Tensors for Physics

216 12 Integral Formulae and Distribution Functions

These functions are orthogonal and normalized according to〈Φ(s)Φ(s

′)
〉 0 =δss′.The
first two vectorial expansion functions are

Φμ(^1 )=

√

2 Vμ,Φ(μ^2 )=

2

√

5

(

V^2 −

5

2

)

Vμ. (12.69)

Here the normalization corresponds to〈Φμ(s)Φ(s

′)
ν 〉 0 =δss′δμν. The first second rank
tensorial expansion functions are

Φ(μν^1 )=

√

2 VμVν,Φμν(^2 )=

2

√

7

(

V^2 −

7

2

)

VμVν. (12.70)

For comparison with [13] notice thats−1 corresponds to the labelrused by
Waldmann, which is the number of zero-values of the Sonine polynomials involved.
Thefirsttwoscalarandfirstvectorialmomentsareessentiallytherelativedeviation
of the number densitynfrom the constant densityn 0 , the relative deviation of the
temperatureTfromT 0 , and the average flow velocity〈v〉,viz.

a(^1 )=n/n 0 − 1 , a(^2 )=

√

3

2

(T/T 0 − 1 ), aμ(^1 )=c 0 −^1 〈vμ〉, c 0 =(kBT 0 /m)^1 /^2.

(12.71)

In pure gases, these are associated with conserved quantities. The number density,
thusa(^1 ),istheonlyconservedquantityforcolloidalparticlesinliquids.Themoments
a(μ^2 )anda(μν^1 )are proportional to the parts of the heat flux and of the friction pressure
tensor which are associated with the translational motion. The specific relations are
stated in Sect.12.3.4.

12.3.3 Kinetic Equations, Flow Term

The time change∂f/∂tof the velocity distributionf=f(t,r,v)contains a contribu-
tion due to the translational motion of the particles, viz.vλ∇λfwhich is referred to as
flow term. The corresponding expression in thekinetic equationfor the dimensionless
distribution functionF=F(t,r,V)is

∂

∂t

F+

√

2 c 0 Vλ∇λF−

(

δF

δt

)

..

= 0. (12.72)

Here(δδFt)..stands either for the collision term(δδFt)collof the Boltzmann equation

or for the damping term(δδFt)FPof Planck’s version of the Fokker-Planck equation.
In any case, these terms guarantee the approach to equilibrium. The Fokker-Planck