Tensors for Physics

220 12 Integral Formulae and Distribution Functions

nowΦhas to be orthogonal to the moments associated with the conserved quantities,
with the number density, the kinetic energy and the velocity. This means, the first and
second scalar, viz. 1 andV^2 − 3 /2, as well as the first vectorial expansion functionV,
must not be included in the expansion. Instead of (12.66), the expansion now reads

Φ=

∑∞

s= 3

a(s)φ(s)+

∑∞

s= 2

a(μs)φμ(s)+

∑∞

= 2

∑∞

s= 1

aμ(s 1 )μ 2 ···μφμ(s 1 )μ 2 ···μ. (12.90)

As before, the expansion coefficients

a(μs 1 )μ 2 ···μ=〈φμ(s 1 )μ 2 ···μ〉 (12.91)

are the moments of the velocity distribution.
Notice, the variableVdepends on the timetand on the positionrvia the temper-
ature and the average velocity. This is a second fundamental point which has to be
observed in applications, e.g. see [41].
In many applications, the most relevant terms of the expansion (12.90)involve

the translational or kinetic partsqkinμ and pkinμν of the heat flux vector and of the
symmetric traceless friction pressure tensor. These quantities are given by

qkinμ =n

〈(

m

2

c^2 −

5

2

kBT

)

cμ

〉

=nkBT(kBT/m)^1 /^2

√

2

〈(

V^2 −

5

2

)

Vμ

〉

,

(12.92)

or, equivalently,

qkinμ =nkBT(kBT/m)^1 /^2

√

5

2

〈φμ〉, (12.93)

and

pkinμν =nm〈cμcν〉=nkBT 2 〈VμVν〉=nkBT

√

2 〈φμν〉. (12.94)

The relevant vectorial and tensorial expansion functions occurring here are

φμ≡φμ(^2 )=

2

√

5

(

V^2 −

5

2

)

Vμ,φμν≡φμν(^1 )=

√

2 VμVν. (12.95)

In the approximation where only these two expansion functions and the correspond-
ing moments characterize the deviationΦfrom the local equilibrium, one has

Φ=〈φμ〉φμ+〈φμν〉φμν

=(nkBT)−^1

[

(kBT/m)−^1 /^2

√

2

5

qkinμ φμ+

√

2

2

pkinμνφμν

]

. (12.96)