\documentstyle[12pt]{article} \newcommand{\be}{\begin{equation}} \newcommand{\ee}{\end{equation}} \newcommand{\bea}{\begin{eqnarray}} \newcommand{\eea}{\end{eqnarray}} \newcommand{\nn}{\nonumber \\} %%%%%%%%%%%%%%% %\hfill {ver. 3.1} %%%%%%%%%%%%%%% \title{ Statistical Mechanics by Examples } \author{Based on Y. Takahashi's textbook, Kiyoshi Shiraishi } \date{ver. 3.1} \begin{document} \maketitle \begin{abstract} Yamaguchi University \end{abstract} {\Large\bf Thermodynamical quantities} %%%%%%%%%%%%%%%%%%%% \begin{center} \begin{tabular}{|c|c|c|}\hline & microcanonical & canonical \\ \hline\hline %0 {\small indep. variables} & $N, E, V$ & $N, \beta, V$ \\ \hline %1 {\small partition function} & $W(N,E,V,\delta E)=\Omega(N,E,V)\delta E$ & $Z=\sum_{\ell_{N}}e^{-\beta E_{\ell_{N}}}$ \\ \hline %3 {\footnotesize prob. distribution fn.} & $f_{MC}=\Omega(N,E,V)\delta E/W$ & $f_{C}=e^{-\beta E_{\ell_{N}}}/Z$ \\ \hline %4 { entropy} & $S=k\ln W$ & $S=k\left(1+T\frac{\partial}{\partial T}\right)\ln Z$ \\ \hline %5 {\footnotesize free energy (Helmholtz)}& $F=E-kT\ln W$ & $F=-kT\ln Z$ \\ \hline %6 {\footnotesize free energy (Gibbs)}& $G=E-kT\left(1-V\frac{\partial}{\partial V}\right)\ln W$ & $G=-kT\left(1-V\frac{\partial}{\partial V}\right)\ln Z$ \\ \hline %6 { internal energy} & $E$ & $E=-\frac{\partial}{\partial \beta}\ln Z$ \\ \hline %7 { temperature} & $T=\left(k\frac{\partial}{\partial E}\ln W\right)^{-1}$ & $T$ \\ \hline %8 { pressure} & $p=kT\frac{\partial}{\partial V}\ln W$ & $p=kT\frac{\partial}{\partial V}\ln Z$ \\ \hline %9 {\small chemical potential} & $\mu=-kT\frac{\partial}{\partial N}\ln W$ & $\mu=-kT\frac{\partial}{\partial N}\ln Z$ \\ \hline %10 {\footnotesize fluctuation of energy} & $\Delta E=0$ & $\Delta E=\frac{\partial^{2}}{\partial \beta^{2}}\ln Z$ \\ \hline %11 \end{tabular} \end{center} \vskip 2cm \begin{center} \begin{tabular}{|c||c|l|l|}\hline & indep. variables & {\tiny thermodynamical var.} & partition function \\ \hline %0 { MC} & $N, E, V$ & $S=k\ln \Omega$ & $\Omega(N,E,V)=\sum_{\ell_{N}}\delta(E-E_{\ell_{N}})$ \\ & & &$=\frac{1}{2\pi i}\int_{-i \infty}^{i\infty}d\beta Z(N,\beta,V) e^{\beta E}$ \\ \hline %1 { C} & $N, \beta, V$ & $F=-\frac{1}{\beta}\ln Z$ & $Z(N,\beta,V)=\sum_{\ell_{N}}e^{-\beta E_{\ell_{N}}}$ \\ & & &$=\int_{0}^{\infty}dE \Omega(N,E,V) e^{-\beta E}$ \\ & & &$=\frac{1}{2\pi i}\int_{-i \infty}^{i\infty}d\alpha Z_{G}(\alpha,\beta,V) e^{\alpha N}$ \\ \hline %2 { GC} & $\alpha=-\frac{\mu}{kT}, \beta, V$ & $pV=\frac{1}{\beta}\ln Z_{G}$ & $Z_{G}(\alpha,\beta,V)=\sum_{N=0}^{\infty}\sum_{\ell_{N}}e^{-\alpha N}e^{-\beta E_{\ell_{N}}}$ \\ & & &$=\sum_{N=0}^{\infty}e^{-\alpha N} Z(N,\beta,V)$ \\ & & &$=\sum_{N=0}^{\infty}\int_{0}^{\infty}dE \Omega(N,E,V) e^{-\alpha N}e^{-\beta E}$ \\ \hline %3 \end{tabular} \end{center} \newpage %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Classical Statistical Mechanics} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{free particles} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{non-relativistic particles} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \be \varepsilon=\frac{\mbox{\boldmath p}^2}{2m} \ee \be \Omega(N,E,V)=e^{5N/2}\left(\frac{V}{N}\right)^{N}\left(\frac{4\pi mE}{3h^{2}N}\right)^{3N/2} \ee \be \alpha=\frac{3}{2}\ln\left[\frac{2\pi m}{h^{2}\beta}\left(\frac{V}{N}\right)^{2/3}\right] \ee \be \beta=\frac{3N}{2E} \ee \be Z(N,\beta,V)=e^{N}\left(\frac{V}{N}\right)^{N}\left(\frac{2\pi m}{h^{2}\beta}\right)^{3N/2} \ee \be Z_{G}(\alpha,\beta,V)=\exp\left[e^{-\alpha}\frac{V}{h^3}\left(\frac{2\pi m}{\beta}\right)^{3/2}\right] \ee %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{phonons} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \be \varepsilon=c |p| \ee \be \Omega(N,E,V)=e^{4N}\left(\frac{V}{N}\right)^{N}\left(\frac{2\pi^{1/3}E}{3hcN}\right)^{3N} \ee \be \alpha=\ln\left[\frac{V}{N}\left(\frac{2\pi^{1/2}}{hc\beta}\right)^{3}\right] \ee \be \beta=\frac{3N}{E} \ee \be Z(N,\beta,V)=e^{N}\left(\frac{V}{N}\right)^{N}\left(\frac{2\pi^{1/3}}{hc\beta}\right)^{3N} \ee \be Z_{G}(\alpha,\beta,V)=\exp\left[e^{-\alpha}\frac{V}{h^3}\left(\frac{2\pi^{1/3}}{c\beta}\right)^{3}\right] \ee %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{harmonic oscillators in classical mechanics} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \be \Omega(f,E)=e^{f}\left(\frac{E}{\hbar\omega f}\right)^{f} \ee \be \alpha=-\ln\left(\hbar\omega\beta\right) \ee \be \beta=\frac{f}{E} \ee \be Z(f,\beta)=\left(\frac{1}{\hbar\omega\beta}\right)^{f} \ee \be Z_{G}(\alpha,\beta)=\frac{\hbar\omega\beta}{\hbar\omega\beta-e^{-\alpha}} \ee %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{harmonic oscillators in quantum mechanics} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \be \Omega(f,E)=\left(\frac{f\varepsilon_{0}+E}{f\varepsilon_{0}}\right)^{f} \left(\frac{f\varepsilon_{0}+E}{E}\right)^{E/\varepsilon_{0}} \ee \be \alpha=\ln\frac{f\varepsilon_{0}+E}{f\varepsilon_{0}} \ee \be \beta=\frac{1}{\varepsilon_{0}}\ln\frac{f\varepsilon_{0}+E}{E} \ee \be Z(f,\beta)=\left(\frac{1}{1-e^{-\beta\varepsilon_{0}}}\right)^{f} \ee %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{rotators in classical mechanics} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \be \Omega(N,E)=e^{N}\left(\frac{2IE}{\hbar^{2}N}\right)^{N} \ee \be \alpha=\ln\left(\frac{2I}{\hbar^{2}\beta}\right) \ee \be \beta=\frac{N}{E} \ee \be Z(N,\beta)=\left(\frac{2I}{\hbar^{2}\beta}\right)^{N} \ee \be Z_{G}(\alpha,\beta)=\frac{\frac{\hbar^{2}\beta}{2I}}{\frac{\hbar^{2}\beta}{2I}-e^{-\alpha}} \ee %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{rotators in quantum mechanics} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \be \Omega(N,E)=\left\{ \begin{array}{ll} \left(\frac{\hbar^{2}N}{\hbar^{2}N-EI}\right)^{N} \left(3\frac{\hbar^{2}N-EI}{EI}\right)^{EI/\hbar^{2}} & \frac{E}{N}<<\frac{\hbar^{2}}{2I} \\ e^{N}\left(\frac{2IE}{\hbar^{2}N}\right)^{N} & \frac{E}{N}>>\frac{\hbar^{2}}{2I} \end{array} \right. \ee \begin{eqnarray} Z(N,\beta)&=&\left[\sum_{\ell=0}^{\infty} (2\ell+1) e^{-\beta\ell(\ell+1)\hbar^{2}/2I}\right]^N\\ &=&\left\{ \begin{array}{ll} \left(1+3 e^{-2\Theta/T}+\cdots\right)^{N} & T<<\Theta \\ \left(T/\Theta\right)^{N} & T>>\Theta \end{array} \right. \end{eqnarray} \be \Theta=\hbar^{2}/2Ik \ee %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{two-level system} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \be \Omega(f,E)=\left(\frac{f\varepsilon_{0}}{f\varepsilon_{0}-E}\right)^{f} \left(\frac{f\varepsilon_{0}-E}{E}\right)^{E/\varepsilon_{0}} \ee \be \alpha=\ln\left(1+e^{-\beta\varepsilon_{0}}\right) \ee \be \beta=\frac{1}{\varepsilon_{0}}\ln\left(\frac{f\varepsilon_{0}}{E}-1\right) \ee \be Z(f,\beta)=\left(1+e^{-\beta\varepsilon_{0}}\right)^{f} \ee \be Z_{G}(\alpha,\beta)=\frac{e^{\alpha}}{e^{\alpha}-e^{-\beta\varepsilon_{0}}-1} \ee \newpage %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Quantum Statistical Mechanics} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{free particles} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \be N=\sum_{s} n_{s} \ee \be E=\sum_{s} \varepsilon_{s} n_{s} \ee %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{Bose particles} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \be W(N,E,V)=\prod_{s}\frac{(n_{s}+g_{s}-1)!}{n_{s}!(g_{s}-1)!} \ee \be Z_{G}(\alpha,\beta,V)=\prod_{\mbox{\boldmath p}} \frac{1}{1-e^{-(\alpha+\beta\varepsilon_{\bf p})}} \ee %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{Fermi particles} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \be W(N,E,V)=\prod_{s}\frac{g_{s}!}{n_{s}!(g_{s}-n_{s})!} \ee \be Z_{G}(\alpha,\beta,V)=\prod_{\mbox{\boldmath p}} \left[1+e^{-(\alpha+\beta\varepsilon_{\bf p})}\right] \ee %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{Bose oscillator} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \be Z_{G}(\alpha,\beta)=\prod_{\ell=0}^{\infty} \frac{1}{1-e^{-(\alpha+\beta\varepsilon_{0}\ell)}} \ee %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{Fermi oscillator} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \be Z_{G}(\alpha,\beta)=\prod_{\ell=0}^{\infty} \left[1+e^{-(\alpha+\beta\varepsilon_{0}\ell)}\right] \ee \newpage %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{ Quantum gas } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{boson gas} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{itemize} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \item {photon gas} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \be \varepsilon=c |p| \ee \be \ln Z_{G}=V\frac{\pi^2}{45}\frac{1}{(\hbar c)^3}\frac{1}{\beta^3} \ee \be E=V\frac{\pi^2}{15}\frac{1}{(\hbar c)^3}\frac{1}{\beta^4} \ee \be p=\frac{\pi^2}{45}\frac{1}{(\hbar c)^3}\frac{1}{\beta^4} \ee %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \item {relativistic massive scalar boson gas} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \be \varepsilon=\sqrt{p^{2}c^{2}+m^{2}c^{4}} \ee %For $\alpha=0$ \be \ln Z_{G}=V\frac{1}{2\pi^2}\frac{m^{2}c^{4}}{\beta (\hbar c)^3}\sum_{n=1}^{\infty} \frac{1}{n^2} K_{2}(\beta nmc^2) \ee %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \item {nonrelativistic scalar boson gas} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \be \varepsilon=\frac{p^{2}}{2m} \ee \be \ln Z_{G}=V\left(\frac{2\pi m}{h^{2}\beta}\right)^{3/2} \ee \end{itemize} \newpage %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{fermion gas} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{itemize} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \item {relativistic massless fermion gas} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% spin $=1/2$ \be \varepsilon=c |p| \ee \be \ln Z_{G}=V\frac{7\pi^2}{360}\frac{1}{(\hbar c)^3}\frac{1}{\beta^3} \ee \be E=V\frac{7\pi^2}{120}\frac{1}{(\hbar c)^3}\frac{1}{\beta^4} \ee \be p=\frac{7\pi^2}{360}\frac{1}{(\hbar c)^3}\frac{1}{\beta^4} \ee \end{itemize} \newpage %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Bose condensation} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{nonrelativistic case} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \be \ln Z_{G}=\frac{V}{h^3}\left(\frac{2\pi m}{\beta}\right)^{3/2}\sum_{n=1}^{\infty} \frac{e^{-\alpha n}}{n^{5/2}} \ee %where $\alpha=\beta\mu$ \be \rho_{0}\approx \rho\left[1-\left(\frac{T}{T_{c}}\right)^{3/2}\right] \ee where \be kT_{c}=\frac{h^{3}}{2\pi m}\left[\frac{\rho}{\zeta(3/2)}\right]^{2/3} \ee %$(\hbar=c=1)$ where $\zeta(3/2)\approx 2.612$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{relativistic case} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% For a charged scalar boson \be \ln Z_{G}=2V\frac{1}{2\pi^2}\frac{m^{2}c^{4}}{\beta (\hbar c)^3}\sum_{n=1}^{\infty} \frac{\cosh n\beta\mu}{n^2} K_{2}(\beta nmc^2) \ee where $\alpha=\beta\mu$ \be \rho_{0}\approx \rho\left[1-\left(\frac{T}{T_{c}}\right)^{2}\right] \ee where \be kT_{c}=\left[\frac{\pi^{2}(\hbar c)^{3}\rho}{2\zeta (2)mc^2}\right]^{1/2} \ee %$(\hbar=c=1)$ \newpage %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{degenerated electron gas} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{nonrelativistic case} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% at low temperature Fermi energy \be \epsilon_{F}=\frac{\hbar^{2}}{2m}\left(3\pi^{2}\rho\right)^{2/3}=\mu_{0} \ee \be E\approx \frac{4\pi V}{h^3}(2m)^{3/2}\frac{2}{5}\mu_{0}^{5/2} \left[1+\frac{5\pi^{2}}{12\beta^{2}\mu_{0}^{2}}+\cdots\right] \ee \be C_{V}=Nk\frac{\pi^{2}}{2}\frac{kT}{\mu_{0}}+\cdots \ee %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{relativistic case} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \be \ln Z_{G}=4V\frac{1}{2\pi^2}\frac{m^{2}c^{4}}{\beta (\hbar c)^3}\sum_{n=1}^{\infty} \frac{(-1)^{n}\cosh n\beta\mu}{n^2} K_{2}(\beta nmc^2) \ee where $\alpha=\beta\mu$ \newpage %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Theory of solids} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{Einstein theory of solids} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Einstein assumed that a solid was composed of $3N$ independent distinguishable quantum oscillators. The Hamiltonian for a single quantum harmonic oscillator is \be \epsilon_{n}=\hbar\omega\left(n+\frac{1}{2}\right) . \ee We have \be Z_{N}(T,V)={\rm Tr} \exp \left[-\beta\hbar\omega \sum_{i=1}^{3N}\left(n_{i}+\frac{1}{2}\right)\right] , \ee or \be Z_{N}(T,V)= \sum_{n_{1}=0}^{\infty}\cdots\sum_{n_{3N}=0}^{\infty} \exp \left[-\beta\hbar\omega \sum_{i=1}^{3N}\left(n_{i}+\frac{1}{2}\right)\right] , \ee which is equivalent to \be Z_{N}(T,V)=\left[\frac{\exp(-\beta\hbar\omega/2)}{ 1-\exp(-\beta\hbar\omega)}\right]^{3N} . \ee The Helmholtz free energy is \bea F(T,V,N)&=&-kT\ln Z_{N} \nn &=&-3NkT\ln\left[\frac{\exp(-\beta\hbar\omega/2)}{ 1-\exp(-\beta\hbar\omega)}\right] \nn &=&\frac{3N\hbar\omega}{2}+3NkT\ln\left[1-\exp(-\beta\hbar\omega)\right] . \eea From this expression, we can find $S$, $C_{V}$, and other state functions. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{Debye theory of solids} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Debye assumed that the oscillators which composed the solids were coupled. We have \be H(\vec{p}^{N},\vec{q}^{N})= \sum_{i=1}^{3N}\frac{p_{i}^{2}}{2m}+\sum_{i,j=1}^{3N}A_{ij}q_{i}q_{j} . \ee The diagonal form is \be H(\vec{P}^{N},\vec{Q}^{N})= \sum_{i=1}^{3N}\frac{P_{i}^{2}}{2m}+\sum_{i=1}^{3N}\frac{m\omega_{i}^{2}}{2}Q_{i}^{2} . \ee Where the $\omega_{i}$'s are all different, the Hamiltonian may be rewritten as \be \hat{H}=\sum_{i=1}^{3N}\hbar\omega_{i}\left(\hat{N}+\frac{1}{2}\right) . \ee Thus, \be Z_{N}(T,V)= \sum_{n_{1}=0}^{\infty}\cdots\sum_{n_{3N}=0}^{\infty} \exp \left[-\beta\hbar \sum_{i=1}^{3N}\left(n_{i}+\frac{1}{2}\right)\omega_{i}\right] , \ee and the Helmholtz free energy is \be F(T,V,N)=\frac{\hbar}{2}\sum_{i=1}^{3N}\omega_{i}+ kT\sum_{i=1}^{3N}\ln\left[1-\exp(-\beta\hbar\omega_{i})\right] . \ee \end{document}

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