Tensors for Physics

12.3 Averages Over Velocity Distributions 215

tensors constructed from the components of the velocity variableV. However, addi-
tional scalar polynomial functions, depending onV^2 , are needed the characterize the
full dependence offon the magnitude and direction of the velocity. These scalar
functions are theSonine polynomials[12, 13], which are closely related to theasso-
ciated Laguerre polynomials. The orthogonal expansion functions are, cf. [13, 41],

φ(μs 1 +μ^12 )···μ∼(− 1 )s+^1 S((s)+ 1 / 2 )(V^2 )Vμ 1 Vμ 2 ···Vμ, (12.61)

where the labelscharacterizes the different Sonine polynomials. The Sonine poly-
nomials are defined by

S((sk))(x)=(s!)−^1

∂s

∂zs

( 1 −z)−(^1 +k)exp[zx/(z− 1 )], z→ 0. (12.62)

The first few of these polynomials areS(^0 )=1 and

S((k^1 ))(x)=k+ 1 −x, S((k^2 ))(x)=( 1 / 2 )(k+ 1 )(k+ 2 )−(k+ 2 )x+( 1 / 2 )x^2 .(12.63)

The functions (12.61) are orthogonal and normalized according to

〈φ(μs 1 )μ 2 ···μφ(s

′)

ν 1 ν 2 ···ν′〉^0 =δss′δ′Δ

()

μ 1 μ 2 ···μ,ν 1 ν 2 ···ν. (12.64)

The velocity distribution is written as

F(V)=F 0 (V)( 1 +Φ), (12.65)

whereΦ(t,r,V)characterizes the deviation ofFfrom the MaxwellianF 0 .The
expansion ofΦreads

Φ=

∑∞

= 0

∑∞

s= 1

a(μs) 1 μ 2 ···μφ(μs 1 )μ 2 ···μ. (12.66)

The expansion coefficients

a(μs 1 )μ 2 ···μ=〈φμ(s 1 )μ 2 ···μ〉 (12.67)

are the moments of the velocity distribution.
Just the first few terms of the expansion are of importance in most applications, so
only the first few expansion functions are listed here. The first three scalar expansion
functions are,

Φ(^1 )= 1 ,Φ(^2 )=

√

2

3

(

V^2 −

3

2

)

,Φ(^3 )=

√

2

15

(

V^4 − 5 V^2 +

15

4

)

.

(12.68)