Statistical Mechanics - Physics at Oregon State University

2 CHAPTER 1. FOUNDATION OF STATISTICAL MECHANICS.
Do we have to know why it works?
In a number of ways, statistical mechanics is like quantum mechanics. In the
latter case, if we assume that a system is described by Schrödinger’s equation,
we can very often find the eigenstates and verify that these eigenstates give a
good description of our experiments. Something similar is true in statistical
mechanics. If we take the standard recipe to calculate the partition function
and find all quantities of interest, often there is a good correspondence with
experimental results. Most textbooks in quantum mechanics follow the approach
mentioned above. They spend perhaps one chapter introducing Schrödinger’s
equation and mainly focus on solutions. Some text books covering statistical
mechanics are based on the same philosophy. They essentially introduce the
partition function and then show many techniques to obtain solutions for specific
problems.
Older textbooks on quantum mechanics spent much more time discussing
the fundamentals of quantum mechanics. There are still essential problems
in this area and one often has to be very careful in interpreting derivations
of Schrödinger’s equation. On the other hand, even today most textbooks on
statistical mechanics still try to justify and explain the fundamental assumptions
in statistical mechanics. Again many problems are present, but not always
discussed.
Traditionally, many textbooks start with classical statistical mechanics. It is
certainly true that most of the development of statistical mechanics was based
on classical physics. It turns out, however, that developing statistical mechanics
is much easier in a discussion based on quantum mechanics. The main focus
of this course is on equilibrium statistical mechanics, which means that we
only need the time-independent Schrödinger equation and talk about stationary
states. Non-equilibrium statistical mechanics and transport theory is in general
a harder subject to study.
Extensive and intensive variables.
Thermodynamics is a study of the macroscopic variables of a large system.
The equations typically contain quantities like the volume V of the system, the
number of particles N, the magnetization M, etc. As we have seen before,
these quantities are extensive parameters. If the size of the system is increased
by a factor of two, but kept in the same state, all these quantities have to
become twice as large. In other words, if we combine two systems, initially we
have to add the quantities of the separate systems. In the last case, it is of
course possible for the total system to go through some process after contact
is made, resulting in a change of the parameters. The combined system is not
necessarily in equilibrium immediately after contact. Note the very important
difference in the usage of N in thermodynamics and statistical mechanics. In
thermodynamics we use N as the number of moles of material. Correspondingly,
the chemical potential is an energy per mole, and the equations of state use the