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

statistical physics


\documentstyle[12pt]{article}
\newcommand{\be}{\begin{equation}}
\newcommand{\ee}{\end{equation}}
\newcommand{\bea}{\begin{eqnarray}}
\newcommand{\eea}{\end{eqnarray}}
\newcommand{\nn}{\nonumber \\}

\newcommand{\vnabla}{{\bf \nabla}}
\newcommand{\vsigma}{{\bf \sigma}}
\newcommand{\vA}{{\bf A}} %vector potential
\newcommand{\vB}{{\bf B}} %
\newcommand{\vD}{{\bf D}}
\newcommand{\vE}{{\bf E}}
\newcommand{\vF}{{\bf F}}
\newcommand{\vg}{{\bf g}}
\newcommand{\vH}{{\bf H}}
\newcommand{\vI}{{\bf I}}
\newcommand{\vi}{{\bf i}}
\newcommand{\vJ}{{\bf J}}
\newcommand{\vj}{{\bf j}}
\newcommand{\vM}{{\bf M}}
\newcommand{\vP}{{\bf P}}
\newcommand{\vS}{{\bf S}}
\newcommand{\vs}{{\bf s}}
\newcommand{\vv}{{\bf v}}
\newcommand{\vx}{{\bf x}}

%%%%%%%%%%%%%%%
%\hfill {ver. 1.0}
%%%%%%%%%%%%%%%
\title{
Statistical Physics 2
}
\author{Based on Tribble's book,
Kiyoshi Shiraishi
}
\date{ver. 1.0}
\begin{document}
\maketitle
\begin{abstract}
Yamaguchi University
\end{abstract}

\section{Quantum gas} %%%%%%%%%%%%%%%%%%%%

\subsection{Partition function}

\be
Z(\beta)=\sum_s e^{-\beta E_s} ,
\ee
where $\beta=(kT)^{-1}$.


\be
P(s)=\frac{e^{-\beta E_s}}{Z}
\ee

\be
Z(\beta)=\sum_{E_s} g(E_s) e^{-\beta E_s}
\ee

\be
P(E_s)=g(E_s)\frac{e^{-\beta E_s}}{Z}
\ee

\be
\langle E\rangle=\frac{\sum_s E_s e^{-\beta E_s}}{Z}=-\frac{\partial\ln Z}{\partial\beta}
\ee

\be
\langle n_s\rangle=-\frac{1}{\beta}\frac{\partial\ln Z}{\partial E_s}
\ee

\be
E=\sum_s n_s E_s
\ee

\be
N=\sum_s n_s
\ee

\be
Z=\sum_{\{n_s\}}\exp\left(-\beta\sum_s n_s E_s\right)
\ee

\subsection{Bose-Einstein gas (bosons)}

\bea
Z&=&\sum_{n_1=0}^{\infty}\sum_{n_2=0}^{\infty}\cdots
\exp\left(-\beta\sum_s n_s E_s\right) \nn
&=&\sum_{n_1=0}^{\infty}\sum_{n_2=0}^{\infty}\cdots
\exp\left(-\beta(n_1 E_1+n_2 E_2+\cdots)\right) \nn
&=&\left\{\sum_{n_1=0}^{\infty}\exp\left(-\beta n_1 E_1\right)\right\}
\left\{\sum_{n_2=0}^{\infty}\exp\left(-\beta n_2 E_2\right)\right\}\cdots \nn
&=&\prod_{s=0}^{\infty}
\left\{\sum_{n_s=0}^{\infty}\exp\left(-\beta n_s E_s\right)\right\} \nn
&=&\prod_{s=0}^{\infty}
\left\{\frac{1}{1-\exp\left(-\beta E_s\right)}\right\} 
\eea

\be
\ln Z=-\sum_{s=0}^{\infty}\ln\left[1-\exp(-\beta E_s)\right]
\ee

\bea
\langle E\rangle&=&-\frac{\partial\ln Z}{\partial\beta} \nn
&=&\frac{\partial}{\partial\beta}\left\{
\sum_{s=0}^{\infty}\ln\left[1-\exp(-\beta E_s)\right]\right\} \nn
&=&\sum_{s=0}^{\infty}\left\{\frac{\partial}{\partial\beta}
\ln\left[1-\exp(-\beta E_s)\right]\right\} \nn
&=&\sum_{s=0}^{\infty}\left\{\frac{E_s}{\exp(\beta E_s)-1}\right\}
\eea

\bea
\langle n_s\rangle&=&-\frac{1}{\beta}\frac{\partial\ln Z}{\partial E_s} \nn
&=&\frac{1}{\beta}\frac{\partial}{\partial E_s}\left\{
\sum_{s'=0}^{\infty}\ln\left[1-\exp(-\beta E_{s'})\right]\right\} \nn
&=&\frac{1}{\beta}\frac{\partial}{\partial E_s}\left\{
\ln\left[1-\exp(-\beta E_{s})\right]\right\} \nn
&=&\frac{\exp(-\beta E_s)}{1-\exp(-\beta E_s)} \nn
&=&\frac{1}{\exp(\beta E_s)-1}
\eea

\subsubsection{Planck's law}

\be
f_{N,BE}(\omega)d\omega=\frac{dN_{\omega}}{dV}=
\frac{1}{\pi^2 c^3}\frac{\omega^2}{\exp(\beta\hbar\omega)-1}d\omega
\ee

\be
f_{E,BE}(\omega)d\omega=\frac{dN_{\omega}}{dV}=
\frac{\hbar}{\pi^2 c^3}\frac{\omega^2}{\exp(\beta\hbar\omega)-1}d\omega
\ee

\subsection{Fermi-Dirac gas (fermions)}

\bea
Z&=&\sum_{n_1=0}^{1}\sum_{n_2=0}^{1}\cdots
\exp\left(-\beta\sum_s n_s E_s\right) \nn
&=&\sum_{n_1=0}^{1}\sum_{n_2=0}^{1}\cdots
\exp\left(-\beta(n_1 E_1+n_2 E_2+\cdots)\right) \nn
&=&\left\{\sum_{n_1=0}^{1}\exp\left(-\beta n_1 E_1\right)\right\}
\left\{\sum_{n_2=0}^{1}\exp\left(-\beta n_2 E_2\right)\right\}\cdots \nn
&=&\prod_{s=0}^{\infty}
\left\{1+\exp\left(-\beta E_s\right)\right\} 
\eea

\be
\ln Z=\sum_{s=0}^{\infty}\ln\left[1+\exp(-\beta E_s)\right]
\ee

\bea
\langle E\rangle&=&-\frac{\partial\ln Z}{\partial\beta} \nn
&=&-\frac{\partial}{\partial\beta}\left\{
\sum_{s=0}^{\infty}\ln\left[1+\exp(-\beta E_s)\right]\right\} \nn
&=&-\sum_{s=0}^{\infty}\left\{\frac{\partial}{\partial\beta}
\ln\left[1+\exp(-\beta E_s)\right]\right\} \nn
&=&\sum_{s=0}^{\infty}\left\{\frac{E_s}{\exp(\beta E_s)+1}\right\}
\eea

\bea
\langle n_s\rangle&=&-\frac{1}{\beta}\frac{\partial\ln Z}{\partial E_s} \nn
&=&-\frac{1}{\beta}\frac{\partial}{\partial E_s}\left\{
\sum_{s'=0}^{\infty}\ln\left[1+\exp(-\beta E_{s'})\right]\right\} \nn
&=&-\frac{1}{\beta}\frac{\partial}{\partial E_s}\left\{
\ln\left[1+\exp(-\beta E_{s})\right]\right\} \nn
&=&\frac{\exp(-\beta E_s)}{1+\exp(-\beta E_s)} \nn
&=&\frac{1}{\exp(\beta E_s)+1}
\eea

\section{Counting statistics} %%%%%%%%%%%%%%%%%%%%

\be
\Omega_{\Delta E}=\sum_{\{n_i\}}W(\{n_i\})
\ee
The sum is over all sets $\{n_i\}$ such that $\sum_i n_i \epsilon_i=E$,
$\sum_i n_i=N$.

\be
\ln\Omega_{\Delta E}(E,V,N)\approx\ln W(\{\bar{n}_i\})
\ee

\be
f(n_i)=\ln W(\{n_i\})-\alpha\sum_i n_i-\beta\sum_i \epsilon_i n_i
\ee

\be
\frac{\partial f}{\partial n_i}=0\rightarrow n_i=\bar{n}_i
\ee

\subsection{Maxwell-Boltzmann gas (distinguishable particles)}

$\prod_i g_i^{n_i}=$number of ways $n_i$ distinguishable objects 
can occupy $g_i$ levels with no restrictions

$\prod_i n_i !=$number of ways to place $N$ objects into cells
such that there are $n_i$ objects in the $i$th cell.

$N !=$number of equivalent permutations

\be
W(\{n_i\})=\frac{N!\prod_i g_i^{n_i}}{\prod_i n_i!}
\ee

\be
\Omega_{\Delta E}=\sum_{\{n_i\}}\frac{N!\prod_i g_i^{n_i}}{\prod_i n_i!}
\ee

\be
f(n_i)=\ln\frac{N!\prod_i g_i^{n_i}}{\prod_i n_i!}-
\alpha\sum_i n_i-\beta\sum_i \epsilon_i n_i
\ee

\be
f(n_i)\approx(N\ln N-N)+\sum_i\left[n_i\ln g_i-(n_i\ln n_i-n_i)-
\alpha n_i-\beta\epsilon_i n_i\right]
\ee

\be
\ln g_i-\ln n_i-\alpha-\beta\epsilon_i=0
\ee

\be
\frac{g_i}{\bar{n}_i}=e^{\alpha}e^{\beta\epsilon_i}
\ee

\be
\bar{n}_i=g_i e^{-\alpha-\beta\epsilon_i}=g_i e^{-\beta(\epsilon_i-\mu)},
\ee
where $-\beta\mu=\alpha$.

\subsection{Bose-Einstein gas (bosons)}

$\frac{(n_i+g_i-1)!}{n_i!(g_i-1)!}=$number of ways to choose $g_i$ objects 
such that the sum is $n_i$ objects.

\be
W(\{n_i\})=\prod_i \frac{(n_i+g_i-1)!}{n_i!(g_i-1)!}
\ee

\be
\Omega_{\Delta E}=\sum_{\{n_i\}}\prod_i \frac{(n_i+g_i-1)!}{n_i!(g_i-1)!}
\ee

\be
f(n_i)=\ln\prod_i \frac{(n_i+g_i-1)!}{n_i!(g_i-1)!}-
\alpha\sum_i n_i-\beta\sum_i \epsilon_i n_i
\ee

\be
f(n_i)=\sum_i\left[\ln (n_i+g_i-1)!-\ln n_i!-\ln (g_i-1)!-
\alpha n_i-\beta\epsilon_i n_i\right]
\ee

\bea
f(n_i)&\approx&\sum_i\left[(n_i+g_i-1)\ln (n_i+g_i-1)-(n_i+g_i-1)\right. \nn
& &-(n_i\ln n_i-n_i)-((g_i-1)\ln (g_i-1)-(g_i-1)) \nn
& &-\left.\alpha n_i-\beta\epsilon_i n_i\right]
\eea

\be
\ln (n_i+g_i)-\ln n_i-\alpha-\beta\epsilon_i=0
\ee

\be
\frac{\bar{n}_i+g_i}{\bar{n}_i}=e^{\beta(\epsilon_i-\mu)}
\ee

\be
\bar{n}_i=\frac{g_i}{e^{\beta(\epsilon_i-\mu)}-1},
\ee
where $-\beta\mu=\alpha$.


\subsection{Fermi-Dirac gas (fermions)}

$\frac{g_i!}{n_i!(g_i-n_i)!}=$number of ways to choose $n_i$ objects 
out of $g_i$ objects.

\be
W(\{n_i\})=\prod_i \frac{g_i!}{n_i!(g_i-n_i)!}
\ee

\be
\Omega_{\Delta E}=\sum_{\{n_i\}}\prod_i \frac{g_i!}{n_i!(g_i-n_i)!}
\ee

\be
f(n_i)=\ln\prod_i \frac{g_i!}{n_i!(g_i-n_i)!}-
\alpha\sum_i n_i-\beta\sum_i \epsilon_i n_i
\ee

\be
f(n_i)=\sum_i\left[\ln g_i!-\ln n_i!-\ln (g_i-n_i)!-
\alpha n_i-\beta\epsilon_i n_i\right]
\ee

\bea
f(n_i)&\approx&\sum_i\left[g_i\ln g_i-g_i\right. \nn
& &-(n_i\ln n_i-n_i)-((g_i-n_i)\ln (g_i-n_i)-(g_i-n_i)) \nn
& &-\left.\alpha n_i-\beta\epsilon_i n_i\right]
\eea

\be
-\ln n_i+\ln (g_i-n_i)-\alpha-\beta\epsilon_i=0
\ee

\be
\frac{g_i-\bar{n}_i}{\bar{n}_i}=e^{\beta(\epsilon_i-\mu)}
\ee

\be
\bar{n}_i=\frac{g_i}{e^{\beta(\epsilon_i-\mu)}+1},
\ee
where $-\beta\mu=\alpha$.

\section{Phase transitions}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Bose-Einstein condensation}

\be
Z_{G}{}_{BE}(T,V,\mu)=\prod_{\ell=0}^{\infty}\left(
\frac{1}{1-e^{-\beta(\epsilon_{\ell}-\mu)}}\right)
\ee

\be
\langle N\rangle=\sum_{\ell=0}^{\infty}\left(
\frac{1}{e^{\beta(\epsilon_{\ell}-\mu)}-1}\right)=
\sum_{\ell=0}^{\infty}\langle n_{pl}\rangle
\ee

\be
\langle n_{pl}\rangle=\frac{1}{e^{\beta\epsilon_{\ell}}z^{-1}-1}
\ee

\be
\langle n_{0}\rangle=\frac{z}{1-z}\stackrel{z\rightarrow 1}{\rightarrow}\infty
\ee

\be
\Omega_{BE}(T,V,\mu)=-kT\ln Z_{BE}(T,V,\mu)
\ee

\be
\Omega_{BE}(T,V,\mu)=kT\sum_{\ell=0}^{\infty}
\ln\left(1-e^{-\beta(\epsilon_{\ell}-\mu)}\right)
\ee

\be
\frac{\Omega}{V}=\frac{kT}{V}\ln (1-z)+\frac{kT}{V}\sum_{\ell=0}^{\infty}
\ln\left(1-e^{-\beta(\epsilon_{\ell}-\mu)}\right)
\ee

\be
\sum_{\ell}\rightarrow\frac{V}{(2\pi)^3}\int d\vec{k}_{\ell}=
\frac{V}{h^3}\int d\vec{p}_{\ell}
\ee

\be
\frac{\Omega}{V}=\frac{kT}{V}\ln (1-z)+\frac{kT}{V}\frac{V}{h^3}\int d\vec{p}_{\ell}
\ln\left(1-e^{-\beta(\epsilon_{\ell}-\mu)}\right)
\ee

\be
\frac{\Omega}{V}=\frac{kT}{V}\ln (1-z)+\frac{kT}{V}\frac{V}{h^3}\int d\vec{p}_{\ell}
\ln\left(1-ze^{-\beta p^2/2m}\right)
\ee

\be
\frac{\Omega}{V}=\frac{kT}{V}\ln (1-z)+\frac{kT}{V}\frac{V}{h^3}\int d\vec{x}
\left(\frac{2m}{\beta}\right)^{1/2}\ln\left(1-ze^{-\beta x^2}\right)
\ee

\be
\frac{\Omega}{V}=\frac{kT}{V}\ln (1-z)+\frac{kT}{h^3}\left(\frac{2m}{\beta}\right)^{1/2}
4\pi\int x^2\ln\left(1-ze^{-\beta x^2}\right)dx
\ee

\be
\frac{\Omega}{V}=\frac{kT}{V}\ln (1-z)-\frac{kT}{\lambda_T^3}g_{5/2}(z),
\ee
where $\lambda_T=\left(\frac{2\pi\hbar^2}{mkT}\right)^{1/2}$ and

\be
g_{5/2}(z)=-\frac{4}{\sqrt{\pi}}=\frac{kT}{V}\int_0^{\infty}
dx x^2\ln\left(1-ze^{-\beta x^2}\right)=\sum_{\alpha=1}^{\infty}
\frac{z^{\alpha}}{\alpha^{5/2}}
\ee

\be
\frac{\langle N\rangle}{V}=\frac{\langle n_0\rangle}{V}+\sum_{\ell=0}^{\infty}
\left(\frac{1}{z^{-1}e^{\beta\epsilon_{\ell}}-1}\right)
\ee

\be
\frac{\langle N\rangle}{V}=\frac{\langle n_0\rangle}{V}+
\frac{1}{\lambda_T^{3}}g_{3/2}(z),
\ee
where
\be
g_{3/2}(z)=z\frac{\partial}{\partial z}g_{5/2}(z)=\sum_{\alpha=1}^{\infty}
\frac{z^{\alpha}}{\alpha^{3/2}}
\ee

\be
\left(\frac{\langle N\rangle}{V}\lambda_T^{3}\right)_{z=0}=
g_{3/2}(0)=0
\ee

\be
\left(\frac{\langle N\rangle}{V}\lambda_T^{3}\right)_{z=1}=
g_{3/2}(1)=\zeta\left(\frac{3}{2}\right)=2.612
\ee

\be
\frac{\langle N\rangle}{V}\lambda_T^{3}=g_{3/2}(z),
\ee
for $\frac{\langle N\rangle}{V}\lambda_T^{3}<2.612$, and

\be
\frac{\langle N\rangle}{V}\lambda_T^{3}=
\frac{\langle n_0\rangle}{V}\lambda_T^{3}+g_{3/2}(1),
\ee
for $\frac{\langle N\rangle}{V}\lambda_T^{3}\ge 2.612$.

\be
\lambda_T^{3}\ge 2.612\frac{V}{\langle N\rangle}
\ee

\be
T_c=\left(\frac{2\pi\hbar^2}{mk}\right)
\left(\frac{\langle N\rangle}{2.612 V}\right)^{2/3}
\ee

\be
\left(\frac{\langle N\rangle}{V}\right)_c=2.612
\left(\frac{mkT}{2\pi\hbar^2}\right)^{3/2}
\ee

\subsection{The Ising model}

\be
E(\sigma_1,\sigma_2,\dots,\sigma_n)=
-J\sum_{j=1}^N \sigma_j\sigma_{j+1}-H\sum_{j=1}^N \sigma_j,
\ee
where $\sigma_j=\pm 1$.

\be
Z_N=\sum_{\sigma}e^{-\beta E(\sigma)}=\sum_{\sigma}\exp\left\{-\beta
\left[-J\sum_{j=1}^N \sigma_j\sigma_{j+1}-H\sum_{j=1}^N \sigma_j
\right]\right\}
\ee

\be
Z_N=\sum_{\sigma_1=\pm 1}\cdots\sum_{\sigma_N=\pm 1}\left[
e^{h\sigma_1/2}e^{K\sigma_1\sigma_2}e^{h\sigma_2/2}
e^{h\sigma_2/2}e^{K\sigma_2\sigma_3}e^{h\sigma_3/2}\cdots
\right],
\ee
where $K=\frac{J}{kT}$, and $h=\frac{H}{kT}$.

\be
Z_N=\sum_{\sigma_1=\pm 1}\cdots\sum_{\sigma_N=\pm 1}
V(\sigma_1,\sigma_2)V(\sigma_2,\sigma_3)\cdots V(\sigma_N,\sigma_1),
\ee
where
\be
V(\sigma,\sigma')=\exp\left[K\sigma\sigma'+\frac{h}{2}(\sigma+\sigma')\right]
\ee

\be
V(\sigma,\sigma')=\left(
\begin{array}{cc}
 e^{K+h} & e^{-K}\\
 e^{-K}  & e^{K-h}
\end{array}
\right)
\ee

\be
\det\left|
\begin{array}{cc}
 e^{K+h}-\lambda & e^{-K}\\
 e^{-K}  & e^{K-h}-\lambda
\end{array}
\right|=
(e^{K+h}-\lambda)(e^{K-h}-\lambda)-e^{-2K}=0
\ee

\be
\lambda^2-e^K(e^h+e^{-h})\lambda+(e^{2K}-e^{-2K})=0
\ee

\be
\lambda=e^K\frac{(e^h+e^{-h})}{2}\pm\frac{1}{2}\left[
e^{2K}(e^h+e^{-h})^2-4(e^{2K}-e^{-2K})\right]^{1/2}
\ee

\be
\lambda=e^K\cosh h\pm\left[
e^{2K}\cosh^2 h-(e^{2K}-e^{-2K})\right]^{1/2}
\ee

\be
\lambda_1=e^K\cosh h+\left[
e^{2K}\sinh^2 h+e^{-2K}\right]^{1/2}
\ee

\be
\lambda_2=e^K\cosh h-\left[
e^{2K}\sinh^2 h+e^{-2K}\right]^{1/2}
\ee

\be
V=P\left(
\begin{array}{cc}
 \lambda_1 & 0\\
 0  & \lambda_2
\end{array}
\right)P^{-1}
\ee

\be
Z_N={\rm Tr}V^N={\rm Tr}\left[
P\left(
\begin{array}{cc}
 \lambda_1 & 0\\
 0  & \lambda_2
\end{array}
\right)P^{-1}
P\left(
\begin{array}{cc}
 \lambda_1 & 0\\
 0  & \lambda_2
\end{array}
\right)P^{-1}
\cdots
P\left(
\begin{array}{cc}
 \lambda_1 & 0\\
 0  & \lambda_2
\end{array}
\right)P^{-1}
\right]
\ee

\be
Z_N={\rm Tr}\left(
\begin{array}{cc}
 \lambda_1 & 0\\
 0  & \lambda_2
\end{array}
\right)^{N}=
\lambda_1^N+\lambda_2^N
\ee

\be
F=-kT\ln Z=-kT\ln\left[\lambda_1^N\left[
1+\left(\frac{\lambda_2}{\lambda_1}\right)^N\right]\right]
\ee

\be
F=-NkT\ln\lambda_1-kT\ln\left[
1+\left(\frac{\lambda_2}{\lambda_1}\right)^N\right]
\ee

\be
\lim_{N\rightarrow\infty}
\left(\frac{\lambda_2}{\lambda_1}\right)^N=0
\ee

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

\section{Ground state properties of the electron gas}%%%%%%%%%%%%

\be
k_i=\frac{2\pi}{L}
\ee

\be
\frac{\Omega}{(2\pi/L)^3}=\frac{\Omega V}{8\pi^3}
\ee

\be
\left(\frac{4\pi k_F^3}{3}\right)\left(\frac{V}{8\pi^3}\right)=
\frac{k_F^3}{6\pi^2}V
\ee

\be
N=2\frac{k_F^3}{6\pi^2}V=\frac{k_F^3 V}{3\pi^2}
\ee

\be
n=\frac{N}{V}=\frac{k_F^3}{3\pi^2}
\ee

\subsection{Definitions}

\begin{itemize}

\item Fermi sphere: Sphere of radius $k_F$ containing the occupied one-electron levels

\item Fermi surface: Surface of the Fermi sphere, separates occupied from unoccupied levels.

\item Fermi momentum: $p_F=\hbar k_F$

\item Fermi energy: $E_F=\frac{\hbar^2 k_F^2}{2m}$

\item Fermi velocity: $v_F=\frac{p_F}{m}$

\item Fermi temperature: $T_F=\frac{E_F}{k}$

\end{itemize}

\end{document} 

߂