EDIT: I somehow missed that John Wentworth and David Lorell ~~are also in the middle of a sequence~~ have written one post on this same topic here. ~~I will see where this goes from here!~~ This sequence will continue!

Introduction to a sequence on the statistical thermodynamics of some things and maybe eventually everything. This will make more sense if you have a basic grasp on quantum mechanics, but if you're willing to accept "energy comes in discrete units" as a premise then you should be mostly fine.

The title of this post has a double meaning:

Forget the thermodynamics you've learnt before, because statistical mechanics starts from information theory.
The main principle of doing things with statistical mechanics is can be summed up as follows:

Forget as much as possible, then find a way to forget some more.

Particle(s) in a Box

All of practical thermodynamics (chemistry, engines, etc.) relies on the same procedure, although you will rarely see it written like this:

Take systems which we know something about
Allow them to interact in a controlled way
Forget as much as possible
If we have set our systems correctly, the information that is lost will allow us to learn some information somewhere else.

For example, consider a particle in a box.

What does it mean to "forget everything"? One way is forgetting where the particle is, so our knowledge of the particle's position could be represented by a uniform distribution over the interior of the box.

Now imagine we connect this box to another box:

If we forget everything about the particle now, we should also forget which box it is in!

If we instead have a lot of particles in our first box, we might describe it as a box full of gas. If we connect this to another box and forget where the particles are, we would expect to find half in the first box and half in the second box. This means we can explain why gases expand to fill space without reference to anything except information theory.

A new question might be, how much have we forgotten? Our knowledge gas particle has gone from the following distribution over boxes 1 and 2

$P (B o x) = {\begin{matrix} 1 & Box 1 0 & Box 2 \end{matrix}$

To the distribution

$P (B o x) = {\begin{matrix} 0.5 & Box 1 0.5 & Box 2 \end{matrix}$

Which is the loss of 1 bit of information per particle. Now lets put that information to work.

The Piston

Imagine a box with a movable partition. The partition restricts particles to one side of the box. If the partition...

Statistical Mechanics