Statistical Mechanics - Physics at Oregon State University

28 CHAPTER 2. THE CANONICAL ENSEMBLE
specify one of them, the other two are determined. We have found two recipes
for these functions. If the energy U is the independent variable, the entropy
S(U) is constructed by counting states with ɛs = U. The temperature T(U)
follows from
∂S
∂U V,N = T −1 . On the other hand, if the temperature T is
the independent variable, we calculate the Boltzmann factor and the partition
function Z(T ). The energy U(T) then follows from the derivative of log(Z),
while the entropy follows from a calculation of the probabilities P(s). Note that
in calculating the partition function we often use the multiplicity function, since
Z(T ) =
E
E − k g(E)e B T (2.24)
which is a simple Laplace transformation. This transformation can be inverted
and used to obtain the multiplicity function g(E) from the partition function,
but that requires an integration in the complex temperature plane. Interesting,
since complex temperature corresponds to time. We obtain factors like e ıEt .
Field theory?? But that is for another course.
2.3 Work and pressure.
A system which remains in a given equilibrium state is rather boring. We would
like to change the state of a system. This can be done by varying some of the
external parameters. For example, if a system is in contact with a reservoir,
we can change the temperature of the reservoir. The system will adjust itself
until it is again in thermal equilibrium with the reservoir. This is an example
of a process in which the system is brought from an equilibrium state with
temperature Ti to an equilibrium state with temperature Tf . What is happening
in between, though? If we suddenly switch the temperature of the reservoir, it
will take some time before thermal equilibrium is restored again. The states in
between are non-equilibrium states and the entropy of the system will increase.
We can also change the temperature very slowly, in such a way that at each time
the system is in thermal equilibrium with the reservoir. A process for which a
system is always in thermal equilibrium in all of the intermediate states is called
reversible. If we reverse the change in temperature of the reservoir, the system
will go back to its initial state. If we suddenly switch the temperature back and
forth, the system will end up in a different state with higher entropy. For the
latter to happen we have to allow other variables to change too, if only T,S, and
U can change the states are completely determined by T. Reversible processes
are nice, since they can easily be described in mathematical terms. It is much
harder to deal with irreversible processes.
An important extensive parameter of a system is its volume. We will in this
chapter still assume that the number of particles does not change. A change
in volume will in general affect the energy eigenvalues of the quantum states in
this system and we write ɛs(V ). If we introduce changes in the volume of the
system, we need a force to produce these changes. We assume that this force is
homogeneous, and hence the only thing we need is an external pressure p. The