Consider a classical gas of indistinguishable particles in equilibr...

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Consider a classical gas of indistinguishable particles in equilibrium in the grand canonical ensemble. Label the set of allowed states as {i], where state i has total energy Ei and total number of particles Ni. You can think of state i as representing some small finite cell of classical phase space. The grand canonical partition function can then be written as = = where B = 1/kBT, u is the chemical potential, and 2 = eBH is the fugacity. a) [5 pts] Write an expression that gives the probability Pi for the system to be found in a particular state i. b) [6 pts] What is the probability P(N) that the system has a given number of particles N? Express your answer in terms of L, 2 and the N-particle canonical partition function QN. c) [7 pts] Find general expressions for the average (N), and variance (N2) - (N)2, of the number of particles in terms of appropriate derivatives of In C. Now suppose that the particles are non-interacting. d) [7 pts] Show that the probability P(N) for the system to have N particles has the form, and express \ in terms of 2 and the single-particle partition function Q1- e) [5 pts] Find (N) and (N2) - (N)2 in terms of 2 and Q1.

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