# 4.6: Statistical Ensembles from Maximum Entropy


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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

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

This page titled 4.6: Statistical Ensembles from Maximum Entropy is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Daniel Arovas.