Statistical Mechanics - Physics at Oregon State University

3.4. GRAND PARTITION FUNCTION. 51
The partial derivatives are − µR
TR
and 1
TR
for the reservoir. Since we assume
thermal and diffusive equilibrium, both µ and T are the same for system and
reservoir, TR = TS = T and µR = µS = µ, and hence we the probability of
finding the system with temperature T and chemical potential µ in a quantum
state s is equal to
P rob(s) = 1 1
k e B T
Z (µns−ɛs)
(3.36)
The exponential factor is called the Gibbs factor in this case and is an obvious
generalization of the Boltzmann factor. The normalization factor Z is called the
grand partition function or grand sum and is defined by
Z(T, V, µ) =
states
e 1
k B T (µns−ɛs)
(3.37)
The probabilities are again a ratio of an exponential factor and a normalization
sum. We saw before that the partition function contains all information needed
to describe a system. The grand partition function also contains all information
needed. Sometimes it is easier to work with the partition function, sometimes
with the grand partition function.
Since both partition functions contain all information, it must be possible
to construct one from the other. The grand partition function is related to the
partition function by a sum over particle numbers:
Z(T, V, µ) =
ˆN
e µ ˆ N
k B T Z(T, V, ˆ N) (3.38)
This follows when we split the summation over all states in the grand sum into
two parts, a summation over all possible number of particles and for each term in
that summation a sum over all states with the corresponding number of particles.
The latter gives the partition function. In a later chapter we will give a more
formal justification why this is allowed even in systems where the number of
particles is not conserved. It is related to the invariance of the trace operator in
quantum mechanics. The transformation above is a standard Laplace transform,
and can be inverted. This requires, however, a analytic continuation of the
grand partition function in the complex µ plane. Mathematically, that is not a
problem, but what does that mean in terms of physics?
How do we extract information from the grand partition function? One case
is easy. The average number of particles in the system, < ns >= nsP rob(s):
∂Z
∂µ
T,V
= 1
kBT
s
nse 1
k B T (µns−ɛs) = 1
kBT Z < ns > (3.39)
Therefore, the average number of particles ( now described by the function
N(T, V, µ) ), is directly related to the grand partition function by
∂ log(Z)
N(T, V, µ) = kBT
(3.40)
∂µ
T,V