Statistical Mechanics - Physics at Oregon State University

VIII INTRODUCTION
Introduction.
Thermodynamics and statistical mechanics are two aspects of the study of
large systems, where we cannot describe the majority of all details of that system.
Thermodynamics approaches this problem from the observational side. We
perform experiments on macroscopic systems and deduce collective properties
of these systems, independent of the macroscopic details of the system. In thermodynamics
we start with a few state variables, and discuss relations between
these variables and between the state variables and experimental observations.
Many theorists find this an irrelevant exercise, but I strongly disagree. Thermodynamics
is the mathematical language of experimental physics, and the nature
of this language itself puts constraints on what kind of experimental results are
valid.
Nevertheless, true understanding needs the details on a microscopic level.
We need to describe the behavior of all particles in the sample. That is an
impossible task, and we need to resort to some kind of statistical treatment.
Discussions on the foundations of statistical mechanics are very interesting, and
help us understand the limitations of statistical mechanical treatments. This
is analogue to the situation in quantum mechanics. Textbooks in quantum
mechanics sue two approaches. One can simply start from Schrödinger’s equation
and discuss the mathematical techniques needed to solve this equation for
important models. Or one can start with a discussion of the fundamentals of
quantum mechanics and justify the existence of the Schrödinger equation. This
is always done in less depth, because if one wants to go in great depth the discussion
starts to touch on topics in philosophy. Again, I find that very interesting,
but most practitioners do not need it.
In statistical mechanics one can do the same. The central role is played by
the partition function, and the rules needed to obtain observable data from the
partition function. If these procedures are done correctly, all limitations set by
thermodynamics are obeyed. In real life, of course, we make approximations,
and then things can go wrong. One very important aspect of that discussion is
the concept of the thermodynamic limit. We never have infinitely large samples,
and hence we see effects due to the finite size, both in space and in time. In
such cases the statistical assumptions made to derive the concept of a partition
function are not valid anymore, and we need alternative models. In many cases
the modifications are small, but sometimes they are large. Quantification of
small and large now depends on intrinsic time and length scales of our system.
The important difference between quantum mechanics and statistical mechanics
is the fact that for all atomic systems quantum mechanics is obeyed,
but for many systems the finite size of a sample is important. Therefore, in
statistical mechanics it is much more important to understand what the assumptions
are, and how they can be wrong. That is why we need to start with
a discussion of the foundations. Again, we can go in greater depth that we