Tensors for Physics

210 12 Integral Formulae and Distribution Functions

in the ordered and the isotropic state is

sa=−kB

∫

fln(f/f 0 )d^2 u=−kB〈ln(f/f 0 )〉. (12.36)

Use off=f 0 ( 1 +Φ),cf.(12.16) yields

sa=−kB〈( 1 +Φ)ln( 1 +Φ)〉 0. (12.37)

With the help of the power series expansion

( 1 +x)ln( 1 +x)=x+

∑∞

n= 2

(− 1 )n

1

n(n− 1 )

xn,

and due to〈Φ〉 0 =0, the entropy (12.37) is equivalent to

sa=−kB

∑∞

n= 2

(− 1 )n

1

n(n− 1 )

〈Φn〉 0. (12.38)

The first few terms in this series are

sa=−kB

(

1

2

〈Φ^2 〉 0 −

1

6

〈Φ^3 〉 0 +

1

12

〈Φ^4 〉 0 ±...

)

. (12.39)

In lowest order inΦ, the orthogonality and the normalization (12.15) of the expansion
tensors imply

sa=−kB

(

1

2

∑∞

= 1

aμ 1 μ 2 ···μaμ 1 μ 2 ···μ±...

)

. (12.40)

The dots stand for terms associated with third and higher powers inΦ. Examples for
the role of higher order terms are discussed in the Exercise15.1.

12.2.7 Fokker-Planck Equation for the Orientational

Distribution

In the absence of any orientating torque and for a spatially homogeneous system, the
distribution function for the orientation of (effectively) uniaxial particles immersed
in a fluid obeys the dynamic equation, frequently called orientationalFokker-Planck
equation,

∂f(u)

∂t

−ν 0 LμLμf(u)= 0. (12.41)