Tensors for Physics

206 12 Integral Formulae and Distribution Functions

Likewise, up to second order inβ 1 , the distribution function is equal to

feq=( 4 π)−^1

(

1 +β 1 Fμφμ+

√

3

10

β^21 FμFνφμν+...

)

. (12.22)

In this high temperature approximation, the first two moments are given by

aμ=〈φμ〉=β 1 Fμ, aμν=〈φμν〉=

√

3

10

β 12 FμFν =

√

3

10

aμaν. (12.23)

These relations imply

〈uμuν〉=c(^2 |^11 )〈uμ〉〈uν〉. (12.24)

The numerical factorc( 2 | 11 )is equal to 3/5 in the high temperature approximation.
For a very strong orienting field which induces a practically perfect orientation,
c(^2 |^11 )=c(^2 |^11 )(β 1 F)approaches 1.
In the case of a dipolar orientation, the distribution and consequently all its
moments have uniaxial symmetry. Letebe a unit vector parallel to the fieldF,
viz.Fμ=Feμ, whereFis the strength of the field. Then the energy can be writ-
ten asHdip=−dFeμuμ =−dFx, wherex =eμuμ|is the cosine of the angle
betweenuand the direction of the field. Furthermore, one haseμφμ=

√

3 xand

−Hdip/kBT=

√

3 β 1 Fx. The average〈xn〉is evaluated according to

〈xn〉=

1

2

Z−^1

∫ 1

− 1

xnexp

[√

3 β 1 Fx

]

dx,

with the state functionZgiven by

Z=L

(√

3 β 1 F

)

, L(z)=

1

2 z

(exp[z]−exp[−z])= 1 +z^2 / 6 +z^4 / 120 +....

(12.25)

LetLkbe thek-derivative ofL(z)with respect toz,viz.Lk=dkL/dzk. The average
〈xk〉is then determined byLk(z)/L(z), withz=

√

3 β 1 F=βdF. The results for the
lowest moments are

〈x〉=

1 +z+exp[ 2 z](z− 1 )

zexp[ 2 z]−z

=z/ 3 −z^3 / 45 ±... , (12.26)

〈x^2 〉−

1

3

=

1 +z+exp[ 2 z](z− 1 )

zexp[ 2 z]−z

−

1

3

= 2 z^2 / 45 − 4 z^4 / 945 ±...(12.27)

Forz→∞, both〈x〉and〈x^2 〉approach 1.