the Pope, which guides you in matters of faith, in form of worship, and morality. Accept with docility and obedience the decisions of the Holy See, and conform to them your opinions and thoughts. Do not follow the changeable and novel opinions of our infidel age. Prayer. O most loving Jesus, etc., etc. Fourteenth Day. The true Christian must not only profess the faith, but also the laws of Christ. He is [pg 082] anxious to observe them exactly and to observe them all, knowing that he is guilty of damnation who violates the law in one important point. This law bids us to love all men as fellow creatures, to love our relatives, our country, but above all, and before all, we must love God, the Author of our being, the great loving Father of heaven. The honor of God is the very first duty of man, who as a rational being knows God and His infinite goodness; we wish to serve Him as His subjects, render Him the homage due to His immensity, a worship which our infinite littleness renders to God. Never profane the word of God yourself, and prevent curses and oaths in others as far as possible. By acts of praise and benediction let us repair the offences against God when we cannot prevent them. At least pray for those who use the name of God in vain, and thus endeavor to ward off from them the eternal punishment due to that wicked practice.
- 75. Prayer. O most loving Jesus, etc., etc. Fifteenth Day. Contribute your share towards the glorification of the name of God, by observing the Sundays and festivals of the Church. God has expressly reserved these days to Himself, and has pointed them out by the authority of the Church. In the Old and in the New Law, God has had days of rest and of religious practices; and for the observances of these He has promised publicly that there should be many rewards. Every good, God-fearing man will give a [pg 083] just tribute of respect to God, because God wills it, and because he is looking for some benefit from God. Abstain then from servile works on those days, no matter what temporal gains may be expected, and be careful that others, too, keep holy the Sabbath of the Lord, particularly those who are intrusted to your care and command. Do not have work done on Sunday, and allow none to be done about your premises. Prayer. O most loving Jesus, etc., etc. Sixteenth Day. Respect your parents, superiors, and masters, and all those who hold positions of trust towards you. They hold the authority of God, and he who despises them despises God Himself. Honor and respect
- 76. those superiors as representatives of God, and obey them in all things that are not against His law; and so we come to that great second commandment, which is like unto the first: “Thou shalt love thy neighbor as thyself.” We should do unto others as we would wish that others should do unto us; that is, we should look upon one another as children of one great family, of which God is the heavenly Father. “By this shall all men know that you are My disciples, if you have love one for another,” not showing this love through politeness only, but through a real, downright feeling of interest in others, and without a selfish regard of our own interests. Try to perform spiritual and corporal works of mercy for your neighbor. [pg 084] Prayer. O most loving Jesus, etc., etc. Seventeenth Day. In all your intercourse with others respect their persons and property; do not look for unjust gains; be faithful to your bargains and contracts; never look to your own selfish interests solely. Never speak ill of anybody, nor circulate detractions, nor reveal secrets and defects that might lessen the esteem in which any one is held; excuse the faults of others, and find some excuse for the intention with which even an evil action is committed. We are all temples of the Holy Ghost, sanctified and ennobled by the blood of Christ. We are the dwellings of the Holy Trinity, called to a heavenly inheritance. Do not desecrate this sacred temple by impurity; guard against impure thoughts and immodest desires, flee from dangerous occasions. Avoid foolishly pursuing the luxuries and vanities of the world, improper company, and bad conversation. Do not enter theatres or places of amusement where your morals are
- 77. endangered, and from which you carry nothing but pictures of immoral objects. Arm yourself most effectually against the approach of evil by the powerful shield of prayer, and walk in the presence of God. Prayer. O most loving Jesus, etc., etc. Eighteenth Day. So far we have considered the law of God practically interpreted in our every-day life; let us [pg 085] go still further and endeavor to learn the laws of the Church, for the Lord has said: “He that heareth you, heareth Me; and he that despiseth you despiseth Me,” and “If he will not hear the Church let him be to thee as the heathen and publican.” On Sundays and holydays of obligation, we ought to hear Mass. We should observe the feasts of the Church and the restriction from flesh meat on Fridays and other days of abstinence. Remember that these little acts of mortification are a great benefit to us, since the Lord has commanded, “That we should bring forth fruits worthy of penance.” Make your Easter duty, for the Church has laid down the law that you should fulfil those duties, not from routine or human respect, but with the knowledge of the needs of your soul. Prayer. O most loving Jesus, etc., etc. Nineteenth Day.
- 78. What an exalted opinion we should have of Christian piety! It inspires the Christian man and woman with lofty ideas, and prepares them for noble undertakings. These lessons of piety should be planted early in the hearts of the young, that they may take root and grow up to a magnificent fruit of mature virtue. Serve God as a loving and dutiful child, cherish a great devotion to the Blessed Virgin, and have recourse to her in all your wants, being sure that all your petitions will meet with a ready and hearty response. Never forget these three truths, which should be the main considerations of [pg 086] the true Christian on every occasion: First, That sin is the only evil which should be feared; Second, The grace of God is the real good for which we should strive with all our heart; Third, The salvation of the soul is the all-important business of our lives, for which we were created, and which should be looked after with that care which it deserves. Prayer. O most loving Jesus, etc., etc. Twentieth Day. The husband, as head of the family, owes to his wife fidelity, love, and support. Fidelity is that constancy of affection which he has promised at his marriage, and which must be preserved inviolable until death; it means that purity of soul and body which will not permit itself to be degraded by impurity and adultery. Conjugal fidelity is a great and holy duty, in which matrimony is held sacred. There you find peace, happiness, and the blessing of almighty God. Our Lord was very stringent upon this point, for He says: “Whosoever shall look on a woman to lust after her, hath already committed adultery with her in his heart.”—Matt. v. 28. Whosoever then commits adultery transgresses a most important divine
- 79. commandment. In the Old Law this crime was punished by stoning the guilty person to death, and in the primitive Church by a severe public penance of many years' duration. What fidelity does not the husband demand of the wife! With the same strictness is he obliged to be faithful to her whom he has chosen as his life companion. [pg 087] Prayer. O most loving Jesus, etc., etc. Twenty-first Day. The second duty of the husband and wife is conjugal love. The word conjugal means joined together, because husband and wife are united to bear the same burden. The Apostle says: “Husbands, love your wives, as Christ also loved the Church, and delivered Himself up for it.”—Ephes. v. 25. Husband and wife are individuals whom God has joined in inseparable companionship. The greatest bond between mankind, and the sweetest one, is conjugal love, of which we are thinking on this day consecrated to the Holy Family. Keep this great duty before your eyes and never forget it, for it is easily destroyed. It is from that love, too, that should spring your children, who are to grow up to take your place in the Church and the State. These children you are to bring up in the fear and love of God, faithful to the Church and their fatherland. A tremendous responsibility! Prayer. O most loving Jesus, etc., etc.
- 80. Twenty-second Day. The husband is the main worker in the family, so that his duty is to provide for his family by his industry and economy. He must look for employment [pg 088] and strive to keep it, so as to have a never- failing source of income, by which his people may live in comparative comfort. There are husbands who will allow their wives and children to work, while they themselves live idle lives, which is the fashion of the untutored Indian. Not only must the husband work, but he must live in economy, and not throw his money away foolishly in gaming or in drunkenness. The most frequent cause of a husband and father's failure to provide for his family is drunkenness. Drunkenness causes woe, sin, sorrow and shame. Drunkenness besots the mind, and makes of an intelligent being a brute in his passions, and a fool in his actions; it extinguishes the spirit of God in him, all sentiments of religion are lost, the Church of God is despised and disgraced. Drunkenness does as much harm as the greatest vices, bringing ruin with it whenever indulged in. Prayer. O most loving Jesus, etc., etc. Twenty-third Day. The husband has to share the care of the children and he should look after the instruction of the child. Children are a great treasure, worth more than all the wealth of the world. The Lord said of them to the apostles: “Suffer the little children to come unto Me, and forbid them not: for of such is the kingdom of God.” And why is the Lord so anxious for the welfare of the innocent child? Because it is a weak human being, unable to help itself, [pg 089] destined for
- 81. heaven, redeemed by the blood of Jesus, and a temple of the Holy Ghost, more fitted to be such on account of the purity of its soul. Now the care of this child is given to the father and the mother, under whose care it is to grow up a true Christian, an exemplary member of the Church of God; to live on until it has fulfilled its days and the duties of its state of life, when, like yourself, having come to the fulness of maturity, it is gathered in to render an account of its life work to almighty God. The success of the child's life depends chiefly on the manner in which its parents fulfil their duty. Prayer. O most loving Jesus, etc., etc. Twenty-fourth Day. In what special duties are you to instruct your children? First of all, let the young children learn early to pray, make them think of God, speak to them of His love for mankind, teach them to adore Him, because He has created them, to thank Him for all His benefits which flow to them so abundantly, to ask Him with confidence for all the graces that they need. Correct the children for their faults: Lying, stealing, cursing, stubbornness, disobedience, fighting, and cruelty. Have an eye very early on their morals, for little children learn to do wicked things, by which they lose the love and grace of God. Be not a tyrant, but a sensible, religious father or mother, and see to it that the children are free from these vices. Give them no bad example, [pg 090] especially by using profanity, or by getting intoxicated. All this presupposes perpetual vigilance; remember you will have to render a strict account of these things before God. Prayer.
- 82. O most loving Jesus, etc., etc. Twenty-fifth Day. Having considered the husband's duties, we must now look at the duty of a wife. The Scriptures say of the wife: “A good wife is a good portion; she shall be given in the portion of them that fear God, to a man for his good deeds.”—Eccles. xxvi. 3. There is nothing in the whole world more precious than a good wife. “A wise woman buildeth her house: but the foolish will pull down with her hands that also which is built.”—Prov. xiv. 1. A wife must love her husband, and she owes him the most scrupulous fidelity; if the first duty of the husband is to love his wife, so also is there a corresponding duty to return that love. She must be patient with him when he comes home murmuring against his fate; she must make the home agreeable to him by cleanliness and cheerfulness. She must bear the burdens of this life with her husband, and encourage him, that he may not be despondent. The wife must be sober, not given to scolding and fault- finding. “Let women be subject to their husbands, as to the Lord; because the husband is the head of the wife, as Christ is the head of the Church.”—Ephes. v. 22. [pg 091] Prayer. O most loving Jesus, etc., etc. Twenty-sixth Day.
- 83. The conscientious observance of the marriage vows is to be a supreme law to the wife. Purity must be the virtue principally looked to in marriage, according to the laws of the Sacrament of Matrimony; the wife's motto must be that of Susanna of old, who said: “I will rather die than sin before God.” The wife owes her husband a compliance at least to his wishes; not exactly an abject obedience, but that the husband and wife consult with each other, and that she comply in all lawful and sensible things. This subjection is founded on the Scriptures. God Himself said to Eve, “Thou shalt be under thy husband's power, and he shall have dominion over thee” (Gen. iii. 16), and St. Paul declares, “Let women be subject to their husbands as to the Lord.”—Ephes. v. 22. The loving, true, and obedient wife exerts an unbounded influence for good over her husband. She will make him great in the eyes of men, she will make him respectable and presentable in society; in short, she will make the married life a truly happy one from beginning to end. Prayer. O most loving Jesus, etc., etc. Twenty-seventh Day. The wife has a sublime calling to be a mother. What is more beautiful than motherhood? [pg 092] what is more useful to the new-born humanity than the mother? All the world recognizes her dignity, and respects her. A Christian mother will consider the child a gift from God, which though given to her, still belongs to God; hers is the first care of the newly-born infant, her care and love will not relax for all time to come. She is always the mother. Have the child presented for Baptism at the very earliest moment; if it be in danger of death see that it receives private baptism. Then start out in patience and kindness to rear it, giving it a secular but above all a
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