Tensors for Physics

222 12 Integral Formulae and Distribution Functions

quantities become anisotropic. The basics and examples for the use of tensors needed
to characterize the anisotropic properties are presented here. For more information
on the physics of liquids e.g. consult [48–52].

12.4.1 Two-Particle Density, Two-Particle Averages

Consider a fluid ofNspherical particles with positionsri,i= 1 , 2 ,...N, contained
withinavolumeV.Thetwo-particleprobabilitydensitytofindoneparticleatposition
raand another one atrbis thetwo-particle density n(^2 )(ra,rb). It is given by

n(^2 )(ra,rb)=

〈

∑

I

∑

j=i

δ(ra−ri)δ(rb−rj)

〉

, (12.97)

where the bracket〈...〉indicates aN-particle average. It can e.g. be a canonical
average or a time-average, but it need not to be specified here. The integral ofn(^2 )
over bothraandrbyields the number of pairs{i,j}viz.N(N− 1 ).
Consider a quantity

Ψ=

∑

I

∑

j=i

ψ(ri,rj),

whereψis a function which depends on the position vectors of two particles. Then
its average is given by

〈Ψ〉=

∫

ψ(r 1 ,r 2 )n(^2 )(r 1 ,r 2 )d^3 r 1 d^3 r 2 , (12.98)

where now the integration variables are denoted byr 1 ,r 2 rather thanra,rb. When
the functionψdepends on the difference between two position vectors only, just like
the binary interaction potential, the average〈Ψ〉can be written as

〈Ψ〉=

N^2

V

∫

ψ(r)g(r)d^3 r. (12.99)

Here thepair-correlation function g=g(r)is defined by

g(r)=

V

N^2

∫

n(^2 )(r 2 +r,r 2 )d^3 r 2. (12.100)

The vector

r=r 1 −r 2