Tensors for Physics

212 12 Integral Formulae and Distribution Functions

In the presence of an external torqueT, which can be derived from a potential

functionH=H(u)according to

Tλ=−LλH(u), (12.45)

the orientational Fokker-Planck equation contains an additional term. More specifi-

cally, the kinetic equation now reads

∂f(u)

∂t

−ν 0 Lμ

(

Lμf(u)+βf(u)LμH(u)

)

= 0 ,β=

1

kBT

. (12.46)

In this case, the stationary solution of the Fokker-Planck equation is the equilib-

rium distributionf=feqwhich is proportional to exp[−βH]=exp[−H/kBT],cf.

Sect.12.2.4.

Multiplication of the (12.46) by a functionψ =ψ(u), integration overd^2 u

and integrations by part leads to themoment equationfor the time change of the

average〈ψ〉:

d

dt

〈ψ〉−ν 0

(

〈LμLμψ〉−β〈(Lμψ)(LμH)〉

)

= 0. (12.47)

Examples for such moment equations, however for the caseH=0, are the relaxation

(12.43). ForH =0, the relaxation equation for theth rank tensoraμ 1 μ 2 ···μis

coupled with tensors of different ranks, in general. In particular, whenH(u)involves

the first and second rank tensorsuμand uμuν, as considered in Sect.12.2.4,the

tenor of rankis coupled with tensors of ranks±1 and±2, respectively, quite in

analogy to the quantum mechanical selection rules for electric dipole and quadrupole

radiation, cf. Sect.12.5. Applications of a generalized Fokker-Planck equation to the

dynamics of liquid crystals is discussed in Sect.16.4.4.

12.2 Exercise: Prove that the Fokker-Planck Equation Implies an Increase of
the Orientational Entropy with Increasing Time
Hint: The time change of an orientational average is d〈ψ〉/dt=

∫

∂(ψf)/∂td^2 u,
use the expression (12.36) for the orientational entropy.

12.3 Averages Over Velocity Distributions

Atoms and molecules never sit still. In equilibrium, they move in all directions with

equal probability and their average speed is higher, at higher temperatures. In non-

equilibrium situations, preferential direction can be favored. The velocity distribution

function characterizes this behavior. The distribution is isotropic in equilibrium and,

in general, anisotropic in non-equilibrium. Here the relevant tools needed in the