Summary Problems
Consolidation and Final Exam
The problems in this chapter are mostly repeats to summarize and consolidate the few
basics of equilibrium statistical mechanics. We can consider entropy to be the central
quantity.* The governing principle is the maximization of entropy under constraints (the
MaxEnt principle) that determines probabilities and all associated averages.
Thermodynamic Manipulations
Many thermodynamic relations are established by comparing a physical statement like
dU  TdS  PdV  dN with a mathematical identity like
 U 
 U 
 U 
dU  
 dS  
 dV  
 dN . Variants such as dF   SdT  PdV  dN are
 S V ,N
 V  S ,N
 N  S ,V
treated similarly.
1.
Chemists favor using the Gibbs free energy G because it is conserved under laboratory
conditions of constant temperature, pressure, and particle number. It is defined by
G  F  PV where F  U  TS .
(a) Show that G depends on independent variables T, P, and N.
 G 
 G 
 G 
(b) Derive S  
 , and   
 . Some authors define 
 , V 
 T  P ,N
 P T ,N
 N  P ,T
from the latter expression.
Micro Canonical Distribution
Information theory revealed that alternative outcomes should be taken as equally probable
when there are no constraints on a system other than the normalization condition  pi  1. Equalprobable outcomes are a central premise in the micro-canonical distribution where the multiplicity
of states W plays a central role. Relations with thermal properties are established through the
Boltzmann form of entropy, S  k ln W .
.
2.
Consider a rubber band to be comprised of N links of length a that may point left or
right. The whole chain has a length L as shown.
a
L
L
*
Boltzmann introduced the statistical form of entropy but committed suicide before it was
widely accepted. His gravestone bears the inscription S  k ln W .
Summary Problems
2
(a) Let NR and NL represent the number of links pointing right and left respectively.
Express the multiplicity of states W in terms of N, NR, and NL . Use the expression
S  k ln W and apply the Sterling approximation assuming all numbers are large.
(b) Note that the length and total number of links are constrained. In particular,
L  N R  N L  a
N  NR  NL
Now NR and NL can be expressed in terms of N and L,
N R  12 N  L a 
N L  12 N  L a 
Substitute these into S from part (a). This is the full expression for the entropy of the
rubber band as a function of N and L. The following sections wring physical results
from this entropy.
(c) A fundamental thermodynamic relation applied to this system is
dU  TdS  fdL
where we take f to be the tension applied by the rubber band on the external
environment. Demonstrate that
 S 
f  T 
 L U
(d) Use the last expression to evaluate f.
Canonical Distribution Probabilities
The canonical distribution is by far the most used distribution in statistical physics. It
applies to systems that can exchange energy with the environment but maintain an average energy
overall  Ei pi  U . The following problem calculates a probability based on the canonical
distribution. In general,
pi  g j exp  Ei  / Z where Z   g j exp  E j 
j
and the g’s are statistical weights or multiplicities.
3.
Gene expression begins when a protein, RNA polymerase, binds to a specific site (the
promoter) on the long DNA molecule (the gene). The gene can be modeled as a long
chain of N boxes each of which can bind one of L molecules of RNA polymerase.
Only the promoter site, however, will result in gene expression.
L
RNA polymerase
N sites
promoter site
(a) Write the statistical weights for (i) none of the L molecules to bind the promoter
and for (ii) one molecule to bind the promoter.
Summary Problems
3
(b) Use the statistical weights determine Z and to express the probability that the
promoter will be bound. Since only energy differences can enter, you can take the
non-promoter attachments to have energy 0 and the promoter attachment to have
energy .
(c) Demonstrate the following:
probabilit y of gene expression 
1
N
1    exp   
L
Entropy in the Canonical Distribution
The following problem illustrates the determination of entropy based on probabilities.
4.
Find the entropy for the rubber band problem (problem 1) using the form
S  kN  pi ln pi . The probability of a link pointing to the right is N R N . Apply
2x
N x
the approximation ln 
for x  N to see the entropy behavior for

N
N x
small L extensions. (Stretching the band orders the molecular structure.)
Partition Function Manipulations
The partition function Z and Helmholtz free energy F were seen to be particularly useful
entities for computation. In the following problem we require integrals over phase space and
manipulations to construct the partition function.
5.
An increment of work done by a one-dimensional gas is f dL The combined first and
second law for reversible processes is dE  TdS  f dL .
(a) Show that for this system,
 F 
 F 
f    and S   
 L  T
 T  L

(b) Given  exp(  a p 2 )dp 
1
2
 / a , show that the partition function for a (fictitious) one-
0
dimensional ideal gas of N identical particles in a “box” of length L is
N
1  2m  2 L 

Z
 
N !    h 


(c) Derive the state equation for this one-dimensional gas using F  kT ln Z .
f L  NkT
 F 
(d) Convert the partition function of part (b) to three dimensions and apply 
  P
 V T
1
Summary Problems
4
Radiation and Occupation Number
Planck’s treatment of radiation is often cited as the origin of quantum mechanics. The
description of radiation as a photon gas remains one of the most important statistical mechanical
problems. Here the concept of occupation number is central.
6.
(a) Show that a differential expression for the number of photons in an angular frequency
V
2d
range d is given by dN   2 3
. (A factor of 2 is included because
 c exp  / kT   1
there are two independent directions of polarization and a factor of 1/8 is included to
exclude negative frequencies.
(b) Calculate the energy of the blackbody using
 dN

and show it is proportional to

VT4. Use the integral
x 3dx  4
k 4 2
4
U

aVT

.
[ans.
where
]
a

 e x  1 15
3
15
(

c
)
0
(c) Find the heat capacity at constant volume CV from the result of part (b). Use this to
C dT
4
calculate entropy from the thermodynamic form dS  V
. [ans. S  aVT 3 ]
T
3
P
 S 
(d) Show that 
  and derive the state equation for radiation PV  13 U .
 V U T
(e) Derive the adiabatic condition for radiation.
Chemical Potentials
Chemical potentials are essential to describe diffusive equilibrium. They are also the
signature quantity in the grand canonical distribution.
7.
Free positrons (charge +e mass m) at a temperature T are separated into two compartments
with respective potentials Vhigh and Vlow. Use chemical potentials to derive an expression
for the ratio of positron concentrations n2/n1 where n2 refers to the compartment at lower
potential.
The point counts for the problems are as follows:
1. 10
2. 15
3. 15
4. 10
5. 15
6. 20
7. 15