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hw02soln

# hw02soln - 1 Homework 2(due October 6 Problem 2-1(3 pts The...

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1 Homework 2 (due October 6) Problem 2-1 (3 pts.) The classical expression for entropy, S = h log p k i = X states p k log p k is derived under assumption that all microstates of the system have the same statistical weight. Generalize this result for the case of states with arbitrary individual statistical weights, w 1 , w 2 , ..., w K . Find the probability distribution which corresponds to the maximum entropy. Note: that the only constrain is, P p k = 1 . Solution: W ( n 1 , n 2 , ..... , n k ) = N ! Y states w n k k n k ! (1) By using the Stirling formula, h S i = log W N = 1 N " log ( N !) + X states log μ w n k k n k ! # = X states n i N log μ n k Nw k = X states p k log p k w k (2) Apply the maximum entropy principle to fi nd the probabilities: δ X states μ p k log p k w k + αp k = 0 , therefore, p k = const · w k = w k P w k Problem 2-2 (3 pts.) Find the free energy and heat capacity C V of the system of N single-atom molecules of mass m each, in the presence of uniform gravitational fi eld g (in 3D).

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