An introduction to Boltzmann statistics.

- The Microcanonical Ensemble
- The Canonical Ensemble
- The Grand Canonical Ensemble
- Two-Dimensional Electrostatics

### The Microcanonical Ensemble

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… Read More »The Microcanonical Ensemble

### The Canonical Ensemble

The canonical ensemble is a probability space which models physical systems at a fixed temperature, but for which the energy can vary. In this situation we imagine that our system, described by states $(\Omega, \mathcal H, \lambda)$, is allowed to exchange energy with some other system usually referred to as a heat bath. The internal… Read More »The Canonical Ensemble

### The Grand Canonical Ensemble

The grand canonical ensemble is a probability space which models physical systems with a variable number of particles and variable energy at a fixed temperature. Here we imagine our physical system is in contact with a heat reservoir and a particle reservoir. That is, it can exchange energy and particles with some outside system. Much… Read More »The Grand Canonical Ensemble

### Two-Dimensional Electrostatics

In two dimensional electrostatics we imagine infinite parallel wires in three dimensions carrying specified charge densities. By inserting a plane perpendicular to the charged wires we may identify each wire with its intersection point in the plane. We usually refer to these points as particles and not wires, and we often identify the plane with… Read More »Two-Dimensional Electrostatics