Statistical Mechanics - Physics at Oregon State University
Statistical Mechanics - Physics at Oregon State University
Statistical Mechanics - Physics at Oregon State University
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VIII INTRODUCTION<br />
Introduction.<br />
Thermodynamics and st<strong>at</strong>istical mechanics are two aspects of the study of<br />
large systems, where we cannot describe the majority of all details of th<strong>at</strong> system.<br />
Thermodynamics approaches this problem from the observ<strong>at</strong>ional side. We<br />
perform experiments on macroscopic systems and deduce collective properties<br />
of these systems, independent of the macroscopic details of the system. In thermodynamics<br />
we start with a few st<strong>at</strong>e variables, and discuss rel<strong>at</strong>ions between<br />
these variables and between the st<strong>at</strong>e variables and experimental observ<strong>at</strong>ions.<br />
Many theorists find this an irrelevant exercise, but I strongly disagree. Thermodynamics<br />
is the m<strong>at</strong>hem<strong>at</strong>ical language of experimental physics, and the n<strong>at</strong>ure<br />
of this language itself puts constraints on wh<strong>at</strong> kind of experimental results are<br />
valid.<br />
Nevertheless, true understanding needs the details on a microscopic level.<br />
We need to describe the behavior of all particles in the sample. Th<strong>at</strong> is an<br />
impossible task, and we need to resort to some kind of st<strong>at</strong>istical tre<strong>at</strong>ment.<br />
Discussions on the found<strong>at</strong>ions of st<strong>at</strong>istical mechanics are very interesting, and<br />
help us understand the limit<strong>at</strong>ions of st<strong>at</strong>istical mechanical tre<strong>at</strong>ments. This<br />
is analogue to the situ<strong>at</strong>ion in quantum mechanics. Textbooks in quantum<br />
mechanics sue two approaches. One can simply start from Schrödinger’s equ<strong>at</strong>ion<br />
and discuss the m<strong>at</strong>hem<strong>at</strong>ical techniques needed to solve this equ<strong>at</strong>ion for<br />
important models. Or one can start with a discussion of the fundamentals of<br />
quantum mechanics and justify the existence of the Schrödinger equ<strong>at</strong>ion. This<br />
is always done in less depth, because if one wants to go in gre<strong>at</strong> depth the discussion<br />
starts to touch on topics in philosophy. Again, I find th<strong>at</strong> very interesting,<br />
but most practitioners do not need it.<br />
In st<strong>at</strong>istical mechanics one can do the same. The central role is played by<br />
the partition function, and the rules needed to obtain observable d<strong>at</strong>a from the<br />
partition function. If these procedures are done correctly, all limit<strong>at</strong>ions set by<br />
thermodynamics are obeyed. In real life, of course, we make approxim<strong>at</strong>ions,<br />
and then things can go wrong. One very important aspect of th<strong>at</strong> discussion is<br />
the concept of the thermodynamic limit. We never have infinitely large samples,<br />
and hence we see effects due to the finite size, both in space and in time. In<br />
such cases the st<strong>at</strong>istical assumptions made to derive the concept of a partition<br />
function are not valid anymore, and we need altern<strong>at</strong>ive models. In many cases<br />
the modific<strong>at</strong>ions are small, but sometimes they are large. Quantific<strong>at</strong>ion of<br />
small and large now depends on intrinsic time and length scales of our system.<br />
The important difference between quantum mechanics and st<strong>at</strong>istical mechanics<br />
is the fact th<strong>at</strong> for all <strong>at</strong>omic systems quantum mechanics is obeyed,<br />
but for many systems the finite size of a sample is important. Therefore, in<br />
st<strong>at</strong>istical mechanics it is much more important to understand wh<strong>at</strong> the assumptions<br />
are, and how they can be wrong. Th<strong>at</strong> is why we need to start with<br />
a discussion of the found<strong>at</strong>ions. Again, we can go in gre<strong>at</strong>er depth th<strong>at</strong> we