1549380323-Statistical Mechanics Theory and Molecular Simulation

Spatial distribution functions 163

E=

3

2

NkT+〈U〉. (4.6.37)

Moreover, since 3NkT/2 =〈

∑N

i=1p

2

i/^2 mi〉, we can write eqn. (4.6.37) as

E=

〈N

∑

i=1

p^2 i

2 mi

+U(r 1 ,...,rN)

〉

=〈H(r,p)〉, (4.6.38)

which is just the sum of the average kinetic and average potential energies over the
canonical ensemble. In eqns. (4.6.37) and (4.6.38), we have expressed a thermody-
namic quantity as an ensemble average of a phase space function. Such a phase space
function is referred to as an instantaneousestimatorfor the corresponding thermo-
dynamic quantity. For the internal energyE, it should come as no surprise that the
corresponding estimator is just the HamiltonianH(r,p).
Let us now apply eqn. (4.6.37) to a pair potential, such as that of eqn. (3.14.3).
Taking the general form of the potential to be

Upair(r 1 ,...,rN) =

∑N

i=1

∑N

j>i

u(|ri−rj|), (4.6.39)

the ensemble average ofUpairbecomes

〈Upair〉=

1

Z

∑N

i=1

∑N

j>i

∫

dr 1 ···drNu(|ri−rj|)e−βUpair(r^1 ,...,rN). (4.6.40)

Note, however, that every term in the sum overiandjin the above expression can
be transformed into
∫
dr 1 ···drNu(|r 1 −r 2 |)e−βUpair(r^1 ,...,rN)

by simply relabeling the integration variables. Since there areN(N−1)/2 such terms,
the average potential energy becomes

〈Upair〉=

N(N−1)

2 Z

∫

dr 1 ···drNu(|r 1 −r 2 |)e−βUpair(r^1 ,...,rN)

=

1

2

∫

dr 1 dr 2 u(|r 1 −r 2 |)

×

[

N(N−1)

Z

∫

dr 3 ···drNe−βUpair(r^1 ,...,rN)

]

. (4.6.41)

However, the quantity in the square brackets is nothing more thatthe pair correlation
functiong(2)(r 1 ,r 2 ). Thus,

〈Upair〉=

ρ^2

2

∫

dr 1 dr 2 u(|r 1 −r 2 |)g(2)(r 1 ,r 2 ). (4.6.42)