Statistical Mechanics - Physics at Oregon State University

2.3. WORK AND PRESSURE. 29
pressure p is the natural intensive parameter which has to be combined with the
extensive volume. The work W performed on the system to reduce the volume
of the system from V to V − ∆V (with ∆V small) is p∆V .
An example of a reversible process in which the volume of the system is
reduced is a process where the system starts in a given state ˆs and remains in
that state during the whole process. It is clearly reversible, by definition. If we
reverse the change in volume we will still be in the same quantum state, and
hence return to the same state of the system at the original value of the volume.
The energy of the system is ɛˆs(V ) and the change in this energy is equal to the
work done on the system
∂ɛˆs
p∆V = ∆ɛˆs(V ) = −∆V
(2.25)
∂V N
Now consider an ensemble of systems, all with initial volume V and initial
temperature Ti. The number of subsystems in a state s is proportional to the
Boltzmann factor at Ti. We change the volume in a similar reversible process
in which each subsystem remains in the same quantum state. The change in
energy is given by
∆U =
∆ɛsP (s) = p∆V
P (s) = p∆V (2.26)
s
because the probabilities P(s) do not change. Since ∆V is a reduction in
volume we find p = −
∂U
∂V . We are working with two independent variables,
however, and hence we have to specify which variable is constant. Since P(s)
does not change, the entropy does not change according to the formula we
derived before. The temperature, on the other hand, will have to change! Hence
∂U
p = −
(2.27)
∂V
If we take other extensive variables into account we have to keep those the
same too while taking this derivative. The last formula is identical to the formula
for pressure derived in thermodynamics, showing that the thermodynamical and
the statistical mechanical description of volume and pressure are the same. In
thermodynamics we assumed that mechanical work performed on the outside
world was given by ¯ dW = −pdV . In general we have dU = T dS − pdV , but at
constant entropy this reduces to dU = −pdV and the formula for the pressure
follows immediately. In our present statistical mechanical description we used
the same formula, −pdV , for the work done by a system in a specific quantum
state ˆs. But now we assume that the probabilities for the quantum states do
not change, again leading to dU = −pdV . In thermodynamics the entropy is
always used as a mathematical tool, and in order to arrive at the formula for the
pressure we simply set dS = 0. In statistical mechanics the entropy is a quantity
connected to the quantum states of a system. The fact that the entropy did not
change in derivation for the formula for the pressure is a simple consequence of
the requirement that the quantum states keep their original probabilities.
s
S,N