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

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