Statistical Physics

14 1 Thermal Equilibrium and the Principle of Equal ProbabilityThe position of the peak of W N (n) is determined by the conditionddn ln W N (n) ≃ d { (n ln p N )( )}N+(N − n)ln qdn nN − n( ) ( )pN N=ln − 1 − ln q +1nN − n=0. (1.13)Namely,pn = qN − n , (1.14)orNp =(p + q)n = n. (1.15)Therefore the peak is positioned at n = 〈n〉.Furthermore,d 2dn 2 ln W N (n) ≃ d { ( ) ( )}pN qNln − lndn n N − n= − 1 n − 1N − n = − 1Np − 1Nq= − 1Npq . (1.16)Therefore, the Taylor expansion of ln W N (n) around the peak isln W N (n) =lnW N (〈n〉) − 1 12 Npq (n −〈n〉)2 + O[(n −〈n〉) 3 ] . (1.17)That is, the distribution function is a Gaussian around the peak: 6[W N (n) ≃ W N (〈n〉)exp − 1 ]12 Npq (n −〈n〉)2 . (1.18)In summary, we have found the following for the probability distribution ofgas molecules:• The distribution has a peak at the mean value.• There is a fluctuation around the peak.• However, the fluctuation is so small that a nonuniform distribution is notobserved in practice.We can imagine that these properties will not be restricted to the spatialdistribution of gas molecules, but may be possessed by any distribution inthermal equilibrium in which a macroscopic number of molecules are involved.6 For details of the Gaussian distribution function, see Appendix B. The result(1.17) is an example of the central limit theorem, described in Sect. B.1.