% miko:type: complete
% miko:title: Complete Statistical Mechanics
% miko:summary: A structured route from ensembles to response and phase transitions.
% miko:tags: statmech
% miko:difficulty: undergrad
% miko:related: microcanonical-ensemble, canonical-ensemble, grand-canonical-ensemble, partition-function, ising-model
% miko:keywords: ensemble, entropy, partition function, fluctuation

\documentclass{article}
\usepackage{amsmath}
\title{Complete Statistical Mechanics}
\begin{document}
\maketitle

\section{Microcanonical Ensemble}

The microcanonical ensemble describes an isolated system at fixed $E,V,N$.
All accessible microstates compatible with these constraints are assigned
equal probability, and entropy is

$$
S=k_B\ln \Omega.
$$

\section{Boltzmann Distribution}

When a system exchanges energy with a heat bath, a state of energy $E_i$ has
weight

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

The exponential weight is the basic bridge from microscopic energy levels to
thermal equilibrium.

\section{Canonical Ensemble}

The canonical ensemble fixes $T,V,N$. Its partition function

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

generates thermodynamics through $F=-k_BT\ln Z$.

\section{Equipartition Theorem}

For classical systems in equilibrium, each independent quadratic degree of
freedom contributes $\frac12 k_BT$ to the mean energy. This explains, within
its classical domain, the heat capacity of monatomic gases.

\section{Grand Canonical Ensemble}

The grand canonical ensemble fixes $T,V,\mu$ and lets particle number
fluctuate. States are weighted by

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

\section{Fluctuation-Response Relation}

Macroscopic response functions are encoded in fluctuations. Heat capacity,
compressibility, and susceptibility measure how equilibrium changes when a
control variable is varied.

\section{Ising Model}

The Ising model places spins $s_i=\pm1$ on a lattice with interactions
between neighbors. It is the canonical minimal model for spontaneous symmetry
breaking and critical behavior.

\end{document}
