Statistical Mechanics - Physics at Oregon State University

30 CHAPTER 2. THE CANONICAL ENSEMBLE
Another subtle point is that here we defined pressure via p = − ∂ɛs
∂V N .
This is a definition in a microscopic sense, and is the direct analogue of what
we would expect based on macroscopic considerations. Because we derive the
same work term as in thermodynamics, this microscopic definition is equivalent
to the macroscopic, thermodynamic one.
Very, very simple model.
An example of a process outlined above is the following. This model is much
too simple, though, and should not be taken too serious. Assume that the energy
eigenvalues of our system depend on volume according to ɛs(V ) = ɛs(V0) V
V0 .
During the process of changing the volume we also change the temperature
and for each volume set the temperature by T (V ) = T0 V . In this case the
V0
probabilities are clearly unchanged because the ratio ɛs
T is constant. Hence in
this very simple model we can describe exactly how the temperature will have
to change with volume when the entropy is constant. The energy is given by
U(V, T (V )) = ɛs(V )P (s) = V
V0 U0(T0) and hence the pressure is p = − U0
V0 .
This is negative!
Why this is too simple.
The partition function in the previous example is given by
and hence the internal energy is
Z(T, V ) = sumse − V ɛs(V 0 )
V 0 k B T = F ( V
U = −kBV F ′ ( V
T )
F ( V
T )
) (2.28)
T
(2.29)
Also, from the formula for the entropy 2.23 we see that S = G( V
T ) and combining
this with the formula for energy we get
U = V H(S) (2.30)
which gives p = − U
V and hence for the enthalpy H = U +pV = 0. This is exactly
why this model is too simple. There are only two intensive state variables, and
only one (T in this case) can be independent. We will need an extra dependence
of the energy on another extensive state variable to construct a realistic system.
The easiest way is to include N, the number of particles.
In general, beware of simple models! They can be nice, because they allow
for quick calculations, and some conclusions are justified. In general, though,
we need a dependence of the energy on at least two extensive state variables to
obtain interesting results which are of practical importance.
Very simple model, revised.