% miko:type: atomic
% miko:title: Boltzmann Distribution
% miko:summary: The equilibrium probability weight for states in contact with a heat bath.
% miko:tags: statmech
% miko:difficulty: undergrad
% miko:related: canonical-ensemble, partition-function, complete-statistical-mechanics
% miko:keywords: probability, beta, heat bath

\documentclass{article}
\usepackage{amsmath}
\title{Boltzmann Distribution}
\begin{document}
\maketitle

\section{Weight}

For a system in thermal contact with a reservoir at temperature $T$, the
probability of a microstate with energy $E_i$ is

$$
p_i=\frac{e^{-\beta E_i}}{Z},
\qquad
\beta=\frac{1}{k_BT}.
$$

\section{Partition Function}

The normalization constant

$$
Z=\sum_i e^{-\beta E_i}
$$

is the partition function. It is more than normalization: derivatives of
$\ln Z$ give thermodynamic quantities.

\end{document}
