## Question

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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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