Tensors for Physics

12.4 Anisotropic Pair Correlation Function and Static Structure Factor 223

is not an ordinary position vector but it is the difference vector in the{r 1 ,r 2 }pair-
space. Due to (12.97),gis also given by theN-particle average

N

V

g(r)=

1

N

〈

∑

I

∑

j=i

δ(r−rij)

〉

, rij=ri−rj. (12.101)

Clearly,rijis the difference vector between the position vectors of particlesiandj.
Notice thatgas given by (12.100) does not depend on the absolute positions of
two particles within the volumeV, but rather on their relative position vector. The
definition (12.100) and consequently (12.101) also apply to spatially inhomogeneous
systems. For a spatially homogeneous case, where the two-particle densityn(^2 )=
n(^2 )(r 1 ,r 2 )depends on the differencer=r 1 −r 2 only,g(r)can also be defined by

n(^2 )(r)=

(

N

V

) 2

g(r)=n^2 g(r), (12.102)

where it is understood thatn=N/Vis the spatially constant number density. For
a ‘pure system’, i.e. a substance composed of one type of particles, the interchange
of the labels 1,2 of two particles, which implies the replacement ofrby−r, should
not make any difference forg. Thusgis an even function ofr:

g(r)=g(−r). (12.103)

For particles which cannot penetrate each other due to their short range repulsion one
hasg( 0 )=0. On the other hand, particles in isotropic media without long-ranged
correlations are uncorrelated when they are separated by distancesrlarge compared
with their size. Theng→1 holds true forr→∞. Typically, the orientationally
averaged part ofghas a maximum at a valuerwhich corresponds to the first neighbor
distance. In dense systems there are several additional maxima at larger distances,
with smaller height, however.
Examples for averages which can be evaluated as integrals overg(r)are the
potential contributions to the energy and to the pressure tensor as well as the static
structure factor.

12.4.2 Potential Contributions to the Energy

and to the Pressure Tensor

Assuming that the total potential energy of the particles is given by the sum of the
binary interaction potentialφ=φ(r)=φ(−r), the total potential energy is

Φ=

1

2

∑

i

∑

j=i

φ(ri−rj). (12.104)