Statistical Physics, Second Revised and Enlarged Edition

Appendix C Some useful integrals

1 MAXWELL–BOLTZMANN INTEGRALS

To calculate the properties of an MB gas, as in Chapter 6, we need to evaluate the
definiteintegralsoftheform:

IIIn=

∫∞

0

∫∫

ynexp(−by^2 )dy (C.1)

wherenis any positiveinteger. This canbedoneinthree stages.

(i) Equation (C.1) maybe integrated byparts togive

IIIn=[−yn−^1 exp(−by^2 )/ 2 b]∞ 0 +[(n− 1 )/ 2 b]IIIn− 2

Forn≥2, the first term is zero since it vanishes at both limits, giving a simple

recurrence relationbetweenIIInandIIIn− 2 :

IIIn=[(n− 1 )/ 2 b]IIIn− 2 (C.2)

For some purposes, for example the calculation of the rms speed of gas

molecules, the recurrence relation contains enoughinformationbyitself.But

equation (C.2)is usefulin everycase, sinceits application reduces anyintegral

IIInto a known multiple of eitherI 1 orIII 0.

(ii)TheintegralI 1 isobtainedbysimpleintegration, the resultbeing

I 1 = 1 / 2 b (C.3)

(iii) The integralIII 0 takes a little longer to evaluate. A quick method is to considera
two-dimensional problem, to integrate exp(−br^2 )over thewholex−yplane,r
beingthedistancefrom theorigin. Weknowfrom thedefinition (C.1) that

2 III 0 =

∫∞

−∞

∫∫

exp(−bx^2 )dx=

∫∞

−∞

∫∫

exp(−by^2 )dy

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