1Fundamentals of statistical mechanics

II Statistical Physics 1.5 The chemical potential and the grand canonical en-
semble
So far we have considered situations where we had fixed energy or fixed volume.
However, there are often other things that are fixed. For example, the number
of particles, or the electric charge of the system would be fixed. If we measure
these things, then this restricts which microstates are accessible.
Let’s consider
N
. This quantity was held fixed in the microcanonical and
canonical ensembles. But these quantities do depend on N , and we can write
S(E, V, N ) = k log Ω(E, V, N).
Previously, we took this expression, and asked what happens when we varied
E
,
and we got temperature. We then asked what happens when we vary
V
, and we
got pressure. Now we ask what happens when we vary N.
Definition
(Chemical potential)
.
The chemical potential of a system is given
by
µ = T
S
N
E,V
.
Why is this significant? Recall that when we only varied
E
, then we figured
that two systems in equilibrium must have equal temperature. Then we varied
V
as well, and found that two systems in equilibrium must have equal temperature
and pressure. Then it shouldn’t be surprising that if we have two interacting
systems that can exchange particles, then we must have equal temperature,
pressure and chemical potential. Indeed, the same argument works.
If we want to consider what happens to the first law when we vary
N
, we
can just straightforwardly write
dS =
S
E
V,N
dE +
S
V
E,N
dV +
S
N
E,V
dV.
Then as before, we get
dE = T dS p dV + µ dN.
From this expressions, we can get some feel for what
µ
is. This is the energy
cost of adding one particle at fixed
S, V
. We will actually see later that
µ
is
usually negative. This might seem counter-intuitive, because we shouldn’t be
able to gain energy by putting in particles in general. However, this
µ
is the cost
of adding a particle at fixed entropy and volume. In general, adding a particle
will cause the entropy to increase. So to keep
S
fixed, we will have to take out
energy.
Of course, we can do the same thing with other sorts of external variables.
For example, we can change
N
to
Q
, the electric charge, and then we use Φ, the
µ
. The theory behaves in exactly the same way.
From the first law, we can write
µ =
E
N
S,V
.
In the canonical ensemble, we have fixed
T
, but the free energy will also depend
on energy:
F (T, V, N) = E T S.
Again, we have
dF = dE d(T S) = S dT p dV + µ dN.
So we have
µ =
F
N
T,V
.
But in this case, the canonical ensemble is not the most natural thing to consider.
Instead of putting our system in a heat reservoir, we put it in a “heat and
particle” reservoir
R
. In some sense, this is a completely open system it can
exchange both heat and particles with the external world.
As before,
µ
and
T
are fixed by their values in
R
. We repeat the argument
with the canonical ensemble, and we will find that the probability that a system
is in state n is
p(n) =
e
β(E
n
µN
n
)
Z
,
where
N
n
is the number of particles in
|ni
, and we can define the grand canonical
partition function
Z =
X
n
e
β(E
n
µN
n
)
Of course, we can introduce more and more quantities after
V, N
, and then get
more and more terms in the partition function, but they are really just the same.
We can quickly work out how we can compute quantities from
Z
. By writing
out the expressions, we have
Proposition.
hEi µhNi =
Z
β
µ,V
.
Proposition.
hNi =
X
n
p(n)N
n
=
1
β
log Z
µ
T,V
.
As in the canonical ensemble, there is a simple formula for variance:
Proposition.
N
2
= hN
2
i hNi
2
=
1
β
2
2
log Z
µ
2
T,V
=
1
β
hN i
µ
T,V
N.
So we have
N
hNi
1
N
.
So again in the thermodynamic limit, the fluctuations in
hNi
are negligible.
We can also calculate the Gibbs entropy:
Proposition.
S = k
T
(T log Z)
µ,N
.
With the canonical ensemble, we had the free energy. There is an analogous
thing we can define for the grand canonical ensemble.
Definition (Grand canonical potential). The grand canonical potential is
Φ = F µN = E TS µN.
Then we have
Proposition.
dΦ = S dT p dV N dµ.
We thus see that it is natural to view Φ as a function of
T
,
V
and
µ
. Using
the formula for E µN, we find
Φ =
log Z
β
µ,V
= kT log Z,
which is exactly the form we had for the free energy in the canonical ensemble.
In particular, we have
Z = e
βΦ
.