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Physics 418: Statistical Mechanics I
Prof. S. Teitel stte@pas.rochester.edu  Spring 2002
Problem Set 2
Due Tuesday, March 5, in lecture
 Problem 1
Consider a system of N distinguishable noninteracting
objects, each of which can be in one of two possible states,
"up" and "down", with energies +e and e. Assume that N is large.
(a) Working in the microcanonical ensemble, find the entropy
of the system S(E, N) as a function of fixed total energy E and number N (Hint: it
is useful to consider the numbers N_{+} and N_{} of up and
down objects). [5 points]
(b) Find the temperature T as a function of energy E and number N. Show
that T will be negative if E>0. [5 points]
(c) What happens if such a system (1) with T_{1}<0 comes into thermal
contact with another such system (2) with T_{2}>0? Does T_{1}
increase or decrease? Does T_{2} increase or decrease? In
which direction does the heat flow? [5 points]
(d) Working in the canonical ensemble, find the Helmholtz free
energy A(T, N) as a function of temperature T and number N. [5 points]
(e) By making the appropriate transformations on A(T, N), find the
canonical entropy as a function of the average energy E and number N.
Show that your result agrees with your answer for part (a), in the large N
limit. [5 points]
 Problem 2
Consider a classical gas of N indistinguishable noninteracting
particles with ultrarelativistic energies, i.e. their kinetic energy 
momentum relation is given by e = pc, with c the speed of light and p the
magnitude of the particle's momentum.
(a) Compute the canonical partition function for this system. [5 points]
(b) Show that this system obeys the usual ideal gas law, pV =
Nk_{B}T. [5 points]
(c) Show that the total average energy is, E = 3Nk_{B}T
(and hence with (a), E/V = 3P). [5 points]
(d) Show that the ratio of specific heats is, C_{p}/C_{V} =
4/3. [5 points]
 Problem 3
The grand canonical ensemble may be described as one in which the
total energy and total number of particles fluctuate, but in which the
total average energy, E, and the total average number of particles, N,
are fixed. Starting with Shannon's definition of the entropy for a
porbability distribution,
S =  k_{B} Sum_{i} p_{i} ln p_{i}
determine the grand canonical probabilities p_{i},
by maximizing the above S subject to the constraints of fixed average E
and N. [5 points]
