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\begin{document}

\begin{center}{\textbf{\large Thermal and Statistical Physics Cheatsheet} }\\
    \href{https://fc472.user.srcf.net/}{Feiyang}
\end{center}
\begin{multicols*}{3}

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\begin{topic}
    [Thermodynamic potentials]
    \textbf{Gibbs-Duhem}
    \begin{align*}
        U &= TS - pV + \mu N\\
        \dd{U} &= T \dd{S} - p\dd{V} + \mu \dd{N}\\
        \dd{\mu} &= - s \dd{T} + v \dd{p}
    \end{align*}
    \textbf{Availability}
    \begin{gather*}
        \begin{aligned}
            \dd{A} &= \dd{U} - T_R \dd{S} +p_R \dd{V} - \mu_R \dd{N}\\
            &= \pqty{T-T_R} \dd{S} - \pqty{p-p_R} \dd{V} + \pqty{\mu - \mu_R} \dd{N}\\
            &= -T_R \dd{S}_\text{tot}
        \end{aligned}\\
        \dd{A} \leq 0 \iff \dd{S}_\text{tot} \geq 0
    \end{gather*}
    \begin{align*}
        H &= U + pV\\
        F &= U - TS=  - k_BT \ln Z\\
        G &= U- TS +pV = \mu N\\
        \Phi &= U - TS - \mu N = - pV = - k_B T \ln \Xi\\
        \pqty{\dd{}}_\text{natural} &= \dd{A} \\
        \Xi &= \sum e^{-\beta\pqty{E_i - \mu N_i}}
    \end{align*}
\end{topic}

\columnbreak
\begin{topic}
    [Statistical physics]
    \textbf{Statistics}
    \[
        \left\langle n_k \right\rangle = \frac{1}{e^{\beta \pqty{\epsilon_k - \mu}} - \eta}
    \]
    \textbf{Quantum}
    \begin{align*}
        \lambda &=  \pqty{ \frac{2\pi \hbar^2}{ m k_B T} }^{\frac{1}{2}}\\
        n_c &= \frac{1}{\lambda^3}\\
        Z_N &= \frac{1}{N!} Z_1^N\\
        Z &= \frac{1}{N!} \pqty{\frac{V}{\lambda^3}}^N
    \end{align*}
    \textbf{Bose condensation}
    \begin{gather*}
        N = \int n(\epsilon) g(\epsilon) \dd{\epsilon}
    \end{gather*}
    At some low, finite temperature, \(\mu=0\).\\
    \textbf{Virial expansion}
    \begin{align*}
        \mean{\nu} &= \mean{ - \frac{1}{2} \sum_i \mathbf{r}_i \dotproduct \mathbf{f}_i } = \mean{\nu_\text{ext}} = \mean{\nu_\text{int}}\\
        \mean{\nu_i} &= \frac{1}{2} \int_0^\infty 4\pi r^2 n e^{-\beta \phi} r \dv[]{\phi}{r} \dd{r}\\
        \mean{\nu} &= \frac{3}{2} k_B T\\
        \frac{p}{k_B T} &= n+ B_2 (T) n^2 + \dots \\
        B_2 (T) &= \int_0^\infty 2\pi r^2 \pqty{1 - e^{-\beta \phi}} \dd{r}
    \end{align*}
\end{topic}

\columnbreak
\begin{topic}
    [Equilibrium fluctuation]
    \begin{align*}
        \mean{x^2} &=  \frac{1}{Z \beta^2} \pdv[2]{Z}{f} \qquad E = - fx\\
        \mean{ \Delta  M^2} &=  k_B T \pqty{ \pdv[]{\mean{M}}{B}}_T\\
        \mean{\Delta M^2} &= k_B T^2 \pdv[]{U}{T}
    \end{align*}
    \textbf{Fluctuation-dissipation theorem}
    \[
        \mean{\Delta  x^2} = \frac{k_B T_R}{\pqty{\pdv[2]{A}{x}}_{x = x_0}} = \frac{k_B T}{ \pqty{\pdv[]{f}{x}}_{x = x_0}} 
    \]
\end{topic}


\end{multicols*}
\end{document}