12.2 Orientational Distribution Function 211
HereLμLμ, with
Lμ=εμνλuν∂
∂uλ,
is the Laplace operator on the unit sphere. The properties ofLas used here are
equivalent to those discussed in Sect.7.6where the unit vector̂rparallel to the
position vectorroccurred instead of the unit vectoruwhich is parallel to the figure
axis of a non-spherical particle. The relaxation frequency coefficientν 0 >0 has
the dimension one over time. The (12.41) essentially describes a diffusional motion
on the unit sphere. For this reason, the coefficientν 0 is referred to as orientational
diffusion coefficient.
By analogy to (11.31), the irreducible tensorsφμ 1 μ 2 ···μdefined by (12.14), are
eigenfunctions of the orientational Laplace operator with the eigenvalues−(+ 1 ),
viz.
LλLλφμ 1 μ 2 ···μ=−(+ 1 )φμ 1 μ 2 ···μ. (12.42)Multiplication of the dynamic equation (12.41)byφμ 1 μ 2 ···μ, subsequent integration
overd^2 u, use of the expansion (12.16) for the distribution function and of the ortho-
normalization (12.1) yields the relaxation equations
daμ 1 μ 2 ···μ
dt+νaμ 1 μ 2 ···μ= 0 ,≥ 1 , (12.43)for the tensorial momentsaμ 1 μ 2 ···μ. The relaxation coefficientsνare given by
ν=(+ 1 )ν 0. (12.44)Equation(12.43) implies an exponential relaxation
aμ 1 μ 2 ···μ(t)=exp(−t/τ)aμ 1 μ 2 ···μ( 0 ), t> 0 ,with the relaxation timeτ=ν−^1. Clearly, higher moments pertaining to larger
values ofrelax faster. In the long time limit all moments approach zero and the
distribution is isotropic, corresponding to the equilibrium state of an isotropic fluid.
Some Historical Remarks
A dynamic equation for an orientational distribution function as discussed here was
first introduced by Adriaan Fokker in his thesis in 1913 and published 1914 [43].
So the (12.41) should actually be calledFokker equation. How got Planck involved?
In 1917, Max Planck was asked by colleagues to explain the work of Fokker. So
at a meeting of the Prussian Academy of Science, in 1917, he presented what he
was inspired to, viz. a dynamic equation for the velocity distribution function of a
Brownian particle immersed in a liquid [44]. So his equation might be calledPlanck
equation. However, it is common practice to refer to both types of these equations
and generalizations thereof asFokker-Planck equation.