Introduction to Boltzmann Statistics

The microcanonical ensemble is a probability space which models physical systems at a fixed energy.

We imagine a physical system whose *states* are described by some set $\Omega$ (which we will assume is a measurable space equipped with $\sigma$-algebra $\mathcal H$). An energy function is a measurable function $E: \Omega \rightarrow \mathbb R$ that is bounded below. Motivated by physics, we expect higher energy states to be less likely than low energy states. States with lowest possible energy (if they exist) are called *ground states*. In physics, only differences in energy are relevant (that is total energy is only defined up to an arbitrary constant) and it can be useful to take $E: \Omega \rightarrow [0, \infty)$ so that ground states are those with zero energy.

Conservation of energy implies that, assuming the system cannot exchange energy with some other system, that it may only transition through states with a given energy. The states with fixed energy $u \in [0,\infty)$ are given by the level set $L_u = E^{-1}(u)$.

**Fundamental Axiom of Statistical Physics:** All states with the same energy are equally likely.

In order to make this mathematically sound, we need to clarify what ‘equally likely’ means. In many common situation, the set of states comes equipped with a natural measure $\mu$. For instance, if the set of states is described by a subset of $\mathbb R^N$ then we naturally have Lebesgue measure $\lambda^N$ on $\mathbb R^N$ at our disposal. In this situation we think of $\lambda^N(L_u) = V_u$ as the volume of $L_u$. If $V_u$ is finite and non-zero, then the uniform probability measure on $L_u$ is given by $V_u^{-1} \lambda^N$, and states chosen with respect to this probability measure are said to be equally likely.

More generally, if $\lambda$ is a given reference measure on $(\Omega, \mathcal H)$, and $V_u = \lambda(L_u) \in (0, \infty)$ then $\mathbb P_u := V_u^{-1} \lambda$ is a probability measure on $L_u$, and states chosen with respect to this probability measure are said to be equally likely. Strictly speaking, in order to have a probability measure, we need a $\sigma$-algebra, and the natural choice is the restriction of $\mathcal H$ to $L_u$ which we denote $\mathcal H_u$.

**Definition:** The microcanonical ensemble with energy $u$ is the probability space $(L_u, \mathcal H_u, \mathbb P_u)$.

#### Particle Models

We will generally be interested in systems consisting of interacting particles. While there are many variations on this theme, a common one is where we have $N$ particles located in some fixed volume $V$ (often a subset of a Euclidean space) equipped with a reference measure $\lambda$ and $\sigma$-algebra $\mathcal H$ (usually Lebesgue or Haar measure) then the set of states $\Omega$ can be identified with $(V^N, \mathcal H^{\otimes N}, \lambda^N)$.

Because of this, the microcanonical ensemble is sometimes thought of an ensemble with fixed energy $u$ and fixed particle number $N$. By allowing either or both to vary, we get the canonical and grand canonical ensembles.