4.6: Statistical Ensembles from Maximum Entropy

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\)
\( \newcommand\Vpi

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\)
\( \newcommand\Vvarpi

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\)
\( \newcommand\Vrho

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\)
\( \newcommand\Vvarrho

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\)
\( \newcommand\Vsigma

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\)
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\)
\( \newcommand\Vtau

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\)
\( \newcommand\Vupsilon

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\)
\( \newcommand\Vphi

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\)
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\)
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\)
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\)
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\)
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\)
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\)
\( \newcommand\VTheta{\vec\Theta}\)
\( \newcommand\VLambda{\vec\Lambda}\)
\( \newcommand\VXi{\vec\Xi}\)
\( \newcommand\VPi{\vec\Pi}\)
\( \newcommand\VSigma{\vec\Sigma}\)
\( \newcommand\VUps{\vec\Upsilon}\)
\( \newcommand\VPhi{\vec\Phi}\)
\( \newcommand\VPsi{\vec\Psi}\)
\( \newcommand\VOmega{\vec\Omega}\)
\( \newcommand\BA{\mib A}\)
\( \newcommand\BB{\mib B}\)
\( \newcommand\BC{\mib C}\)
\( \newcommand\BD{\mib D}\)
\( \newcommand\BE{\mib E}\)
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\( \newcommand\BG{\mib G}\)
\( \newcommand\BH{\mib H}\)
\( \newcommand\BI{\mib I}}\)
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\( \newcommand\BK{\mib K}\)
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\( \newcommand\BP{\mib P}\)
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\( \newcommand\BT{\mib T}\)
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\( \newcommand\BGamma{\mib\Gamma}\)
\( \newcommand\BDelta{\mib\Delta}\)
\( \newcommand\BTheta{\mib\Theta}\)
\( \newcommand\BLambda{\mib\Lambda}\)
\( \newcommand\BXi{\mib\Xi}\)
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\( \newcommand\BSigma{\mib\Sigma}\)
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\( \newcommand\Bxhi{\raise.35ex\hbox{$\Bchi$}}\)
\( \newcommand\RGamma{ \Gamma}\)
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\( \newcommand\RA{ A}\)
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\( \newcommand\RL{ L}\)
\( \newcommand { M}\)
\( \newcommand\RN{ N}\)
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\( \newcommand\RV{ V}\)
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\( \newcommand\RY{ Y}\)
\( \newcommand\RZ{ Z}\)
\( \newcommand\Ra{ a}\)
\( \newcommand\Rb{ b}\)
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\( \newcommand\Ri{ i}\)
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\( \newcommand\Rk{ k}\)
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\( \newcommand { m}\)
\( \newcommand\Rn{ n}\)
\( \newcommand\Ro{ o}\)
\( \newcommand\Rp{ p}\)
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\( \newcommand\BBA{\boldsymbol\RA}\)
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\( \def\texpect#1#2#3{\langle #1 |   #2  |  #3 \rangle}\)
\( \def\sket#1{|  \, #1 \,  \rangle}\)
\( \def\sbra#1{\langle \,  #1 \, |}\)
\( \def\sbraket#1#2{\langle \, #1  \, |  \, #2 \,  \rangle}\)
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\(\def\ket#1{\big| \, #1\, \big\rangle}\)
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\( \newcommand\pz{\partial}\)
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\( \newcommand\svph{\vphantom{\int}}\)
\( \newcommand\vph{\vphantom{\sum_i}}\)
\( \newcommand\bvph{\vphantom{\sum_N^N}}\)
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\(\newcommand \gtwid{\approx}\)

\( \newcommand\index{\textsf{ind}}\)
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\( \newcommand\bsqcap{\mbox{\boldmath{$\sqcap$}}}\)

\( \def\pabc#1#2#3{\left({\pz #1\over\pz #2}\right)\ns_{\!\!#3}}\)
\( \def\spabc#1#2#3{\big({\pz #1\over\pz #2}\big)\ns_{\!#3}}\)
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\( \def\rabc#1#2#3#4{(\pz #1,\pz #2)\over (\pz #3,\pz #4)}\)
\( \newcommand\subA{\ns_\ssr{A}}\)
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\( \newcommand\subDA{\ns_\ssr{DA}}\)
\( \def\lmapright#1{\ \ \smash{\mathop{\hbox to 55pt{\rightarrowfill}}\limits^{#1}}\ \ }\)
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\( \newcommand\longrightleftharpoons{ \mathop{\vcenter{\hbox{\ooalign{\raise1pt\hbox{$\longrightharpoonup\joinrel$}\crcr  \lower1pt\hbox{$\longleftharpoondown\joinrel$}}}}}}\)
\( \newcommand\longrightharpoonup{\relbar\joinrel\rightharpoonup}\)
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\( \newcommand\XFD{\RXi\ns_\ssr{FD}}\)
\( \newcommand\OBE{\Omega\ns_\ssr{BE}}\)
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\( \newcommand\veF{\ve\ns_\RF}\)
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\( \newcommand\kFu{k\ns_{\RF\uar}}\)
\( \newcommand\SZ{\textsf Z}}\) \( \newcommand\kFd{k\ns_{\RF\dar}\)
\( \newcommand\muB{\mu\ns_\ssr{B}}\)
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\( \newcommand\xoN{\Bx\ns_1\,,\,\ldots\,,\,\Bx\ns_N}\)
\( \newcommand\rok{\Br\ns_1\,,\,\ldots\,,\,\Br\ns_k}\)
\( \newcommand\xhiOZ{\xhi^\ssr{OZ}}\)
\( \newcommand\xhihOZ

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\)
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\( \newcommand\nda{\nd_\alpha}\)
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The basic principle: maximize the entropy,

\[S=-\kB\sum_n P\ns_n\ln P\ns_n\ .\]

\(\mu\)CE

We maximize \(S\) subject to the single constraint

\[C=\sum_n P\ns_n-1=0\ .\]

We implement the constraint \(C=0\) with a Lagrange multiplier, \({\bar\lambda}\equiv\kB\,\lambda\), writing

\[S^*=S-\kB\,\lambda\,C\ ,\]

and freely extremizing over the distribution \(\{P\ns_n\}\) and the Lagrange multiplier \(\lambda\). Thus,

\[\begin{aligned} \delta S^*&=\delta S - \kB \lambda\,\delta C- \kB\,C\,\delta\lambda\nonumber\\ &=-\kB\sum_n\Big[\ln P\ns_n + 1 + \lambda\Big]\delta P\ns_n-\kB\,C\,\delta\lambda\equiv 0\ .\end{aligned}\]

We conclude that \(C=0\) and that

\[\ln P\ns_n=-\big(1+\lambda\big)\ ,\]

and we fix \(\lambda\) by the normalization condition \(\sum_n P\ns_n=1\). This gives

\[P\ns_n={1\over \ROmega}\qquad,\qquad\ROmega=\sum_n\RTheta(E+\RDelta E-E\ns_n)\,\RTheta(E\ns_n-E)\ .\]

Note that \(\ROmega\) is the number of states with energies between \(E\) and \(E+\RDelta E\).

OCE

We maximize \(S\) subject to the two constraints

\[C\ns_1=\sum_n P\ns_n-1=0\qquad,\qquad C\ns_2=\sum_n E\ns_n\,P\ns_n - E=0\ .\]

We now have two Lagrange multipliers. We write

\[S^*= S-\kB\sum_{j=1}^2\lambda\ns_j\,C\ns_j\ ,\]

and we freely extremize over \(\{P\ns_n\}\) and \(\{C\ns_j\}\). We therefore have

\[\begin{split} \delta S^*&=\delta S - \kB\sum_n \big(\lambda\ns_1 + \lambda\ns_2\,E\ns_n\big)\,\delta P\ns_n -\kB\sum_{j=1}^2 C\ns_j\,\delta\lambda\ns_j\\ &=-\kB\sum_n\Big[\ln P\ns_n + 1 + \lambda\ns_1 + \lambda\ns_2\,E\ns_n\Big]\delta P\ns_n-\kB\sum_{j=1}^2 C\ns_j\,\delta\lambda\ns_j\equiv 0\ . \end{split}\]

Thus, \(C\ns_1=C\ns_2=0\) and

\[\ln P\ns_n=-\big(1+\lambda\ns_1+\lambda\ns_2\,E\ns_n\big)\ .\]

We define \(\lambda\ns_2\equiv\beta\) and we fix \(\lambda\ns_1\) by normalization. This yields

\[P\ns_n={1\over Z}\,e^{-\beta E\ns_n}\qquad,\qquad Z=\sum_n e^{-\beta E\ns_n}\ .\]

GCE

We maximize \(S\) subject to the three constraints

\[C\ns_1=\sum_n P\ns_n-1=0\quad,\quad C\ns_2=\sum_n E\ns_n\,P\ns_n -E=0\quad,\quad C\ns_3=\sum_n N\ns_n\,P\ns_n - N=0\ .\]

We now have three Lagrange multipliers. We write

\[S^*=S-\kB\sum_{j=1}^3 \lambda\ns_j\,C\ns_j\ ,\]

and hence

\[\begin{split} \delta S^*&=\delta S - \kB\sum_n \big(\lambda\ns_1 + \lambda\ns_2\,E\ns_n + \lambda\ns_3\,N\ns_n\big)\,\delta P\ns_n -\kB\sum_{j=1}^3 C\ns_j\,\delta\lambda\ns_j\\ &=-\kB\sum_n\Big[\ln P\ns_n + 1 + \lambda\ns_1 + \lambda\ns_2\,E\ns_n+ \lambda\ns_3\,N\ns_n\Big]\delta P\ns_n -\kB\sum_{j=1}^3 C\ns_j\,\delta\lambda\ns_j\equiv 0\ . \end{split}\]

Thus, \(C\ns_1=C\ns_2=C\ns_3=0\) and

\[\ln P\ns_n=-\big(1+\lambda\ns_1+\lambda\ns_2\,E\ns_n+\lambda\ns_3\,N\ns_n\big)\ .\]

We define \(\lambda\ns_2\equiv \beta\) and \(\lambda\ns_3\equiv-\beta\mu\), and we fix \(\lambda\ns_1\) by normalization. This yields

\[P\ns_n={1\over \Xi}\,e^{-\beta (E\ns_n-\mu N\ns_n)}\qquad,\qquad \Xi=\sum_n e^{-\beta (E\ns_n-\mu N\ns_n)}\ .\]