As a student, more years ago than I care to count, I was bothered by something we were taught in statistical mechanics. Actually, I was bothered by a lot of what we were taught in stat mech — but I’m going to focus on one particular issue in this post.

If you’re a physics student, you almost certainly have encountered something called the “assumption of equal apriori probabilities”. This says that every microstate in the microcanonical ensemble has an equal probability.

Putting aside whether such an assumption is philosophically or physically justified, we can ask whether the statement itself has any meaning at all. I.e. can we define a natural notion of uniform probabilities for the set of microstates?

It turns out that the answer is quite complicated. In these notes, I’ll focus on the case of a classical system with a finite number of degrees of freedom. In getting to the answer (spoiler: it’s “yes” with a lot of caveats), we’ll get to explore various aspects of measure and probability theory, symplectic manifolds, and the Riesz representation theorem.

These notes are meant to be fairly self-contained. They assume some familiarity with basic topology, manifolds, and measure theory, but anything beyond that is included in the discussion.

I’m sure there are typos and possibly even omissions or mistakes. If you spot one, please let me know. I’d like to improve these notes in any way I can. They are primarily for my future self, blessed as I am with an incredibly poor memory. However, I hope that they may of interest and use to others as well.

I’ve included two versions: one with color-coded note-boxes, and one without. The former is useful for visually distinguishing side-notes from core concepts, and the latter is optimized for black and white printing and use with ereaders.

Assumption of Equal apriori Probabilities — Color version (PDF)

Assumption of Equal apriori Probabilities — B&W version (PDF)