Statistical Physics

12 1 Thermal Equilibrium and the Principle of Equal ProbabilityIn the course of this calculation, we have used the binomial theorem again.From this variance, the relative deviation is√〈(n −〈n〉)2 〉= 1 √ q√〈n〉 N p ≪ 1 . (1.10)This deviation is unbelievably small. Let us see how small this value is. Considera sample of gas consisting of a macroscopic number of molecules. Forexample, there are about 2.7×10 25 molecules in a volume of 1 m 3 of air in thestandard state. Let us take p =0.1, or V 1 =0.1m 3 .Then〈n〉 =2.7 × 10 24 ,and the standard deviation √ Npq ≃ 10 12 . So the number fluctuates by ±10 12around the mean value. This fluctuation itself is quite large. However, it isnegligibly small compared with the mean value, ≃ 10 24 . To realize how smallFig. 1.2. The probability distribution of the number of molecules in a small part ofa system. The probability of finding n molecules in a volume V 1 =0.1m 3 , W N (n),when there are 2.7×10 25 molecules in V =1m 3 is plotted. The vertical axis shows√ 2πNpqWN (n), normalized such that the peak value is approximately unity. (a)The whole range of possible variation 0