From dissipation-induced instability to the dynamics of engines
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The document is an invitation letter for Prof. Alejandro Jenkins to speak at a workshop on nonlinear phenomena in Bolivia. It provides context on Jenkins' work relating dissipation-induced instability to the dynamics of engines. Specifically, it discusses how self-oscillators maintain periodic motion by extracting energy from a power source, and how this relates to concepts in thermodynamics and nonequilibrium systems.
![Electron shuttle
7
From: L. Y. Gorelik et al., PRL 80, 4526 (1998)
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As
ha-
of them is located near a positively and the other near
a negatively biased electrode. Because of the Coulomb
blockade phenomenon an integer number of electrons are
loaded onto the grain close to one turning point, and the
same number of electrons are unloaded close to the other,
as illustrated in Fig. 1. The result is that in each cycle
the shuttle moves a discrete number of electrons from one
electrode to the other. It follows that the current is propor-
tional to the mechanical vibration frequency of the grain.
-V/2V/2
a)
R1 m R2
k k
2N electrons
"loading" of
electrostatic field
"unloading" of
2N electrons
q=-Ne
-V/2V/2
q=Ne
b)
FIG. 1. (a) Simple model of a soft Coulomb blockade system
in which a metallic grain (center) is linked to two electrodes by
elastically deformable organic molecular links. (b) Dynamic
instabilities occur since in the presence of a sufficiently
large bias voltage V the grain is accelerated by the same
electrostatic force towards first one, then the other electrode. A
cyclic change in direction is caused by the repeated “loading”
of electrons near the negatively biased electrode and the
subsequent “unloading” of the same at the positively biased
electrode. As a result, the sign of the net grain charge alternates
leading to an oscillatory grain motion and a novel “electron
shuttle” mechanism for charge transport.
Mechanical model:
resistances R1sxd, R2sxd which are assumed to be exponen-
tial functions of the grain coordinate x. In order to avoid
unimportant technical complications we study the symmet-
ric case for which R1,2 ≠ Re6xyl. When the position of
the grain is fixed, the electrical potential of the grain and
its charge q are given by current balance between the grain
and the leads [1]. As a consequence, at a given bias volt-
age V the charge q is completely controlled by the ratio
R1sxdyR2sxd. In addition, the bias voltage generates an
electrostatic field E ≠ aV in the space between the leads
[7], and hence a charged grain will be subjected to an elec-
trostatic force Fq ≠ aVq.
The central point of our considerations is that the
grain—because of the “softness” of the organic molecular
links connecting it to the leads—may move and change
its position. The grain motion disturbs the current bal-
ance and as a result the grain charge will vary in time.
This variation affects the work W ≠ aV
R
Ÿxqstd dt per-
formed on the grain during, say, one period of its oscilla-
tory motion. It is significant that this work is nonzero and
positive, i.e., the electrostatic force, on the average, accel-
erates the grain. The nature of this acceleration is best
understood by considering a grain oscillating with a large
amplitude A . l. In this case the charge fluctuations be-
tween the grain and the most distant electrode is expo-
nentially suppressed when the displacement of the grain
is maximal. As a result, the grain at such a turning point
gains extra charge from the nearby electrode, as shown
in Fig. 1(b). The added charge is positive at the turning
point near the positively charged electrode and negative
at the turning point close to the negatively charged one.
The sign change of the charge takes place mainly in the
immediate vicinity of the electrodes, while along the ma-
jor part of the path between them the charge on the grain
brations will occur when a finite voltage bias V is applied
to our system.
In real systems a certain amount Q of energy is dis-
sipated due to mechanical damping forces which al-
ways exist. In order to get to the self-excitation regime
more energy must be pumped into the system from the
electrostatic field than can be dissipated; W must ex-
ceed Q. Since the electrostatic force increases with the
bias voltage this condition can be fulfilled if V exceeds
some critical value Vc. If the electrostatic and damp-
ing forces are much smaller than the elastic restoring
force self-excitation of vibrations with a frequency equal
to the eigenfrequency of elastic oscillations arise. In
this case Vc can be implicitly defined by the relation
V/2 -V/2
2L
q
x
q= e[VC/e + 1/2]
q= -e[VC/e + 1/2]
δx
2δq
FIG. 2. Charge response to a cyclic grain motion. The dashed
arrows describe an imagined, particularly simple trajectory in
the charge-position plane that allows the work done on the
grain by the electrostatic field to be easily calculated and
shown to be positive, hence leading to an instability: For
times t , 2t when the grain is at rest at x ≠ 2dxy2 the
charge exchange with the positively biased electrode dominates
and the grain is positively charged q ≠ dq; at the instant
t ≠ 2t it instantaneously moves to the point x ≠ 1dxy2
where the charge relaxes to a new equlibrium value q ≠ 2dq;
then the negatively charged grain instantaneously moves back
at t ≠ 1t. During this cyclic process the electrostatic force
acts only along the direction of the grain displacement. The
Cycle:](https://image.slidesharecdn.com/ajenkinslawnp19-191105102641/85/From-dissipation-induced-instability-to-the-dynamics-of-engines-7-320.jpg)
![critical wind speed V = v
“What Zel’dovich knew”
11
Galileo:
@⇠
@t
!
@⇠
@t
+ V
@⇠
@x
= i!⇠ + V · ik⇠
Zel’dovich, JETP Lett. 14, 180 (1971);
Sov. Phys. JETP 35, 1085 (1971)
⇠ ⇠ exp [ik(x vt)] ! = kv
Air
Water
x
*
* K. Thorne, 2013
= ik⇠(v V )<latexit sha1_base64="ZqUUAp7GO5gJX0BDF2a4SN1498c=">AAAB+3icbVBNS8NAEJ3Ur1q/Yj16WSxCPbQkIuhFKHrxWMF+QBvKZrtpl242YXdTWkL/ihcPinj1j3jz37htc9DWBwOP92aYmefHnCntON9WbmNza3snv1vY2z84PLKPi00VJZLQBol4JNs+VpQzQRuaaU7bsaQ49Dlt+aP7ud8aU6lYJJ70NKZeiAeCBYxgbaSeXbxFFYZGqDthqDxGFdS86Nklp+osgNaJm5ESZKj37K9uPyJJSIUmHCvVcZ1YeymWmhFOZ4VuomiMyQgPaMdQgUOqvHRx+wydG6WPgkiaEhot1N8TKQ6Vmoa+6QyxHqpVby7+53USHdx4KRNxoqkgy0VBwpGO0DwI1GeSEs2nhmAimbkVkSGWmGgTV8GE4K6+vE6al1XXqbqPV6XaXRZHHk7hDMrgwjXU4AHq0AACE3iGV3izZtaL9W59LFtzVjZzAn9gff4AHg6R5A==</latexit><latexit sha1_base64="ZqUUAp7GO5gJX0BDF2a4SN1498c=">AAAB+3icbVBNS8NAEJ3Ur1q/Yj16WSxCPbQkIuhFKHrxWMF+QBvKZrtpl242YXdTWkL/ihcPinj1j3jz37htc9DWBwOP92aYmefHnCntON9WbmNza3snv1vY2z84PLKPi00VJZLQBol4JNs+VpQzQRuaaU7bsaQ49Dlt+aP7ud8aU6lYJJ70NKZeiAeCBYxgbaSeXbxFFYZGqDthqDxGFdS86Nklp+osgNaJm5ESZKj37K9uPyJJSIUmHCvVcZ1YeymWmhFOZ4VuomiMyQgPaMdQgUOqvHRx+wydG6WPgkiaEhot1N8TKQ6Vmoa+6QyxHqpVby7+53USHdx4KRNxoqkgy0VBwpGO0DwI1GeSEs2nhmAimbkVkSGWmGgTV8GE4K6+vE6al1XXqbqPV6XaXRZHHk7hDMrgwjXU4AHq0AACE3iGV3izZtaL9W59LFtzVjZzAn9gff4AHg6R5A==</latexit><latexit sha1_base64="ZqUUAp7GO5gJX0BDF2a4SN1498c=">AAAB+3icbVBNS8NAEJ3Ur1q/Yj16WSxCPbQkIuhFKHrxWMF+QBvKZrtpl242YXdTWkL/ihcPinj1j3jz37htc9DWBwOP92aYmefHnCntON9WbmNza3snv1vY2z84PLKPi00VJZLQBol4JNs+VpQzQRuaaU7bsaQ49Dlt+aP7ud8aU6lYJJ70NKZeiAeCBYxgbaSeXbxFFYZGqDthqDxGFdS86Nklp+osgNaJm5ESZKj37K9uPyJJSIUmHCvVcZ1YeymWmhFOZ4VuomiMyQgPaMdQgUOqvHRx+wydG6WPgkiaEhot1N8TKQ6Vmoa+6QyxHqpVby7+53USHdx4KRNxoqkgy0VBwpGO0DwI1GeSEs2nhmAimbkVkSGWmGgTV8GE4K6+vE6al1XXqbqPV6XaXRZHHk7hDMrgwjXU4AHq0AACE3iGV3izZtaL9W59LFtzVjZzAn9gff4AHg6R5A==</latexit><latexit sha1_base64="ZqUUAp7GO5gJX0BDF2a4SN1498c=">AAAB+3icbVBNS8NAEJ3Ur1q/Yj16WSxCPbQkIuhFKHrxWMF+QBvKZrtpl242YXdTWkL/ihcPinj1j3jz37htc9DWBwOP92aYmefHnCntON9WbmNza3snv1vY2z84PLKPi00VJZLQBol4JNs+VpQzQRuaaU7bsaQ49Dlt+aP7ud8aU6lYJJ70NKZeiAeCBYxgbaSeXbxFFYZGqDthqDxGFdS86Nklp+osgNaJm5ESZKj37K9uPyJJSIUmHCvVcZ1YeymWmhFOZ4VuomiMyQgPaMdQgUOqvHRx+wydG6WPgkiaEhot1N8TKQ6Vmoa+6QyxHqpVby7+53USHdx4KRNxoqkgy0VBwpGO0DwI1GeSEs2nhmAimbkVkSGWmGgTV8GE4K6+vE6al1XXqbqPV6XaXRZHHk7hDMrgwjXU4AHq0AACE3iGV3izZtaL9W59LFtzVjZzAn9gff4AHg6R5A==</latexit>](https://image.slidesharecdn.com/ajenkinslawnp19-191105102641/85/From-dissipation-induced-instability-to-the-dynamics-of-engines-11-320.jpg)
![Bogie hunting I
21
• Mathematical theory of train stability debated since 1840s
see Knothe, Veh. Syst. Dyn. 46, 9 (2008)
• In practice, stable for V < Vc, unstable for V > Vc
• Vc = f 𝛌K, where f is observed hunting frequency
diameters and the other one on increasingly larger diameters. Both wheels will
reach the same level at the precise moment when the axle center is situated at the
maximum distance from the rail longitudinal axis. From now on, the movement
will repeat itself in reverse. The axle center’s trajectory is a sinuous curve.
This phenomenon of kinematical motion was described for the first time
by Stephenson, and Klingel [1] determined the hunting wavelength according to
his famous formulae
e
re
L 2 , (1)
where r is the rolling radius, 2e – the distance between the wheel/rail contact
points and e – the effective conicity.
This movement is passed over to the bogie and to the vehicle body through
suspension elements. During the circulation, this hunting movement is also
sustained by rail alignment irregularities; therefore, its intensity will be influenced
by the size of these irregularities. In addition to that, the regime of this hunting
movement depends on the running speed - at low speeds, this movement is stable
and at high speeds, this movement becomes unstable. The value of speed when
Source: Mazilu, U.P.B. Sci. Bull. (ser. D) 71, 63 (2009)
Klingel wavelength: K = 2⇡
r
e0 r0
<latexit sha1_base64="rcq80CR/t95Z2YjkNL4fnoe0nwY=">AAACHHicbVDLSgMxFM3UV62vqks3wSK4kDLTCroRim4ENxXsAzpluJNm2tBkZkwyQhnmQ9z4K25cKOLGheDfmD4W2nogcDjnXG7u8WPOlLbtbyu3tLyyupZfL2xsbm3vFHf3mipKJKENEvFItn1QlLOQNjTTnLZjSUH4nLb84dXYbz1QqVgU3ulRTLsC+iELGAFtJK9YdbkJ98C7wRe4gt2YYVfdS51iN5BAUurZ2D3B0rOz1O2DEJBlXrFkl+0J8CJxZqSEZqh7xU+3F5FE0FATDkp1HDvW3RSkZoTTrOAmisZAhtCnHUNDEFR108lxGT4ySg8HkTQv1Hii/p5IQSg1Er5JCtADNe+Nxf+8TqKD827KwjjRNCTTRUHCsY7wuCncY5ISzUeGAJHM/BWTAZhStOmzYEpw5k9eJM1K2amWK7enpdrlrI48OkCH6Bg56AzV0DWqowYi6BE9o1f0Zj1ZL9a79TGN5qzZzD76A+vrB5Y3oHo=</latexit>
2e0: rail distance
r0: rolling radius
𝛾: conicity](https://image.slidesharecdn.com/ajenkinslawnp19-191105102641/85/From-dissipation-induced-instability-to-the-dynamics-of-engines-21-320.jpg)
From dissipation-induced instability to the dynamics of engines
- 1. From dissipation-induced instability to the dynamics of engines Alejandro Jenkins U. de Costa Rica LETTER OF INVITATION Dear Prof. Alejandro Jenkins Escuela de Física, Universidad de Costa Rica 16th Latin American Workshop on Nonlinear Phenomena Universidad Mayor de San Andrés La Paz, Bolivia 22-26 October, 2019
- 2. Self-oscillation • Self-oscillators maintain regular, periodic motion at expense of power source with no corresponding periodicity • Make waves, by positive feedback between oscillation & energy source • also known as: maintained, sustained, self-induced, self-excited, spontaneous, autonomous, etc. • Usually approached from theory of differential eqs. (limit cycle, Hopf bifurcation) or as instabilities in feedback control 2
- 3. Rayleigh 3 ¨q ↵ ˙q + ˙q3 /3 + !2 q = 0 Combines linear anti-damping α 2nd ed. of Theory of Sound (1894-6) models “maintained oscillations”, including wind musical instruments, by Ploss = ˙qFdamp = m ˙q4 /3 Pgain = ˙qFanti damp = ↵m ˙q2 with non-linear damping β
- 4. Positive feedback • Vout amplified & fed back to Vin • Resistance effectively negative • Exponential growth limited by amplifier’s saturation (non-linearity) • Clocks work on this principle Vin = g · Vout ¨Vout + 1 g RC ˙Vout + 1 LC Vout = 0 4 Vin Vout R L C
- 5. Engines • Le Corbeiller: autonomous engines are self-oscillators (“prime movers”) • Mathematical theory of self- oscillators may benefit from thermodynamics and vice-versa • See: “The non-linear theory of the maintenance of oscillations”, Proc. Inst. Electr. Engrs. 11, 292 (1936); “Theory of prime movers”, in Non- Linear Mechanics (Brown U., 1943), ch. 2 5 Philippe Emmanuel Le Corbeiller (1891-1980) © Collections École polytechnique (Palaiseau)
- 6. Thermodynamics • Time not a variable in classical thermodynamics • Engine’s output W > 0 requires active, non-conservative force acting on piston or turbine • Carnot’s bound on efficiency of heat engines only achievable at zero power (non-conservative force → 0) • Irreversible dynamics of cyclic engines is of great practical importance • …but it’s largely ignored in theoretical physics! 6 Nicolas Léonard Sadi Carnot (1796 – 1832) (oil on canvas, Louis-Léopold Boilly, 1813)
- 7. Electron shuttle 7 From: L. Y. Gorelik et al., PRL 80, 4526 (1998) s in d it ons. rily in zed cale gies the sed ow ely site ade pect ork ters heir the ists ular or ary der ely with rge ese ted n of for- As ha- of them is located near a positively and the other near a negatively biased electrode. Because of the Coulomb blockade phenomenon an integer number of electrons are loaded onto the grain close to one turning point, and the same number of electrons are unloaded close to the other, as illustrated in Fig. 1. The result is that in each cycle the shuttle moves a discrete number of electrons from one electrode to the other. It follows that the current is propor- tional to the mechanical vibration frequency of the grain. -V/2V/2 a) R1 m R2 k k 2N electrons "loading" of electrostatic field "unloading" of 2N electrons q=-Ne -V/2V/2 q=Ne b) FIG. 1. (a) Simple model of a soft Coulomb blockade system in which a metallic grain (center) is linked to two electrodes by elastically deformable organic molecular links. (b) Dynamic instabilities occur since in the presence of a sufficiently large bias voltage V the grain is accelerated by the same electrostatic force towards first one, then the other electrode. A cyclic change in direction is caused by the repeated “loading” of electrons near the negatively biased electrode and the subsequent “unloading” of the same at the positively biased electrode. As a result, the sign of the net grain charge alternates leading to an oscillatory grain motion and a novel “electron shuttle” mechanism for charge transport. Mechanical model: resistances R1sxd, R2sxd which are assumed to be exponen- tial functions of the grain coordinate x. In order to avoid unimportant technical complications we study the symmet- ric case for which R1,2 ≠ Re6xyl. When the position of the grain is fixed, the electrical potential of the grain and its charge q are given by current balance between the grain and the leads [1]. As a consequence, at a given bias volt- age V the charge q is completely controlled by the ratio R1sxdyR2sxd. In addition, the bias voltage generates an electrostatic field E ≠ aV in the space between the leads [7], and hence a charged grain will be subjected to an elec- trostatic force Fq ≠ aVq. The central point of our considerations is that the grain—because of the “softness” of the organic molecular links connecting it to the leads—may move and change its position. The grain motion disturbs the current bal- ance and as a result the grain charge will vary in time. This variation affects the work W ≠ aV R Ÿxqstd dt per- formed on the grain during, say, one period of its oscilla- tory motion. It is significant that this work is nonzero and positive, i.e., the electrostatic force, on the average, accel- erates the grain. The nature of this acceleration is best understood by considering a grain oscillating with a large amplitude A . l. In this case the charge fluctuations be- tween the grain and the most distant electrode is expo- nentially suppressed when the displacement of the grain is maximal. As a result, the grain at such a turning point gains extra charge from the nearby electrode, as shown in Fig. 1(b). The added charge is positive at the turning point near the positively charged electrode and negative at the turning point close to the negatively charged one. The sign change of the charge takes place mainly in the immediate vicinity of the electrodes, while along the ma- jor part of the path between them the charge on the grain brations will occur when a finite voltage bias V is applied to our system. In real systems a certain amount Q of energy is dis- sipated due to mechanical damping forces which al- ways exist. In order to get to the self-excitation regime more energy must be pumped into the system from the electrostatic field than can be dissipated; W must ex- ceed Q. Since the electrostatic force increases with the bias voltage this condition can be fulfilled if V exceeds some critical value Vc. If the electrostatic and damp- ing forces are much smaller than the elastic restoring force self-excitation of vibrations with a frequency equal to the eigenfrequency of elastic oscillations arise. In this case Vc can be implicitly defined by the relation V/2 -V/2 2L q x q= e[VC/e + 1/2] q= -e[VC/e + 1/2] δx 2δq FIG. 2. Charge response to a cyclic grain motion. The dashed arrows describe an imagined, particularly simple trajectory in the charge-position plane that allows the work done on the grain by the electrostatic field to be easily calculated and shown to be positive, hence leading to an instability: For times t , 2t when the grain is at rest at x ≠ 2dxy2 the charge exchange with the positively biased electrode dominates and the grain is positively charged q ≠ dq; at the instant t ≠ 2t it instantaneously moves to the point x ≠ 1dxy2 where the charge relaxes to a new equlibrium value q ≠ 2dq; then the negatively charged grain instantaneously moves back at t ≠ 1t. During this cyclic process the electrostatic force acts only along the direction of the grain displacement. The Cycle:
- 8. Non-equilibrium thermo. • Following Onsager’s work in 1930s, non-equilibrium thermodynamics has focused on currents & forces given by gradients of state functions • Cannot describe macroscopic cycles • e.g., no generalization of Ohm’s law gives W > 0 in electric circuit • Electric work seen only by equating to 𝚫t × power dissipated in load, assuming steady state • Connected to failure of Prigogine’s attempted general theory of “dissipative structures” 8 Lars Onsager (1903-1976)
- 9. This work • AJ, Phys. Rep. 525, 167 (2013) • C. D. Díaz-Marín & AJ, arXiv:1806.01527 • AJ, The Physical Theory of Self- Oscillators & Engines, (in preparation) • discussions with Carlos Díaz (MIT), John McGreevy (UCSD), Juan Sabuco (Oxford), and Pol Spanos (Rice) 9 This article appeared in a journal published by Elsevier. The attached
- 10. 10 Seen on Prof. Spanos’s door at Rice U.:
- 11. critical wind speed V = v “What Zel’dovich knew” 11 Galileo: @⇠ @t ! @⇠ @t + V @⇠ @x = i!⇠ + V · ik⇠ Zel’dovich, JETP Lett. 14, 180 (1971); Sov. Phys. JETP 35, 1085 (1971) ⇠ ⇠ exp [ik(x vt)] ! = kv Air Water x * * K. Thorne, 2013 = ik⇠(v V )<latexit sha1_base64="ZqUUAp7GO5gJX0BDF2a4SN1498c=">AAAB+3icbVBNS8NAEJ3Ur1q/Yj16WSxCPbQkIuhFKHrxWMF+QBvKZrtpl242YXdTWkL/ihcPinj1j3jz37htc9DWBwOP92aYmefHnCntON9WbmNza3snv1vY2z84PLKPi00VJZLQBol4JNs+VpQzQRuaaU7bsaQ49Dlt+aP7ud8aU6lYJJ70NKZeiAeCBYxgbaSeXbxFFYZGqDthqDxGFdS86Nklp+osgNaJm5ESZKj37K9uPyJJSIUmHCvVcZ1YeymWmhFOZ4VuomiMyQgPaMdQgUOqvHRx+wydG6WPgkiaEhot1N8TKQ6Vmoa+6QyxHqpVby7+53USHdx4KRNxoqkgy0VBwpGO0DwI1GeSEs2nhmAimbkVkSGWmGgTV8GE4K6+vE6al1XXqbqPV6XaXRZHHk7hDMrgwjXU4AHq0AACE3iGV3izZtaL9W59LFtzVjZzAn9gff4AHg6R5A==</latexit><latexit sha1_base64="ZqUUAp7GO5gJX0BDF2a4SN1498c=">AAAB+3icbVBNS8NAEJ3Ur1q/Yj16WSxCPbQkIuhFKHrxWMF+QBvKZrtpl242YXdTWkL/ihcPinj1j3jz37htc9DWBwOP92aYmefHnCntON9WbmNza3snv1vY2z84PLKPi00VJZLQBol4JNs+VpQzQRuaaU7bsaQ49Dlt+aP7ud8aU6lYJJ70NKZeiAeCBYxgbaSeXbxFFYZGqDthqDxGFdS86Nklp+osgNaJm5ESZKj37K9uPyJJSIUmHCvVcZ1YeymWmhFOZ4VuomiMyQgPaMdQgUOqvHRx+wydG6WPgkiaEhot1N8TKQ6Vmoa+6QyxHqpVby7+53USHdx4KRNxoqkgy0VBwpGO0DwI1GeSEs2nhmAimbkVkSGWmGgTV8GE4K6+vE6al1XXqbqPV6XaXRZHHk7hDMrgwjXU4AHq0AACE3iGV3izZtaL9W59LFtzVjZzAn9gff4AHg6R5A==</latexit><latexit sha1_base64="ZqUUAp7GO5gJX0BDF2a4SN1498c=">AAAB+3icbVBNS8NAEJ3Ur1q/Yj16WSxCPbQkIuhFKHrxWMF+QBvKZrtpl242YXdTWkL/ihcPinj1j3jz37htc9DWBwOP92aYmefHnCntON9WbmNza3snv1vY2z84PLKPi00VJZLQBol4JNs+VpQzQRuaaU7bsaQ49Dlt+aP7ud8aU6lYJJ70NKZeiAeCBYxgbaSeXbxFFYZGqDthqDxGFdS86Nklp+osgNaJm5ESZKj37K9uPyJJSIUmHCvVcZ1YeymWmhFOZ4VuomiMyQgPaMdQgUOqvHRx+wydG6WPgkiaEhot1N8TKQ6Vmoa+6QyxHqpVby7+53USHdx4KRNxoqkgy0VBwpGO0DwI1GeSEs2nhmAimbkVkSGWmGgTV8GE4K6+vE6al1XXqbqPV6XaXRZHHk7hDMrgwjXU4AHq0AACE3iGV3izZtaL9W59LFtzVjZzAn9gff4AHg6R5A==</latexit><latexit sha1_base64="ZqUUAp7GO5gJX0BDF2a4SN1498c=">AAAB+3icbVBNS8NAEJ3Ur1q/Yj16WSxCPbQkIuhFKHrxWMF+QBvKZrtpl242YXdTWkL/ihcPinj1j3jz37htc9DWBwOP92aYmefHnCntON9WbmNza3snv1vY2z84PLKPi00VJZLQBol4JNs+VpQzQRuaaU7bsaQ49Dlt+aP7ud8aU6lYJJ70NKZeiAeCBYxgbaSeXbxFFYZGqDthqDxGFdS86Nklp+osgNaJm5ESZKj37K9uPyJJSIUmHCvVcZ1YeymWmhFOZ4VuomiMyQgPaMdQgUOqvHRx+wydG6WPgkiaEhot1N8TKQ6Vmoa+6QyxHqpVby7+53USHdx4KRNxoqkgy0VBwpGO0DwI1GeSEs2nhmAimbkVkSGWmGgTV8GE4K6+vE6al1XXqbqPV6XaXRZHHk7hDMrgwjXU4AHq0AACE3iGV3izZtaL9W59LFtzVjZzAn9gff4AHg6R5A==</latexit>
- 12. Feedback • Air pressure acts on wave, • wave modulates air pressure. • For V > v, wind and current velocities opposite in wave’s frame • Bernoulli’s theorem air pressure high in throughs, low in peaks • If resulting “suction” overcomes gravity, conservative Kelvin- Helmholtz instability results • Predicts unrealistically large critical wind speeds to make ocean waves 12 Images: E. Mollo-Christensen, “Flow Instabilities”, National Committee for Fluid Mechanics Films (1972)
- 13. Non-potential flow • Air viscosity introduces phase delay 𝛟 of air pressure with respect to wave • Resulting airflow is non-potential; air pressure not conservative • Since wave moves backwards in air’s frame, air pressure leads water wave sin > 0 ) net W > 0 13 • Air’s viscosity is destabilizing, while water’s viscosity is stabilizing
- 14. Non-potential flow when the air is moving in the direction in which the wave-form is travelling, but with a greater velocity, there will evidently be an excess of pressure in the rear slopes, as well as a tangential drag on the exposed crests… Hence the tendency will be to increase the amplitude of the waves to such a point that the dissipation balances the work done by the surface forces. — Lamb, Hydrodynamics, 6th ed. (1932), sec. 350 14
- 15. Irreversibility 15 ˙Ewind ˙Ewave > 0 available for dissipation in the air Ewind = p2 2m ; See: Bekenstein & Schiffer, PRD 58, 064014 (1998) ˙pwind = ˙pwaveMom. conservation: ˙Ewind = p m · ˙pwave = V · (f~k) ˙Ewave = f~! = f~vk ; V > v , ˙Ewind > ˙Ewave “Quantum mechanics helps understand classical mechanics” — ‘Paradoksov’, Sov. Phys. Uspekhi 9, 618 (1967)
- 16. Dissipation-induced instabilities • treated as paradox in dynamical systems literature (e.g., Ziegler, 1952; Krechetnikov & Marsden, 2007) • damping potentially destabilizing whenever moving part “carries its dissipative mechanisms around with it” - Pippard, Physics of Vibration (1979), ch. 8 • irreversibility & connection to damping often overlooked in theory of flow instabilities 16
- 17. D’Alembert’s paradox • Consider submerged solid held in place against drag Fd • In frame with flow at rest far from the solid, solid is pulled against drag Fd • W = 𝓁 Fd either transmitted to the flow or dissipated by viscosity • Fd = 0 for steady potential flow • Flow impingement must act non- conservatively 17 Source: Wikipedia
- 18. Tidal acceleration 18 Circulatory force: dissipation-induced time lag: ˙Eorbital = 3Gm2 R5 r6 (! ˙↵) ˙↵⌧ x y tidal bulge Moon (mass m) G Earth O S F See: R. Brito, V. Cardoso y P. Pani, Superradiance (2015), sec. 2.6
- 19. Active chaos I • In one dimension, active non-conservative force appears linearly as negative damping (Hopf bifurcation) • Physically, this results from feedback with another degree of freedom, in the presence of external disequilibrium • Non-linear damping then gives limit cycle (Poincaré- Bendixson theorem) • In higher dimensions, positional non-conservative (i.e., circulatory) forces may appear • Also, non-linearity can lead to strange attractors 19
- 20. Active chaos II 20 Malkus-Lorenz water wheel: M͑t͒ = ͑Q/͒͑1 − e−t ͒. ͑2b͒ the whe Fig. 1. Twelve-cup water wheel. The wheel’s axis is horizontal; water is added at the top, and the hanging cups leak. Fig. 3. W of the wh stream of of the wa Source: Matson, Am. J. Phys. 75, 1114 (2007) Chua circuit:
- 21. Bogie hunting I 21 • Mathematical theory of train stability debated since 1840s see Knothe, Veh. Syst. Dyn. 46, 9 (2008) • In practice, stable for V < Vc, unstable for V > Vc • Vc = f 𝛌K, where f is observed hunting frequency diameters and the other one on increasingly larger diameters. Both wheels will reach the same level at the precise moment when the axle center is situated at the maximum distance from the rail longitudinal axis. From now on, the movement will repeat itself in reverse. The axle center’s trajectory is a sinuous curve. This phenomenon of kinematical motion was described for the first time by Stephenson, and Klingel [1] determined the hunting wavelength according to his famous formulae e re L 2 , (1) where r is the rolling radius, 2e – the distance between the wheel/rail contact points and e – the effective conicity. This movement is passed over to the bogie and to the vehicle body through suspension elements. During the circulation, this hunting movement is also sustained by rail alignment irregularities; therefore, its intensity will be influenced by the size of these irregularities. In addition to that, the regime of this hunting movement depends on the running speed - at low speeds, this movement is stable and at high speeds, this movement becomes unstable. The value of speed when Source: Mazilu, U.P.B. Sci. Bull. (ser. D) 71, 63 (2009) Klingel wavelength: K = 2⇡ r e0 r0 <latexit sha1_base64="rcq80CR/t95Z2YjkNL4fnoe0nwY=">AAACHHicbVDLSgMxFM3UV62vqks3wSK4kDLTCroRim4ENxXsAzpluJNm2tBkZkwyQhnmQ9z4K25cKOLGheDfmD4W2nogcDjnXG7u8WPOlLbtbyu3tLyyupZfL2xsbm3vFHf3mipKJKENEvFItn1QlLOQNjTTnLZjSUH4nLb84dXYbz1QqVgU3ulRTLsC+iELGAFtJK9YdbkJ98C7wRe4gt2YYVfdS51iN5BAUurZ2D3B0rOz1O2DEJBlXrFkl+0J8CJxZqSEZqh7xU+3F5FE0FATDkp1HDvW3RSkZoTTrOAmisZAhtCnHUNDEFR108lxGT4ySg8HkTQv1Hii/p5IQSg1Er5JCtADNe+Nxf+8TqKD827KwjjRNCTTRUHCsY7wuCncY5ISzUeGAJHM/BWTAZhStOmzYEpw5k9eJM1K2amWK7enpdrlrI48OkCH6Bg56AzV0DWqowYi6BE9o1f0Zj1ZL9a79TGN5qzZzD76A+vrB5Y3oHo=</latexit> 2e0: rail distance r0: rolling radius 𝛾: conicity
- 22. Bogie hunting II • normal forces exerted by rail contact give f > 0 see, e.g., Shayak, Am. J. Phys. 85, 178 (2017) • sliding friction damps hunting for V < Vc = f 𝛌K • At V = Vc, phase of hunting becomes stationary w.r.t. rail, eliminating sliding • For V > Vc, phase of hunting travels backwards w.r.t. rail; sliding anti-damps hunting • grows until limited by nonlinearities (self- oscillation) 22
- 23. Trainspotting I • Pol Spanos: What drives an electric train forward, if not the momentum of the electrons? • Train pushed forward by friction between wheel & rail • i.e., train’s momentum comes from Earth • But W > 0 is imparted by electric motor • internal magnetic field exerts active, non- conservative force on conducting coils, powering the wheels 23
- 24. Trainspotting II • Coil is an open system: electrons flow in at high potential (catenary) and out at low potential (ground) • Some of the electric potential from catenary is converted to mechanical work, some is resistively dissipated • Magnetic field breaks Onsager’s reciprocal relations • Thermodynamics of this process never discussed in physics or engineering 24
- 25. Outlook I • Blind spot in theoretical physics: • non-conservative forces not regarded as mechanical (see, e.g., Landau & Lifshitz) • classical thermodynamics doesn’t use time as variable (cf. “finite-time thermo.”) • Statistical mechanics treats large systems in thermal & chemical equilibrium, fluctuating stochastically about equilibrium, or steadily relaxing to equilibrium 25
- 26. Outlook II • Onsager reciprocal relations describe only passive dissipation • We still need adequate dynamical theory of work extraction by engines, which involve active, non- conservative forces • A promising starting point is the study of non- conservative instabilities in solid & fluid mechanics • This approach dispels much of the paradox of dissipation-induced instability, and reveals underlying unity across diverse phenomena 26