% miko:type: atomic
% miko:title: Canonical Ensemble
% miko:summary: The equilibrium ensemble at fixed temperature, volume, and particle number.
% miko:tags: statmech
% miko:difficulty: undergrad
% miko:related: boltzmann-distribution, partition-function, heat-capacity
% miko:keywords: temperature, Helmholtz free energy, energy fluctuation

\documentclass{article}
\usepackage{amsmath}
\title{Canonical Ensemble}
\begin{document}
\maketitle

\section{Definition}

The canonical ensemble fixes $T,V,N$ and assigns probabilities

$$
p_i=\frac{e^{-\beta E_i}}{Z}.
$$

The Helmholtz free energy is

$$
F=-k_BT\ln Z.
$$

\section{Mean Energy}

The average energy is obtained from

$$
\langle E\rangle=-\frac{\partial}{\partial\beta}\ln Z.
$$

Energy fluctuates, but relative fluctuations become small for macroscopic
systems.

\end{document}
