Statistical Mechanics - Physics at Oregon State University

50 CHAPTER 3. VARIABLE NUMBER OF PARTICLES
chemical reactions! How do we describe the equilibrium state of a system where
the values of µ, T, and V are specified. A simple observation is based on the
relations between the differentials. If we define Ω = F − µN we find
dΩ = −SdT − pdV − Ndµ (3.33)
Hence Ω has as natural variables T,V, and µ. This is exactly what we need.
Also, for a process at constant µ and T the change in Ω is equal to the work
performed by the system. In other words, Ω measures the amount of energy
available in a system to do work at constant chemical potential and temperature.
The free energy Ω is called the grand potential.
3.4 Grand partition function.
One way of calculating Ω is via the Helmholtz free energy F(T,V,N), obtained
from the partition function. If we solve µ =
∂F
∂N for N(µ, T, V ) we can
T,V
construct Ω(µ, T, V ) = F (T, V, N(µ, T, V )) − µN(µ, T, V ). This assumes that
we are able to invert the formula for µ(T, V, N) and solve for N uniquely. That
this is possible is by no means obvious from a mathematical point of view only.
Of course, in thermodynamics we know that the relation between µ and N
is never decreasing, and hence if we did thermodynamics there would be no
problem. Here we are doing statistical mechanics, though, and we have to show
that the statistical mechanical description is equivalent to the thermodynamical
one.
We will now describe a procedure to calculate the grand potential directly,
and hence in this procedure we find the inverse relation for N. The procedure
is the analogue of what we did to construct the partition function. It is a very
general construction, which works for all types of free energies! Consider a
system S in thermal and diffusive equilibrium with a very large reservoir R.
Both energy and particles can be exchange. The total energy U0 of R + S
and the total number of particles N0 in R + S are constant. Therefore, the
probability of finding the system S in a particular quantum state s with energy
ɛs and number of particles ns is again proportional to the number of quantum
states gR(U0 − ɛs, N0 − ns) the reservoir has available with energy U0 − ɛs and
number of particles N0 − ns. Or
P rob(1)
P rob(2) = gR(U0 − ɛ1, N0 − n1)
gR(U0 − ɛ2, N0 − n2)
(3.34)
The multiplicity function is the exponent of the entropy. If we assume the the
reservoir is very large, the presence of the system causes only small perturbations,
and in first order we find
P rob(1) 1
= exp( (n2 − n1)
P rob(2) kB
∂SR
+
∂N U,V
1
(ɛ2 − ɛ1)
kB
∂SR
) (3.35)
∂U V,N