terry tao on statistical mechanics

This post gives a nice, mathematically clear development of basic terms in statistical mechanics. Highlights:

Think of a physical system as a discrete-time Markov chain with many states.

We assume that the transition matrix is doubly stochastic, i.e., that incoming probabilities sum to 1 just as outgoing probabilities do. This implies that the stationary distribution within each energy band is just the uniform distribution.In physics, the analogue of this fact is Liouville's theorem, and the unitarity of quantum evolution.

Conservation laws mean that the graph is disconnected, since there is (for example) no way to get from a state to another state with different energy.

This distribution on the states of an isolated system is called the microcanonical ensemble of a system $S$ at energy $E$.

If we connect $S$ to a much larger system $S'$, the newly combined system $S \cup S'$ will have some microcanonical ensemble (with some total energy $E$). The canonical ensemble is the marginal distribution of the joint microcanonical ensemble on the original system $S$. Specifically, the probability of a state $x \in S$ with energy $H(x)$ matches the probability of an outer state $x' \in S'$ having energy $H(x') = E - H(x)$, since by conservation of energy those two events must always co-occur.

Let $\Omega(E - H(x))$ be the number of outer states having energy $E - H(x)$, and assume that this grows exponentially in some smooth function $F$ of the energy level: $\Omega(E - H(x)) = \exp(F(E - H(x)))$. By Taylor expanding $F$ around $E$,This is legitimate only if $H(x)$ is very small relative to the total energy $E$, as when $S$ is much smaller than $S \cup S'$.

and dropping the constant $F(E)$, we get the approximation

showing that the canonical ensemble has a single parameter, $\beta = \frac{1}{kT}$.

Tao recommends Schrodinger's "Statistical Thermodynamics" for more.