Statistical Mechanics · Problem 3.4

The Grand Canonical Ensemble — R. K. Pathria

Physics — Statistical Mechanics · entropy-most-probable-distribution

Problem

Verify that the quantity

(k/\mathcal{N}) \ln \Gamma

, where

\Gamma(\mathcal{N}, U) = \sum_{\{n_r\}}' W\{n_r\},

is equal to the (mean) entropy of the given system. Show that we obtain essentially the same result for

\ln \Gamma

if we take, in the foregoing summation, only the largest term of the sum, viz. the term

W\{n_r^*\}

which corresponds to the most probable distribution set. [Surprised? Well, note the following example: For all

N

\sum_{r=0}^{N} {}^N C_r = 2^N;

therefore,

\ln\left\{\sum_{r=0}^{N} {}^N C_r\right\} = N \ln 2. \tag{a}

Now, the largest term in the sum corresponds to

r \simeq N/2

; so, for large

N

, the logarithm of the largest term is very nearly equal to

\begin{aligned} \ln\{N!\} - 2\ln\{(N/2)!\} &\approx N\ln N - 2\frac{N}{2}\ln\frac{N}{2} = N\ln 2, \tag{b} \end{aligned}

which agrees with (a).]