12.3 Averages Over Velocity Distributions 213
Kinetic Theory of Gasesare presented. Physical phenomena associated with the
velocity distribution, both for an equilibrium situation and for non-equilibrium, in
particular for transport processes, are e.g. discussed in [40], see also [13].
12.3.1 Integrals Over the Maxwell Distribution
Letvbe the velocity of an atom or molecule in a gas or liquid. The distribution
of the different velocities is characterized by the velocity distribution function, also
denoted byf=f(v), which is conventionally normalized such that
∫
f(v)d^3 v=n. (12.48)Herenis the number density. The average〈ψ〉of a functionψ(v)is given by
〈ψ〉=1
n∫
ψ(v)f(v)d^3 v. (12.49)In thermal equilibrium, at temperatures and densities, where quantum effects play
no role,fis equal to the Maxwell distributionf 0
f 0 (v)≡n 0(
m
2 πkBT 0) 3 / 2
exp(
−
mv^2
2 kBT 0)
, (12.50)
wheremis the mass of a particle. The constant densityn 0 =N/V,oftheNparticles
confined to the volumeVand the constant temperatureT 0 characterize the absolute
equilibrium state. The Boltzmann constant is denoted bykB. It is convenient to
introduce a dimensionless velocity variableVvia
V^2 =
mv^2
2 kBT 0, (12.51)
which implies
v=√
2 c 0 V, c 0 =√
kBT 0 /m, (12.52)and to use the velocity distributionF=F(V), linked withf(v), such that
f(v)d^3 v=F(V)d^3 V.Instead of (12.49), averages are then evaluated according to
〈ψ〉=1
n∫
ψ(V)F(V)d^3 V. (12.53)