1549380323-Statistical Mechanics Theory and Molecular Simulation

Spatial distribution functions 157

around a given pair does not depend on where the pair is in the system. Thus, we
integrate overR, yielding a new function ̃g(r) defined as

g ̃(r)≡

1

V

∫

D(V)

dR ̃g(2)(r,R)

=

(N−1)

ρZ

∫

D(V)

dRdr 3 ···drNe−βU(R+

(^12) r,R− (^12) r,r 3 ,...,rN)

=

(N−1)

ρ

〈δ(r−r′)〉R′,r′,r′ 3 ,...,r′N, (4.6.17)

where the last line follows from eqn. (4.6.16) by integrating one of theδ-functions
overRand renamingr′ 1 −r′ 2 =r′. Next, we recognize that a system such as a liquid
is spatially isotropic, so that there are no preferred directions in space. Thus, the
correlation function should only depend on the distance between the two particles,
that is, on the magnitude|r|. Thus, we introduce the spherical polar resolution of the
vectorr= (x,y,z)

x=rsinθcosφ

y=rsinθsinφ

z=rcosθ, (4.6.18)

whereθis the polar angle andφis the azimuthal angle. Defining the unit vector
n= (sinθcosφ,sinθsinφ,cosθ), it is clear thatr=rn. Also, the Jacobian is dxdydz=
r^2 sinθdrdθdφ. Thus, integrating ̃g(r) over angles gives a new function

g(r) =

1

4 π

∫ 2 π

0

dφ

∫π

0

dθsinθ ̃g(r)

=

(N−1)

4 πρZ

∫ 2 π

0

dφ

∫π

0

dθsinθ

∫

D(V)

dRdr 3 ···drN

×e−βU(R+

(^12) rn,R− (^12) rn,r 3 ,...,rN)

=

(N−1)

4 πρr^2

〈δ(r−r′)〉r′,θ′,φ′,R′,r′ 3 ,....,r′N, (4.6.19)

known as theradial distribution function. The last line follows from the identity

δ(r−r′) =

δ(r−r′)

rr′

δ(cosθ−cosθ′)δ(φ−φ′). (4.6.20)

From the foregoing analysis, we see that the radial distribution function is a measure
of the probability of finding two particles a distancerapart under the conditions of
the canonical ensemble.
As an example of a radial distribution function, consider a system ofNidentical
particles interacting via the pair-wise additive Lennard-Jones potential of eqn. (3.14.3).