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PHY 418: Statistical Mechanics I
Prof. S. Teitel: stte@pas.rochester.edu ---- Spring 2024

Problem Set 4

Due Monday, March 4, uploaded to Blackboard by 11pm

  • Problem 1 [10 points total]

    Consider N, indistinguishable, non-interacting, non-relativisitic, particles moving in one dimension -- so there is only one coordinate variable x, and one momentum variable p, per particle. The particles move in a box of length L. There is a potential energy in the box U(x) such that U=0 for 0 ≤ x < L/2, and U=U0 for L/2 ≤ x ≤ L. What is the probability that a particular particle will be found in the right half of the box? What is the probability to find M of the particles in the right half of the box?

  • Problem 2 [15 points total]

    Consider the same system as in the previous Problem Set, a system of N distinguishable non-interacting objects, each of which can be in one of two possible states, "up" and "down", with energies +ε and −ε. Assume that N is large.

    (a) Working in the canonical ensemble, find the Helmholtz free energy A(T, N) as a function of temperature T and number N. Note: since these degrees of freedom are distinguishable, there is no Gibbs factor 1/N! when you compute the canonical partition function QN(T).

    Note: Having found Ω(E,N) in the previous Problem Set, you could compute the canoncial QN(T) by taking the Laplace transform of Ω(E,N) with respect to E. Don't do it this way! Instead, compute QN(T) by directly summing the Boltzmann factor over all states in the phase space.

    (b) Starting from A(T, N) of part (a), find the canonical entropy, and then express it as a function of the average energy E and number N. Show that, in the large N limit, your result agrees with your answer for the entropy in the microcanonical ensemble, as computed in the previoius Problem Set.

  • Problem 3 [25 points total]

    Consider a classical gas of N indistinguishable non-interacting particles with ultrarelativistic energies, i.e. their kinetic energy - momentum relation is given by ε = pc, with c the speed of light and p the magnitude of the particle's momentum. The gas is confined to a box of volume V.

    (a) [5 pts] Compute the canonical partition function for this system.

    (b) [5 pts] Show that this system obeys the usual ideal gas law, pV = NkBT.

    (c) [5 pts] Show that the total average energy is, E = 3NkBT (and hence using (b) gives, E/V = 3p).

    (d) [5 pts] Using the canonical partition function of part (a), compute the entropy of the gas.

    (e) [5 pts] Show that the ratio of specific heats is, Cp/CV = 4/3.