II Statistical Physics

0 Introduction
In all of our previous physics courses, we mostly focused on “microscopic” physics.
For example, we used the Schr¨odinger equation to describe the mechanics of
a single hydrogen atom. We had to do quite a bit of hard work to figure out
exactly how it behaved. But what if we wanted to study a much huger system?
Let’s say we have a box of hydrogen gas, consisting of
molecules. If we
tried to solve the Schr¨odinger equation for this system, then, even numerically,
it is completely intractable.
So how can we understand such a system? Of course, given these
molecules, we are not interested in the detailed dynamics of each molecule. We
are only interested in some “macroscopic” phenomena. For example, we might
want to know its pressure and temperature. So what we want to do is to describe
the whole system using just a few “macroscopic” variables, and still capture the
main properties we are interested.
In the first part of the course, we will approach this subject rather “rigorously”.
We will start from some microscopic laws we know about, and use them to deduce
properties of the macroscopic system. This turns out to be rather successful,
at least if we treat things sufficiently properly. Surprisingly, it even applies to
scenarios where it is absolutely not obvious it should apply!
Historically, this was not how statistical physics was first developed. Instead,
we only had the “macroscopic” properties we tried to understand. Back in
the days, we didn’t even know things are made up of atoms! We will try to
understand statistical phenomena in purely macroscopic and “wordy” terms,
and it turns out we can reproduce the same predictions as before.
Finally, we will turn our attention to something rather different in nature
phase transitions. As we all know, water turns from liquid to gas as we raise the
temperature. This transition is an example of a phase transition. Of course, this
is still a “macroscopic” phenomena, fitting in line with the theme of the course.
It doesn’t make sense to talk about a single molecule transitioning from liquid
to gas. Interestingly, when we look at many different systems undergoing phase
transitions, they seem to all behave “in the same way” near the phase transition.
We will figure out why. At least, partially.
Statistical physics is important. As we mentioned, we can use this to make
macroscopic predictions from microscopic laws. Thus, this allows us to put our
microscopic laws to experimental test. It turns out methods of statistical physics
have some far-reaching applications elsewhere. It can be used to study black
holes, or even biology!