% miko:type: atomic
% miko:title: Grand Canonical Ensemble
% miko:summary: The ensemble for systems exchanging both energy and particles.
% miko:tags: statmech
% miko:difficulty: undergrad
% miko:related: chemical-potential, canonical-ensemble, complete-statistical-mechanics
% miko:keywords: chemical potential, particle number, fugacity

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

\section{Weights}

For fixed $T,V,\mu$, a microstate with energy $E_i$ and particle number $N_i$
has probability proportional to

$$
\exp[-\beta(E_i-\mu N_i)].
$$

The grand partition function is

$$
\Xi=\sum_i e^{-\beta(E_i-\mu N_i)}.
$$

\section{Thermodynamic Potential}

The grand potential is

$$
\Omega=-k_BT\ln \Xi.
$$

It is useful for quantum gases, adsorption, and any system where particle
number fluctuates.

\end{document}
