214 12 Integral Formulae and Distribution Functions
In thermal equilibrium,Fis equal to the absolute MaxwellianF 0 , which is the
Gaussian function
F 0 (V)≡n 0 π−^3 /^2 exp(−V^2 ). (12.54)Averages evaluated with this Maxwell velocity distribution function are denoted by
〈...〉 0 ,viz.
〈ψ〉 0 ≡π−^3 /^2∫
ψ(V)exp(−V^2 )d^3 V=1
n 0∫
ψ(V)F 0 (V)d^3 V. (12.55)The equilibrium average of powers of the magnitudeVof the dimensionless velocity
Vare
〈Vn〉 0 = 2 π−^1 /^2 Γ(
n+ 3
2)
, (12.56)
wherenis a positive integer number andΓ(x)is the gamma-function, with the
propertyΓ(n+ 1 )=n!. For the first few even powers, (12.56) implies〈 1 〉 0 =1, and
〈V^2 〉 0 =
3
2
, 〈V^4 〉 0 =
15
4
, 〈V^6 〉 0 =
105
8
, 〈V^8 〉 0 =
945
16
. (12.57)
Clearly, the Maxwell distribution is isotropic. Thus equilibrium averages of the irre-
ducible tensorsVμ 1 Vμ 2 ···Vμ vanish:
〈Vμ 1 Vμ 2 ···Vμ〉 0 = 0. (12.58)This is not the case for products of irreducible tensors of the same rank. In particular,
from (12.1) and (12.56), with 2π−^1 /^2 Γ(^2 2 +^3 )=(^2 + 2 ^1 )!!, follows
〈Vμ 1 Vμ 2 ···Vμ Vν 1 Vν 2 ···Vν′〉 0 =!
2
δ′Δ()μ 1 μ 2 ···μ,ν 1 ν 2 ···ν. (12.59)The special cases=′= 1 ,2 correspond to
〈VμVν〉 0 =1
2
δμν, 〈VμVν VλVκ〉 0 =1
2
Δμν,λκ. (12.60)12.3.4 Expansion About a Local Maxwell Distribution
Clearly, the Maxwell distribution is isotropic. In a non-equilibrium situation, how-
ever, this is not the case. In general, the velocity distribution is anisotropic. The
directional properties of the velocity distribution can be characterized by irreducible