i

= F i δ(r i ). (17.121)

i

Proof The ¬rst step follows from the three-dimensional generalization of

b b

d d

f (ξ ’ x) dx = ’ f (ξ ’ x) dx = ’f (ξ ’ b) + f (ξ ’ a), (17.122)

dξ dx

a a

the last step uses the fact that = 0 for internal forces.

i Fi

If we integrate the stress tensor over the whole (con¬ned) volume of the

system of particles, only the end points in the line integral survive and we

obtain the sum of the dyadic products of forces and positions:

σ d3 r = ’ F i (r i ’ r 0 ) = ’ F i ri , (17.123)

V i i

which is independent of the choice of reference position r 0 .

The introduction of a reference point is

E

undesirable as it may localize the stress Cij

tensor far away from the interacting par-

it tj

ticles. When the forces are pair-additive,

the stress tensor is the sum over pairs; for

k

0

each pair i, j the two contours from the C

C0i

0j

reference position can be replaced by one t

contour between the particles, and the ref- 0

erence position cancels out.

484 Review of statistical mechanics

(a) (c)

(b) (d)

Figure 17.8 Four di¬erent contours to localize the stress tensor due to interaction

between two particles. Contour (b) is the minimal path with optimal localization.

The result is

δ(r ’ r c ) (dr c )β ,

σ±β = Fij± (17.124)

Cij

i<j

with integral

σ ±β d3 r = Fi± (rjβ ’ riβ ) = ’Fi± riβ ’ Fj± rjβ , (17.125)

V

consistent with (17.123). Here Fij± is the ± component of the force acting

on i due to the interaction with j. The path between the interacting particle

pair is irrelevant for the pressure: Fig. 17.8 gives a few examples including

distribution over a collection of paths (a), the minimal path, i.e., the straight

line connecting the particles (b), and curved paths (c, d) that do not con¬ne

the localization to the minimal path. Irving and Kirkwood (1950) chose the

straight line connecting the two particles as a contour, and we recommend

that choice.23

17.7.2 The statistical mechanical pressure

Accepting the mechanical de¬nition of the pressure tensor as the momentum

¬‚ux of (17.116), we ask what the average pressure over an ensemble is. First

we average over the system, and then over the ensemble of systems. The

average over the whole system (we assume our system is con¬ned in space) is

given by the integral over the volume, divided by the volume (using dyadic

vector products):

1 1

=’ σ(r)d3 r + mi v i v i , (17.126)

Π V

V V

V i

23 This choice is logical but not unique, although Wajnryb et al. (1995) argue that additional

conditions make this choice unique. The choice has been challenged by Lovett and Baus (1997;

see also Marechal et al., 1997) on the basis of a local thermodynamic pressure de¬nition, but

further discussion on a physically irrelevant choice seems pointless (Rowlinson, 1993). For

another local de¬nition see Zimmerman et al. (2004).

17.7 Pressure and virial 485

or, with (17.123):

V= F i ri + mi v i v i . (17.127)

Π V

i i

In a dynamic system this instantaneous volume-averaged pressure is a ¬‚uc-

tuating function of time. We remark that (in contrast to the local tensor)

this averaged tensor is symmetric, because in the ¬rst term the di¬erence

between an o¬-diagonal element and its transpose is a component of the

total torque on the system, which is always zero in the absence of external

forces. The second term is symmetric by de¬nition. Finally, the thermo-

dynamic pressure tensor P which we de¬ne as the ensemble average of the

volume-averaged pressure tensor, is given by

PV = F i ri + mi v i v i , (17.128)

i i

where the angular brackets are equilibrium ensemble averages, or because of

the ergodic theorem, time averages over a dynamic equilibrium system. P

is a “sharp” symmetric tensor. For an isotropic system the pressure tensor

is diagonal and its diagonal elements are the isotropic pressure p:

1 1 2

F i · ri +

pV = tr P V = Ekin . (17.129)

3 3 3

i

The ¬rst term on the right-hand side relates to the virial of the force, al-

ready de¬ned by Clausius (see, e.g., Hirschfelder et al., 1954) and valid for

a con¬ned (i.e., bounded) system:

1

def

Ξ=’ F i · ri . (17.130)

2

i

Here F i is the total force on particle i, including external forces. The re-

sulting virial Ξ is the total virial, which can be decomposed into an internal

virial due to the internal forces and an external virial, for example caused

by external forces acting on the boundary of the system in order to maintain

the pressure. Written with the Clausius virial, (17.129) becomes

2

pV = ( Ekin ’ Ξint ). (17.131)

3

This relation follows also directly from the classical virial theorem:

Ξtot = Ekin (17.132)

486 Review of statistical mechanics

which is valid for a bounded system.24 This virial theorem follows also from

the generalized equipartition theorem, treated in Section 17.10 on page 503.

Since the external virial due to an external force ’p dS acting on any surface

element dS equals