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…

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…

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…