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PHY 418: Statistical Mechanics I
Prof. S. Teitel stte@pas.rochester.edu ---- Spring 2013

Problem Set 6

Due Wednesday, April 17, in lecture

  • Problem 1 [5 points]

    The Stefan-Boltzmann law states that a black body at temperature T radiates power per unit surface area equal to σT4 where σ is a universal constant independent of the material properties of the body. Assuming that the sun and the earth are black bodies, and that the earth is in thermal equilibrium with the sun [i.e. energy absorbed = energy emitted] calclate the temperature of the earth in terms of the temperature of the sun. Look up the parameters you need in order to compute a number for this estimate of the earth's temperature.

  • Problem 2 [30 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 ("a" is labeling the full N-particle state),

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

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

    P({ni}) = ∏i [pi(ni)]
    where pi(ni) is the probability that single particle state i has occupation ni, and pi(ni) is given by
    pi(ni) = [e-(εi-µ)ni/kBT]/wi
    wi = ∑ni [e-(εi-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.

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

    S = -kB{ni} [P({ni}) ln P({ni})] = -kBini [pi(ni) ln pi(ni)]

    (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 = kBi [(1+<ni>) ln (1+<ni>) - <ni> ln <ni>]
    fermions: S = kBi [-(1-<ni>) ln (1-<ni>) - <ni> ln <ni>]
    where <ni> = ∑}ni [ni pi(ni)] is the average occupation number of state i.

  • Problem 3 [30 points]

    Consider an ideal quantum gas of non-interacting identical particles in the grand canonical ensemble. The gas is in equilibrium at temperature T and chemical potential μ. In the previous problem part (a) you found the number of particles ni that occupy the single-particle energy eigenstate i is statistically independent of the number of particles nj in that occupy state j, and you found the probability distribution pi(ni) that there are exactly ni particles in state i.

    a) Using this pi(ni) rederive the average occupation number of particles <ni> in state i, for a gas of bosons and a gas of fermions.

    b) Using pi(ni), derive the occupation number fluctuations <ni2> - <ni>2 for a gas of bosons and a gas of fermions.

    c) If the total number of particles is N = ∑ini, show that the fluctuation in N is give by,

    <N2> - <N>2 = ∑i [ <ni2> - <ni>2 ].
    Are the flutucations in N for the quantum gas bigger or smaller than they are for the corresponding clasical gas, for a gas of bosons? for a gas of fermions?