Statistical Mechanics - Physics at Oregon State University

1.1. INTRODUCTION. 3
gas constant R. In statistical mechanics we use N for the number of particles
in the system. The chemical potential is now an energy per particle, and the
equations contain the Boltzmann constant kB. We have the following simple
relation, R = NAkB, where NA is Avogadro’s number, the number of particles
in a mole.
If we want to change the volume of a system we have to apply a pressure
p. The work done during a volume change is of the form p∆V . In equilibrium,
the pressure is the same in the whole system, unless other forces are present. If
we take a system in equilibrium and separate it into two parts, both parts will
have the same pressure. Therefore, the pressure is not additive and it is called
an intensive parameter. There is an intensive parameter corresponding to most
extensive parameters (the energy, an extensive parameter, is the exception).
For example, the chemical potential µ is a partner of N, the magnetic field H
pairs with M, etc. These intensive parameters correspond in many cases to the
external handles we have on a system and which can be used to define the state
of a system. The energy contains terms like pV , µN, and H · M.
How is temperature defined?
In thermodynamics we define the temperature T operationally, by how we
can measure it. It clearly is an intensive parameter. The corresponding extensive
parameter is called the entropy S and the energy contains a term of the form
T S. The transport of this form of energy is called heat. Entropy cannot be
measured directly like volume or number of particles, and this is often seen
as a major problem in understanding entropy. But there are other extensive
quantities that are often measured indirectly too! For example, the value for
a magnetic moment follows from the response of a magnetic system to a small
change in magnetic field. Measuring a magnetic moment this way is identical to
measuring the entropy by submitting a system to a small change in temperature.
It is true, however, that for all extensive quantities except entropy we are able to
find some simple physical macroscopic picture which enables us to understand
the meaning of that quantity.
What is the real entropy?
Entropy is associated with randomness or chaos, and these concepts are
harder to put into a simple picture. Nevertheless, this is the path followed in
statistical mechanics. Entropy is defined first, and then temperature simply
follows as the intensive state variable conjugate to the entropy. But how to
define entropy is not clear, and there are several approaches. In a technical
sense, the quantities defined in statistical mechanics are only entropy analogues.
For each proposed definition one needs to show that the resulting equations
are equivalent to those derived in thermodynamics, which in the end describes
experimental reality. We will only be able to do so in the thermodynamic limit,
where the system becomes large. Also, we need to keep in mind that we always