Spring 2006
PHY 203: Introduction to Statistical Mechanics
Assignment #4
Due at the beginning of class on Wednesday, 2/22/06
Problems:
1. Free energy of a two state system:
(a) Find an expression for the free energy as a function of T of a system with two states,
one at energy 0 and one at energy . [10 points]
(b) Find expressions for the internal energy and the entropy of the system. [10 points]
(c) Find an expression for the heat capacity. (Assume that in passing between the two
states the system does no work, so the heat capacity is just given by ∂U/∂T .) Find the
limiting behavior at low and high temperatures and make a sketch of C(T ). [10 points]
2. The zipper problem:
A zipper has N links; each link has a state in which it is closed with energy 0 and a state in
which it is open with energy . The zipper can only unzip, however, from the left end: the
link number s can only be open if all links to the left (numbers 1, 2, . . . , s − 1) are open.
(a) Find the partition function of the zipper. (You should sum the series.) [10 points]
(b) In the limit of low temperature, kT , find the average number of open links. The
model is a very simplified model of the unwinding of two-stranded DNA molecules. [10
points]
3. Ideal Gas in the canonical ensemble: use the Hamiltonian of the ideal gas
H(pν , qν ) =
3N
X
p2ν
ν=1
2m
with the momenta enumerated from 1 to 3N to calculate
(a) its partition function (including the Gibbs factor) [15 points]
Z(T, V, N ) =
1
N ! h3N
Z
d3N q d3N p exp (−βH(pν , qν ))
(b) its free energy F , entropy S, pressure p and chemical potential µ [15 points]
4. Ultrarelativstic ideal gas in the canonical ensemble:
Consider an ideal Boltmann gas of indistinguishable particles in the extreme ultrarelativistic
limit, in which the energy E of a particle is related to its momentum p by E = cp (c being
the velocity of light). The gas is confined to a volume V and is in contact with a thermal
bath with temperature T .
(a) For a single particle, find the partition function Z1 (T, V ). [5 points]
(b) Generalize this result to the canonical partition function ZN (T, V ) for N particles.
[5 points]
(c) Derive expressions for the free energy F , the energy U and the pressure p and compare
with those for a non-relativistic ideal gas. [10 points]
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