Physics 418: Statistical Mechanics I
Prof. S. Teitel ----- Spring 2006

Problem Set 5

Due Friday, April 14, in lecture

  • Problem 1 [15 points total]

    In lecture we saw that the quantum N particle canonical partition function could be written as a series involving 0-particle, 2-particle, 3-particle, ...,and N-particle exchanges. The first two terms in this series were,

    QN(T,V) = (VN/ N! lambda3N) [ 1 ± V -2 Sum
    int d3ri int d3rj f(ri-rj) f(rj-ri)  ] 

    where f(r)=e-pir2/lambda2 and lambda is the thermal wavelength. The (+) sign is for bosons, and the (-) sign is for fermions. The first term above gives the classical partition function, while the second term can be viewed as the leading quantum correction (in the limit that quantum corrections are small). For the calculations below, you may assume that this quantum correction term is small.

    a) Explicitly evaluate the integrals to compute the above partition function. [5 pts]

    b) Find the corresponding Helmholtz free energy. [5 pts]

    c) Using your result in part (b), find the corresponding equation of state. How does the leading quantum correction change the usual ideal gas law, pV = NkBT? [5 pts]

  • Problem 2 [10 points total]

    Consider photons of a given energy epsilon = hbar omega.

    (a) If <n> is the average number of such photons in equilibrium at temperature T, show that the fluctuation in the number of photons is

    <n2> - <n>2 = - (1/epsilon) (d<n>/dbeta)     where beta = 1/kBT
    [5 pts]

    (b) Using the forumula for the equilibrium value of <n>, apply the above result to determine the relative fluctuation in the number of photons

    [<n2> - <n>2]/<n>2
    Is this large or small? [5 pts]

  • Problem 3 [20 points]

    In the grand canonical ensemble, the probability to have a given state "a" with total energy Ea and total number of particles Na is,

    Pa = [e-(Ea-µNa)/kBT]/L

    L = suma [e-(Ea-µNa)/kBT]       is the grand canonical partition function.

    (a) For a quantum ideal gas, with single particle states i of energy epsiloni, many particle states are specified by the occupation numbers {ni} and have energy E = sumi [epsilonini]. Show that the probability for the state with occupations {ni} is given by

    P({ni}) = prodi [pi(ni)]
    where pi(ni) is the probability that single particle state i has occupation ni

    pi(ni) = [e-(epsiloni-µ)ni/kBT]/wi

    wi = sumni [e-(epsiloni-µ)ni/kBT]
    can be thought of as the partition function for the single particle state i. The above factorization says that the number of particles ni in state i, is independent of the number of particles nj in state j. [5 points]

    (b) Using the above result, show that the Shannon definition of entropy can be written as

    S = -kB sum{ni} [P({ni}) ln P({ni})] = -kB sumi sumni [pi(ni) ln pi(ni)]
    [5 points]

    (c) Using the above result, show that the following expressions apply for the entropies of an ideal gas of bosons and fermions, respectively

    bosons: S = kB sumi [(1+<ni>) ln (1+<ni>) - <ni> ln <ni>]
    fermions: S = kB sumi [-(1-<ni>) ln (1-<ni>) - <ni> ln <ni>]
    where <ni> = sumni [ni pi(ni)] is the average occupation number of state i. [10 points]

Last update: Wednesday, August 22, 2007 at 3:59:42 PM.