Tensors for Physics

218 12 Integral Formulae and Distribution Functions

with the relaxation frequencyν 1 =ν 0. Clearly, the flow term couples the moment
a(μ^1 ), which is essentially the average velocity of the Brownian particles, with the

scalara(^2 )and the second rank tensora(μν^1 ). When these terms can be disregarded, the
(12.76) and (12.77) are equivalent to the following equations for the number density
nand the flux densityj=n〈v〉of the colloidal particles:

∂

∂t

n+∇μjμ= 0 , (12.77)

∂

∂t

jμ+c^20 ∇μn+ν 1 jμ= 0. (12.78)

For time changes which are slow compared with the relaxation timeτ 1 =ν 1 −^1 ,the
second of these equations reduces tojμ=−D∇μnwith the diffusion coefficient

D=c 02 τ 1 =

kBT

mν 1

=

kBT

6 πηR

. (12.79)

The third equality is based on the Stokes friction law (10.58).

12.3.4 Expansion About a Local Maxwell Distribution

The expansion of the velocity distribution function about an absolute Maxwellian, as
used in [13] and [21] is appropriate for small deviations from equilibrium. In general,
it is more advantageous, cf. [41], to expand the velocity distribution of gases about
a local Maxwellian, as treated next.
The difference between the velocity of a particle, at positionr, and the local
average flow velocity〈vμ〉=〈vμ〉(r,t),viz.

cμ=vμ−〈vμ〉, (12.80)

isreferredtoasthepeculiar velocityoftheparticle.ThelocaltemperatureT=T(r,t)
is linked with the average peculiar kinetic energy via(m/ 2 )〈c^2 〉=( 3 / 2 )kBT.
The distribution of the peculiar velocities is also denoted byf. It is normalized
according to
∫
f(c)d^3 c=n, (12.81)

The average〈ψ〉of a functionψ(c)is given by

〈ψ〉=

1

n

∫

ψ(c)f(c)d^3 c. (12.82)