Question 3.5: Evaluate W ⁄ Wmax for N2 in a 1 cm³ volume at standard tempe...

N=\sum\limits_{j}{N_{j} } = 2\times 10^{19}    (3.35)

Consider an average perturbation of \frac{\Delta N_{j} }{N^{*}_{j} }=10^{- 3}

\ln \left(\frac{W}{W_{max} } \right) = – \frac{1}{2}\times 10^{- 6}\times 2\times 10^{19}=- 10^{13}       (3.36)

Thus,

W=W_{\max} \ exp(-10^{13} )      (3.37)

that is an incredibly small number. Thus, a small perturbation away from N^{* }_{j} contributes only a negligibly small number of additional microstates. This indicates that W is a very peaked function, and so \Omega =W_{\max} is a good assumption. Put another way, the macrostates for which \Omega =W_{\max} must very closely follow N_{j}\approx N^{* }_{j}  .