1549380323-Statistical Mechanics Theory and Molecular Simulation

Spatial distribution functions 155

of the locations of the remainingn+1,...,Nparticles. This probability can be obtained
by simply integrating eqn. (4.6.6) over the lastN−nparticles:

P(n)(r 1 ,...,rn)dr 1 ···drn=

1

Z

[∫

D(V)

drn+1···drNe−βU(r^1 ,...,rN)

]

dr 1 ···drn. (4.6.7)

Since the particles are indistinguishable, we are actually interested inthe probability
of findinganyparticle in a volume element dr 1 about the pointr 1 andanyparticle in
dr 2 about the pointr 2 , etc., which is given by the distribution

ρ(n)(r 1 ,...,rn)dr 1 ···drn=

N!

(N−n)!Z

[∫

D(V)

drn+1···drNe−βU(r^1 ,...,rN)

]

dr 1 ···drn. (4.6.8)

The prefactorN!/(N−n)! =N(N−1)(N−2)···(N−n+ 1)! comes from the fact
that the first particle can be chosen inNways, the second particle inN−1 ways, the
third particle inN−2 ways and so forth.
Eqn. (4.6.8) is really a measure of the spatial correlations among particles. If, for
example, the potentialU(r 1 ,...,rN) is attractive at long and intermediate range, then
the presence of a particle atr 1 will increase the probability that another particle will
be in its vicinity. If the potential contains strong repulsive regions,then a particle at
r 1 will increase the probability of a void in its vicinity. More formally, ann-particle
correlation function is defined in terms ofρ(n)(r 1 ,...,rn) via

g(n)(r 1 ,...,rn) =

1

ρn

ρ(n)(r 1 ,...,rn)

=

N!

(N−n)!ρnZ

∫

D(V)

drn+1···drNe−βU(r^1 ,...,rN), (4.6.9)

whereρ=N/Vis the number density. Then-particle correlation function eqn. (4.6.9)
can also be formulated in an equivalent manner by introducingδ-functions for the first
nparticles and integrating over allNparticles:

g(n)(r 1 ,...,rn) =

N!

(N−n)!ρnZ

∫

D(V)

dr′ 1 ···dr′Ne−βU(r

′ 1 ,...,r′N)∏n

i=1

δ(ri−r′i). (4.6.10)

Note that eqn. (4.6.10) is equivalent to an ensemble average of the quantity

∏n

i=1

δ(ri−r′i)

usingr′ 1 ,...,r′Nas integration variables: