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Physics 418: Statistical Mechanics I
Prof. S. Teitel stte@pas.rochester.edu ----- 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! 3N)

  [ 1 ± V -2

  i<j

  d3ri

  d3rj

  f(ri-rj) f(rj-ri)

  ]

  where f(r)=e^-r2/2 and 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 = Nk_BT? [5 pts]

• Problem 2 [10 points total]

  Consider photons of a given energy = .

  (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/) (d<n>/d)     where = 1/k_BT

  [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 E_a and total number of particles N_a is,

  P_a = [e^{-(E_a-µN_a)/k_BT}]/L

  where

  L = _a [e^{-(E_a-µN_a)/k_BT}]       is the grand canonical partition function.

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

  P({n_i}) = _i [p_i(n_i)]

  where p_i(n_i) is the probability that single particle state i has occupation n_i

  p_i(n_i) = [e^{-(_i-µ)n_i/k_BT}]/w_i

  where

  w_i = _ni [e^{-(_i-µ)n_i/k_BT}]

  can be thought of as the partition function for the single particle state i. The above factorization says that the number of particles n_i in state i, is independent of the number of particles n_j in state j. [5 points]

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

  S = -k_{B {ni}} [P({n_i}) ln P({n_i})] = -k_{B i ni} [p_i(n_i) ln p_i(n_i)]

  [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 i [(1+<ni>) ln (1+<ni>) - <ni> ln <ni>]
  fermions:	S = kB i [-(1-<ni>) ln (1-<ni>) - <ni> ln <ni>]

  where <n_i> = _ni [n_i p_i(n_i)] is the average occupation number of state i. [10 points]