Tensors for Physics

226 12 Integral Formulae and Distribution Functions

where it is understood, that the “spherical” partgsof the pair correlation function, as
well as the expansion tensorsg···are function of the inter-particle distancer.They
depend also on the timetwhen the pair correlation function applies to a general
non-equilibrium situation.
Due to the orthogonality relation (12.5), the quantitiesg···are tensorial moments
ofg(r), more specifically, one has

1

4 π

∫

̂rμ 1 ̂rμ 2 ···̂rμg(r)d^2 ̂r=

!

( 2 + 1 )!!

gμ 1 μ 2 ···μ. (12.110)

For=2 and=4 this relation implies, see also (12.1),

gμν=

15

2

1

4 π

∫

̂rμ̂rνg(r)d^2 ̂r, (12.111)

and

gμνλκ=

315

8

1

4 π

∫

̂rμ̂rν̂rλ̂rκg(r)d^2 ̂r. (12.112)

Insertion of the expansion (12.109)into(12.107) leads to expressions which involve
just the integration overr, but no longer over the angles. These relations are

ppot=

2 π

3

n^2

∫

rλFλgs(r)r^2 dr, ppotμν =

4 π

15

n^2

∫

rλFλgμν(r)r^2 dr.(12.113)

Notice thatrλFλ=−rφ′. Here the prime indicates the differentiation with respect
tor.
For a fluid composed of spherical particles, all moments with≥2 vanish
in thermal equilibrium. This, however, is no longer the case in non-equilibrium
situations. In particular, certain components ofgμνare non-zero for a viscous flow.
This is already the case in the limit of small shear rates. At higher shear rates, also
higher moments, e.g. with=4, cf. Sect.12.4.5and Exercise12.4.
The expansion (12.109) is equivalent to an expansion with respect to spherical

harmonicsY(m) =Y(m)(̂r), cf. Sect.9.4.2. The first few terms corresponding to
(12.109)are

g(r)=gs+

∑^2

m=− 2

g2mY 2 (m)+

∑^4

m=− 4

g4mY 4 (m)+..., (12.114)

where the expansion coefficients are functions ofr.TheY(m)obey the ortho-

normalization

∫

Y(m)(Y(m

′)

′ )

∗d (^2) ̂r=δ′δmm′, thus
gm(r)=

∫

(Y(m)(̂r)∗g(r)d^2 ̂r. (12.115)