thermodynamics-and-statistical-physics.tex

\documentclass[main]{subfiles}

\begin{document}

Notation: $\beta = 1/k_BT$

\begin{multicols}{2}

Maxwell-Boltzmann distribution

Bose-Einstein distribution
\begin{gather}
    f_\text{BE}(p) = \frac{1}{e^{E(p)/T} - 1}
\end{gather}
Fermi-Dirac distribution
\begin{gather}
    f_\text{FD}(p) = \frac{1}{e^{E(p)/T} + 1}
\end{gather}
where $E(p)$ is the energy of a particle with the momentum $p$ and the thermal potential is considered to be equal to zero. The average thermal (density) value of some quantity $X$ at temperature $T$ is given by
\begin{gather}
    \langle X\rangle = \frac{g}{(2\pi)^3} \int X(p)f(p) d^3p
\end{gather}



Partition functions
\begin{gather}
    Z[j] := \int\dd{x} P(x) e^{-xj} \implies \langle x^n \rangle = (-1)^n \eval{\pdv[n]{Z[j]}{j}}_{j=0}
\end{gather}

\begin{gather}
    Z = \sum_{\text{conf.}} e^{-\beta H} \implies \langle E \rangle = -\pdv{\log Z}{\beta}
\end{gather}
Sum over all possible states, $H$ being the Hamiltonian of each states.

\end{multicols}

\end{document}