1549380323-Statistical Mechanics Theory and Molecular Simulation

164 Canonical ensemble

Proceeding as we did in derivingg(r) (Section 4.6), we introduce the change of variables
in eqn. (4.6.14), which gives

〈Upair〉=

N^2

2 V^2

∫

drdRu(r) ̃g(2)(r,R). (4.6.43)

Next, assumingg(2)is independent ofR, then integrating over this variable simply
cancels a factor of volume in the denominator, yielding

〈Upair〉=

N^2

2 V

∫

dru(r) ̃g(r). (4.6.44)

Introducing spherical polar coordinates and assuming ̃g(r) is independent ofθandφ,
integrating over the angular variables leads to

〈Upair〉=

N^2

2 V

∫∞

0

dr 4 πr^2 u(r)g(r). (4.6.45)

Finally, inserting eqn. (4.6.45) into eqn. (4.6.37) gives the energy expression

E=

3

2

NkT+ 2πNρ

∫∞

0

dr r^2 u(r)g(r), (4.6.46)

which involves only the functional form of the pair potential and theradial distribution
function. Note that extending the integral overrfrom 0 to∞rather than limiting
it to the physical domain is justified if the potential is short-ranged. Interestingly, if
the potential energyU(r 1 ,...,rN) includes additionalN-body terms, such as 3-body
or 4-body terms, then by extension of the above analysis, an expression analogous to
eqn. (4.6.46) for the average energy would include additional termsinvolving general
N-point correlation functions, e.g.g(3)andg(4), etc.
Let us next consider the pressure, which is given by the thermodynamic derivative

P=kT

∂

∂V

lnQ(N,V,T) =

kT

Z(N,V,T)

∂Z(N,V,T)

∂V

. (4.6.47)

This derivative can be performed only if we have an explicit volume dependence in
the expression forZ(N,V,T). For a Hamiltonian of the standard form

H=

∑N

i=1

p^2 i

2 mi

+U(r 1 ,...,rN), (4.6.48)

the configurational partition function is

Z(N,V,T) =

∫

D(V)

dr 1 ···

∫

D(V)

drNe−βU(r^1 ,...,rN), (4.6.49)

whereD(V) is spatial domain defined by the physical container. It can be seenimme-
diately that the volume dependence is contained implicitly in the integration limits, so