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)