%Kiyoshi Shiraishi:statistical mechanics %
12/07/1997
%

statistical mechanics by example


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%%%%%%%%%%%%%%%
%\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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