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+\documentclass[a4paper,11pt]{article}
+
+\usepackage[portrait,margin=0.5in,top=0.5in,bottom=0.5in]{geometry}
+\usepackage{amsmath,multicol,siunitx,amsfonts,adjustbox}
+\usepackage[tiny]{titlesec}
+\usepackage[version=4]{mhchem}
+\usepackage[overload]{abraces}
+
+\usepackage[utf8]{inputenc}
+\usepackage[spanish]{babel}
+\titlespacing{\section}{0pt}{5pt}{0pt}
+\titlespacing{\subsection}{0pt}{5pt}{0pt}
+\titlespacing{\subsubsection}{0pt}{5pt}{0pt}
+\setlength{\parindent}{0pt}
+\setlength{\parskip}{.5em}
+\pagenumbering{gobble}
+\setlength{\columnseprule}{1pt}
+
+\newcommand{\forceindent}{\leavevmode{\parindent=1em\indent}}
+
+\title{\vspace{-1em} Formulari Física Estadística}
+\author{Adrià Vilanova Martínez}
+\date{}
+
+\def\dbar{{\mathchar'26\mkern-12mu d}}
+\everymath{\displaystyle}
+\DeclareSIUnit{\atmosphere}{atm}
+\def\hrulefilll{\leavevmode\leaders\hrule height 1.4pt\hfill\kern 0pt}
+\def\hrulefillll{\leavevmode\leaders\hrule height 2.1pt\hfill\kern 0pt}
+
+\begin{document}
+ \maketitle
+
+ \begin{multicols*}{3}
+ \section{Microcanonical ensemble $(N, V, E)$}
+
+ \subsection{Introduction}
+
+ $S = K_b \log \Omega$.
+
+ \underline{Basic relations:}
+
+ \begin{adjustbox}{width=\textwidth/3}
+ $\begin{array}{cc}
+ \frac{1}{T} = \left( \frac{\partial S}{\partial E} \right)_{N, V} & \frac{P}{T} = \left( \frac{\partial S}{\partial V} \right)_{E, N} \\
+ \mu = - T \left( \frac{\partial S}{\partial N} \right)_{E, V} & C(T) = \frac{\partial E}{\partial T}
+ \end{array}$
+ \end{adjustbox}
+
+ \underline{Potentials:} $\begin{cases}
+ F = E - TS, \\
+ G = F + PV, \\
+ H = E + PV
+ \end{cases}$
+
+ \subsection{Ideal gas}
+
+ $H(\vec{q}, \vec{p}) = \sum_{i = 1}^N H_i(\vec{q_i}, \vec{p_i}) = \sum_{i = 1}^N \frac{p_i^2}{2m}$
+
+ %$H_i(\vec{q_i}, \vec{p_i}) \, \psi_{\vec{n_i}} = \varepsilon_{\vec{n_i}} \, \psi_{\vec{n_i}}$
+
+ $PV = \frac{2}{3} E$ (valid for ideal classic and quantum gasses)
+
+ For a reversible adiabatic process the entropy is constant, and thus: $PV^{\frac{5}{3}} = \text{const.}$
+
+ $C_V = \frac{3}{2} N K_B$, $C_P = \frac{5}{2} N K_B$
+
+ $\mu = K_B T \log\left( \frac{N \lambda^3}{V} \right)$
+
+ $\lambda = \sqrt{\frac{h^2}{2 \pi m K_B T}}$
+
+ $\begin{array}{l}
+ S(E, N, V) = \frac{3}{2} N K_B \left( - \log(\lambda) \right. \\
+ \left. + \frac{2}{3} \log\left( \frac{V}{N} \right) + \frac{5}{3} \right)
+ \end{array}$
+
+ \section{Canonical ensemble $(N, V, T)$}
+
+ \underline{Partition function:}
+
+ $Z = \sum_{E_i} \Omega(E_i) e^{- \beta E_i}$
+
+ $P(E_i) = \frac{1}{Z} \Omega(E_i) e^{- \beta E_i}$
+
+ $\langle E \rangle = - \left( \frac{\partial \log Z}{\partial \beta} \right)_{N, V}$
+
+ $\sigma_E = \sqrt{K_B T^2 C_v}$
+
+ $F = - K_B T \log Z$
+
+ \underline{Thermodynamic relations:}
+
+ $\begin{array}{ll}
+ P = - \left( \frac{\partial F}{\partial V} \right)_{T, N} & S = \left( \frac{\partial F}{\partial T} \right)_{V, N} \\
+ \mu = \left( \frac{\partial F}{\partial N} \right)_{T, V}
+ \end{array}$
+
+ \underline{Identical part.:} $Z(N) = \frac{[Z(1)]^N}{N!}$
+
+ \underline{Localized part.:} $Z(N) = [Z(1)]^N$
+
+ \underline{Single-particle partition function:}
+ $Z(1) = \frac{1}{h^3} \int e^{- \beta H(\vec{q_i}, \vec{p_i})} d\vec{q_i} d\vec{p_i}$
+
+ \underline{Equipartition theorem:} $\langle x_i \cdot \frac{\partial H}{\partial x_j} \rangle = K_B T \delta_{ij}$
+
+ $H = \sum_{i = 1}^n a_i x_i^{\eta_i} \Rightarrow E = K_B T \sum_{i = 1}^{6n} \frac{1}{\eta_i}$
+
+ \section{Grand canonical ensemble $(\mu, V, T)$}
+
+ $Q = \sum_{N = 0}^\infty \sum_E \Omega(N, E) e^{- \beta (E - \mu N)}$
+
+ $P(E, N) = \frac{1}{Q} \Omega(N, E) e^{- \beta (E - \mu N)}$
+
+ \underline{Fugacity:} $z := e^{\beta \mu}$
+
+ $Q = \sum_{N = 0}^\infty z^N \sum_E \Omega(N, E) e^{- \beta E} = \sum_{N = 0}^\infty z^N Z(N)$
+
+ \begin{adjustbox}{width=\textwidth/3}
+ $\begin{cases}
+ Z(N) = \frac{1}{N!} [Z(1)]^N \implies Q = e^{z Z(1)} \\
+ Z(N) = [Z(1)]^N \implies Q = \frac{1}{1 - z Z(1)} \\
+ \end{cases}$
+ \end{adjustbox}
+
+ $\alpha = - \beta \mu$
+
+ $\langle E \rangle = - \left( \frac{\partial \log Q}{\partial \beta} \right)_{\alpha, V}$
+
+ \begin{adjustbox}{width=\textwidth/3}
+ $\sigma_E^2 = \left( \frac{\partial^2 \log Q}{\partial \beta^2} \right)_{\alpha, V} = - \left( \frac{\partial \langle E \rangle}{\partial \beta} \right)_{\alpha, V}$
+ \end{adjustbox}
+
+ \begin{adjustbox}{width=\textwidth/3}
+ $\langle N \rangle = - \left( \frac{\partial \log Q}{\partial \alpha} \right)_{\beta, V} = z \left( \frac{\partial \log Q}{\partial z} \right)_{T, V}$
+ \end{adjustbox}
+
+ \begin{adjustbox}{width=\textwidth/3}
+ $\sigma_N^2 = \left( \frac{\partial^2 \log Q}{\partial \alpha^2} \right)_{\beta, V} = - \left( \frac{\partial \langle N \rangle}{\partial \alpha} \right)_{\beta, V}$
+ \end{adjustbox}
+
+ $\Xi = U - TS - \mu N = - K_B T \log Q$
+
+ \underline{Thermodynamic relations:}
+
+ $\begin{array}{ll}
+ PV = - \Xi & N = - \left( \frac{\partial \Xi}{\partial \mu} \right)_{T, V} \\
+ S = - \left( \frac{\partial \Xi}{\partial T} \right)_{V, \mu}
+ \end{array}$
+
+ (isolating $\mu$ from the first 2 eqs. we get the eq. of state)
+
+ \section{Quantum statistical mechanics}
+
+ $E_k = \sum_i n_i \varepsilon_i$, $Z = \sum_k e^{- \beta E_k}$
+
+ $Z = \sum_{\{ n_i \}} f(\{ n_i \}) e^{- \beta E_k}$
+
+ Distinguishable: $g(\{ n_i \}) = \frac{N!}{n_1! n_2! \cdots n_N!}$
+
+ Indistinguishable: $g(\{ n_i \}) = 1$
+
+ $Q = \prod_i \sum_{n_i = 0}^{n_{i, max}} (z e^{- \beta \varepsilon_i})^{n_i}$
+
+ \section{Miscelanea}
+
+ \underline{Vol($d$-sphere):}
+ $V_d(R) = \frac{\pi^{\frac{d}{2}} R^d}{\Gamma\left( 1 + \frac{d}{2} \right)}$
+
+ \underline{Stirling:} $\log n! \approx n \log n - n$
+
+ $\sum_{n = 0}^N x^n = \frac{x^{N + 1} - 1}{x - 1}$
+
+ \end{multicols*}
+\end{document}