Lecture3

Lecture 3. Combinatorics, Probability and
Multiplicity (Ch. 2 )
• Combinatorics and probability
• 2-state paramagnet and Einstein solid
• Multiplicity of a macrostate
– Concept of Entropy (next lec.)
• Directionality of thermal processes
(irreversibility)
– Overwhelmingly probable

Combinatorics is the branch of mathematics studying the
enumeration, combination, and permutation of sets of
elements and the mathematical relations that characterize
their properties.
Combinatorics and probability
Examples: random walk, two-state systems, …
Probability is the branch of mathematics that studies the
possible outcomes of given events together with the
outcomes' relative likelihoods and distributions. In common
usage, the word "probability" is used to mean the chance
that a particular event (or set of events) will occur.
Math 104 - Elementary Combinatorics and Probability

Probability
Multiplication rule for independent events: P (i and j) = P (i) x P (j)
Example: What is the probability of the same face appearing on two successive
throws of a dice?
The probability of any specific combination, e.g., (1,1): 1/6x1/6=1/36 (multiplication
rule) . Hence, by addition rule, P(same face) = P(1,1) + P(2,2) +...+ P(6,6) = 6x1/36 = 1/6
An event (very loosely defined) – any possible outcome of some measurement.
An event is a statistical (random) quantity if the probability of its occurrence, P, in the
process of measurement is < 1.
The “sum” of two events: in the process of measurement, we observe either one of the
events. Addition rule for independent events: P (i or j) = P (i) + P (j)
The “product” of two events: in the process of measurement, we observe
both events.
(independent events – one event does not change the probability for the
occurrence of the other).
 
 
 
N
N A
P
A 




,...,
,..., 1
1


Expectation value of a macroscopic
observable A:
(averaged over all accessible microstates)

Two model systems with fixed positions of particles
and discrete energy levels
- the models are attractive because they can be described in terms of
discrete microstates which can be easily counted (for a continuum of
microstates, as in the example with a freely moving particle, we still need
to learn how to do this). This simplifies calculation of . On the other
hand, the results will be applicable to many other, more complicated
models.
Despite the simplicity of the models, they describe a number of
experimental systems in a surprisingly precise manner.
- two-state paramagnet ....
(“limited” energy spectrum)
- the Einstein model of a solid
(“unlimited” energy spectrum)

The Two-State Paramagnet
The energy of a macrostate:



 N
N
N
B

N - the number of “up” spins
N - the number of “down” spins
- a system of non-interacting magnetic dipoles in an external magnetic field B, each dipole
can have only two possible orientations along the field, either parallel or any-parallel to this
axis (e.g., a particle with spin ½ ). No “quadratic” degrees of freedom (unlike in an ideal gas,
where the kinetic energies of molecules are unlimited), the energy spectrum of the particles
is confined within a finite interval of E (just two allowed energy levels).
 - the magnetic moment of an individual dipole (spin)
E
E1 = - B
E2 = + B
0
an arbitrary choice
of zero energy
- B for  parallel to B,
+B for  anti-parallel to B
The total magnetic moment:
(a macroscopic observable)
The energy of a single dipole in the
external magnetic field: B
i
i




 

A particular microstate (....)
is specified if the directions of all spins are
specified. A macrostate is specified by the total
# of dipoles that point “up”, N (the # of dipoles
that point “down”, N  = N - N ).
   









 N
N
B
N
N
B
B
M
U 2




     
N
N
N
N
N
N
N
M 





 




2






Example
Consider two spins. There are four possible configurations of microstates:
M = 2 0 0 - 2
In zero field, all these microstates have the same energy (degeneracy). Note
that the two microstates with M=0 have the same energy even when B0:
they belong to the same macrostate, which has multiplicity =2. The
macrostates can be classified by their moment M and multiplicity :
M = 2 0 - 2
 = 1 2 1
For three spins:
M = 3    - - - -3
M = 3  -  -3
 = 1 3 3 1
macrostates:

The Multiplicity of Two-State Paramagnet
Each of the microstates is characterized by N numbers, the number of
equally probable microstates – 2N, the probability to be in a particular
microstate – 1/2N.
n !  n factorial =
1·2·....·n
0 !  1 (exactly one way to
arrange zero objects)
)!
(
!
!
!
!
!
)
,
(









N
N
N
N
N
N
N
N
N
For a two-state paramagnet in zero field, the energy of all macrostates is
the same (0). A macrostate is specified by (N, N). Its multiplicity - the
number of ways of choosing N objects out of N :
1
)
0
,
( 
 N N
N 
 )
1
,
(
 
2
1
)
2
,
(




N
N
N
   
2
3
2
1
)
3
,
(







N
N
N
N
   
 
 























n
N
n
N
n
N
n
n
N
N
N
n
N
!
!
!
1
2
3
...
1
...
1
)
,
(
The multiplicity of a
macrostate of a two-state
paramagnet with (N, N):

Stirling’s Approximation for N! (N>>1)
N
e
N
N
e
N
N
N
N
N

 2
2
! 






 
Multiplicity depends on N!, and we need an approximation for ln(N!):
N
N
N
N 
 ln
!
ln
Check:
    N
N
N
x
x
x
x
x
N
N









  ln
ln
d
ln
lnN
·
·
·
ln3
ln2
ln1
lnN! 1
1
More accurately:
because ln N << N for large N
      N
N
N
N
N
N
N
N 




 ln
2
ln
2
1
ln
2
1
ln
!
ln 
N
e
N
N 






!
or

The Probability of Macrostates of a Two-State PM (B=0)
(http://stat-www.berkeley.edu/~stark/Java/Html/BinHist.htm)
- as the system becomes larger, the
P(N,N) graph becomes more
sharply peaked:
N =1  (1,N) =1, 2N=2, P(1,N)=0.5
N
N
N
N
N
N
N
N
N
N
N
P
2
)
,
(
)
,
(
)
,
(
#
)
,
(
)
,
( 











all
s
microstate
all
of
      
   N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
e
N
N
e
N
e
N
N
N
N
N
N
N
P
2
2
2
!
!
!
)
,
(

























N
P(1, N)
0.5
0 1 n
0 0.5·1023 1023
 
N N
P(15, N) P(1023, N) - random orientation
of spins in B=0 is
overwhelmingly
more probable
2nd law!

Multiplicity (Entropy) and Disorder
In general, we can say that small multiplicity implies
“order”, while large multiplicity implies “disorder”. An
arrangement with large  could be achieved by a
random process with much greater probability than an
arrangement with small .


large 
small 

The Einstein Model of a Solid
In 1907, Einstein proposed a model that reasonably predicted the thermal
behavior of crystalline solids (a 3D bed-spring model):
a crystalline solid containing N atoms behaves as if it contained
3N identical independent quantum harmonic oscillators, each of
which can store an integer number ni of energy units  = ħ.
We can treat a 3D harmonic oscillator as if it were oscillating
independently in 1D along each of the three axes:

























 2
2
2
2
2
2
2
2
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
z
k
mv
y
k
mv
x
k
mv
r
k
mv
E z
y
x
classic:
quantum: 

































3
1
,
,
,
2
1
2
1
2
1
2
1
i
i
z
i
y
i
x
i
i n
n
n
n
E 


 


the solid’s internal
energy:





2
3
2
1
2
1 3
1
3
1
3
1
3
1
N
n
n
n
U i
N
i
N
i
i
N
i
i
N
i











 


 



the zero-point
energy
the effective internal
energy:
i
N
i
n
U 


3
1

1 2 3 3N
ħ
all oscillators are identical, the energy quanta are the same

The Einstein Model of a Solid (cont.)
At high kBT >> ħ (the classical limit of large ni):
mole
J/K
9
.
24
3
3
2
1
)
2
(
3
3
1






 

B
B
B
i
N
i
Nk
dT
dU
T
Nk
T
k
N
n
U 
solid
dU/dT,
J/K·mole
Lead 26.4
Gold 25.4
Silver 25.4
Copper 24.5
Iron 25.0
Aluminum 26.4
To describe a macrostate of an Einstein solid, we have
to specify N and U, a microstate – ni for 3N oscillators.
Example: the “macrostates” of an Einstein Model with only one atom
 (1,0) =1
 (1,1) =3
 (1,2) =6
 (1,3) =10
Dulong-Petit’s rule

The Multiplicity of Einstein Solid
 







 







q
N
q
N
q
N
q
q
N
1
!
)
1
(
!
!
1
)
,
(
Proof: let’s consider N oscillators, schematically represented as follows:
   - q dots and N-1 lines, total q+N-1 symbols. For given q
and N, the multiplicity is the number of ways of choosing n of the symbols
to be dots, q.e.d.
The multiplicity of a state of N oscillators (N/3 atoms) with q energy quanta
distributed among these oscillators:
In terms of the total
internal energy U =q:
 
  !
)
1
(
!
/
!
1
/
)
,
(





N
U
N
U
U
N


Example: The multiplicity of an Einstein solid with three atoms and eight units of
energy shared among them
 
  !
)
1
9
(
!
8
!
1
9
8
)
8
,
9
(




 12,870

Multiplicity of a Large Einstein Solid (kBT >> )
 
 
 
 
     
 
     
     
1 ! !
ln ( , ) ln ln ln ! ln ! ln !
!( 1)! ! !
Stirling approxmation: ln ! ln
ln ln ln
ln ln ln
q N q N
N q q N q N
q N q N
N N N N
q N q N q N q q q N N N
q N q N q q N N
   
  
      
 
     

   
 
        
    
q = U/ = N - the total # of energy quanta in a solid.
 = U/( N) - the average # of quanta (microstates) available for each molecule
Dulong-Petit’s rule:
B
High temperature limit: k T q N
 
B
B
B B
U q
q Nk T
q N q N
U Nk T
k T Nk T N


 
 
  

  
 
 
 


 

Multiplicity of a Large Einstein Solid (kBT >> )
q
N
q
q
N
q
N
q 

















 ln
1
ln
)
ln(
 N
N
N
N
q
N
e
N
eq
e
e
q
N 










ln
)
,
(
q = U/ = N - the total # of energy quanta in a solid.
 = U/( N) - the average # of quanta (microstates) available for each molecule
General statement: for any system with N “quadratic” degrees of freedom
(“unlimited” spectrum), the multiplicity is proportional to U N/2.
Einstein solid:
(2N degrees
of freedom)
N
N
U
N
f
N
eU
N
U )
(
)
,
( 









   
N
N
q
N
N
N
q
N
N
q
N
N
N
q
q
q
N
q
N
q
q
N


















ln
ln
ln
ln
ln
ln
)
,
(
ln
2
high temperatures:
(kBT >> ,  >>1, q >> N )

Multiplicity of a Large Einstein Solid (kBT << )
low temperatures:
(kBT << ,  <<1, q << N )


N
q
e
q
eN
q
N 

















 )
,
( (Pr. 2.17)

Microstates of a system (e.g. ideal gas)
Microstate: the state of a
system specified by describing
the quantum state of each
molecule in the system. For a
classical particle – 6
parameters (xi, yi, zi, pxi, pyi,
pzi), for a macro system – 6N
parameters.
The evolution of a system can be represented by a trajectory
in the multidimensional (configuration, phase) space of micro-
parameters. Each point in this space represents a microstate.
During its evolution, the system will only pass through accessible microstates
– the ones that do not violate the conservation laws: e.g., for an isolated
system, the total internal energy must be conserved.
 1
 2
 i

Statistics  Probabilities of Macrostates
Macrostate: the state of a macro system specified
by its macroscopic parameters. Two systems with the
same values of macroscopic parameters are
thermodynamically indistinguishable. A macrostate tells
us nothing about a state of an individual particle.
For a given set of constraints (conservation laws), a
system can be in many macrostates.
The statistical approach: to connect the
macroscopic observables (averages) to the probability
for a certain microstate to appear along the system’s
trajectory in configuration space, P( 1,  2,..., N).

The Phase Space vs. the Space of Macroparameters
V
T
P
 1
 2
 i
the surface
defined by an
equation of
states
some macrostate
 1
 2
 i
 1
 2
 i
 1
 2
 i
numerous microstates
in a multi-dimensional
configuration (phase)
space that correspond
the same macrostate
etc., etc., etc. ...

Examples: Two-Dimensional Configuration Space
motion of a particle in a
one-dimensional box
-L L
-L L x
px
-px
“Macrostates” are characterized by a
single parameter: the kinetic energy K0
K
0
Each “macrostate” corresponds to a continuum of
microstates, which are characterized by specifying the
position and momentum
K=K0
Another example: one-dimensional
harmonic oscillator
x
px
K + U =const
x
U(r)

The Fundamental Assumption of Statistical Mechanics
The ergodic hypothesis: an isolated system in
an equilibrium state, evolving in time, will pass
through all the accessible microstates at the
same recurrence rate, i.e. all accessible
microstates are equally probable.
The average over long times will equal the average over the ensemble of all
equi-energetic microstates: if we take a snapshot of a system with N
microstates, we will find the system in any of these microstates with the same
probability.
Probability for a
stationary system
many identical measurements
on a single system
a single measurement on
many copies of the system
The ensemble of all equi-energetic states
 a microcanonical ensemble.
 1
 2
 i
microstates which
correspond to the
same energy

Probability of a Macrostate, Multiplicity
 
s
microstate
accessible
all
of
macrostate
given
a
to
correspond
that
s
microstate
of
macrostate
particular
a
of
y
Probabilit
#
#



The probability of a certain macrostate is determined by how many
microstates correspond to this macrostate – the multiplicity of a given
macrostate  .
This approach will help us to understand why some of the macrostates are
more probable than the other, and, eventually, by considering the interacting
systems, we will understand irreversibility of processes in macroscopic
systems.
s
microstate
accessible
all
of
#
1
ensemble
ical
microcanon
a
of
microstate
particular
a
of
y
Probabilit


Concepts of Statistical Mechanics
1. The macrostate is specified by a sufficient number of macroscopically
measurable parameters (for an Einstein solid – N and U).
2. The microstate is specified by the quantum state of each particle in a
system (for an Einstein solid – # of the quanta of energy for each of N
oscillators)
3. The multiplicity is the number of microstates in a macrostate. For
each macrostate, there is an extremely large number of possible
microstates that are macroscopically indistinguishable.
4. The Fundamental Assumption: for an isolated system, all
accessible microstate are equally likely.
5. The probability of a macrostate is proportional to its multiplicity. This
will be sufficient to explain irreversibility.

Lecture3

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Lecture3

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