1549380323-Statistical Mechanics Theory and Molecular Simulation

166 Canonical ensemble

The quantity in the angle brackets in eqn. (4.6.56) is an instantaneous estimatorP(r,p)
for the pressure

P(r,p) =

1

3 V

∑N

i=1

[

p^2 i

mi

+ri·Fi

]

. (4.6.57)

Note the presence of the virial in eqns. (4.6.55) and (4.6.56). When,Fi= 0, the
pressure reduces to the usual ideal gas law. In addition, becauseof the virial theorem,
the two terms in eqn. (4.6.57) largely cancel, so that this estimator essentially measures
boundary effects. One final note concerns potentials that have an explicit volume
dependence. Volume dependence in the potential arises, for example, in molecular
dynamics calculations in systems with long-range forces. For such potentials, eqn.
(4.6.57) is modified to read

P(r,p) =

1

3 V

[N

∑

i=1

p^2 i

mi

+

∑N

i=1

ri·Fi− 3 V

∂U

∂V

]

. (4.6.58)

Inddimensions, the “3” eqns. (4.6.57) and (4.6.58) is replaced byd.
We now consider eqn. (4.6.55) for the case of a pair-wise additive potential (with no
explicit volume dependence). For such a potential, it is useful to introduce the vector,
fij, which is the force on particleidue to particlejwith

Fi=

∑

j 6 =i

fij. (4.6.59)

From Newton’s third law
fij=−fji. (4.6.60)

In terms offij, the virial can be written as

∑N

i=1

ri·Fi=

∑N

i=1

∑N

j=1,j 6 =i

ri·fij≡

∑

i,j,i 6 =j

ri·fij. (4.6.61)

By interchanging theiandjsummations in the above expression, we obtain

∑N

i=1

ri·Fi=

1

2





∑

i,j,i 6 =j

ri·fij+

∑

i,j,i 6 =j

rj·fji



, (4.6.62)

so that, using Newton’s third law, the virial can be expressed as

∑N

i=1

ri·Fi=

1

2





∑

i,j,i 6 =j

ri·fij−

∑

i,j,i 6 =j

rj·fij





=

1

2

∑

i,j,i 6 =j

(ri−rj)·fij≡

1

2

∑

i,j,i 6 =j

rij·fij, (4.6.63)

whererij=ri−rj. The ensemble average of this quantity is