Boltzman: Boltzman








Physics 418: Statistical Mechanics I
Prof. S. Teitel ----- Spring 2002

Problem Set 2

Due Tuesday, March 5, in lecture

  • Problem 1

    Consider a system of N distinguishable non-interacting 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 T1<0 comes into thermal contact with another such system (2) with T2>0? Does T1 increase or decrease? Does T2 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 non-interacting 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 = NkBT. [5 points]

    (c) Show that the total average energy is, E = 3NkBT (and hence with (a), E/V = 3P). [5 points]

    (d) Show that the ratio of specific heats is, Cp/CV = 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 = - kB Sumi pi ln pi
    determine the grand canonical probabilities pi, by maximizing the above S subject to the constraints of fixed average E and N. [5 points]

Last update: Monday, August 20, 2007 at 12:09:17 PM.