’1

bosons

’2

0.25 0.5 0.75 1 1.25 1.5

interparticle distance r/Λ

Figure 17.6 E¬ective exchange potential between fermions or bosons. The solid

black curves are the ¬rst form of (17.112); the dashed black curves are the approx-

imate second form. The distance is expressed in units of the de Broglie wavelength

√

Λ = h/ 2πmkB T . For comparison, the Lennard“Jones interaction for 4 He atoms,

also expressed as a function of r/Λ, is drawn in gray for temperatures of 5, 10, 15,

20, 25 and 30 K.

surface element dS, moving the surface an in¬nitesimal distance δ inwards,

then an amount of work p Sδ = ’p dV is done on the system, increasing its

internal energy. But these de¬nitions are not equivalent in the sense that

the mechanical pressure can be de¬ned locally and can have a tensorial char-

acter, while the thermodynamic pressure is a global equilibrium quantity.

In statistical mechanics we try to average a detailed mechanical quantity

(based on an atomic description) over an ensemble to obtain a thermody-

namic quantity. The question to be asked is whether and how a mechanical

pressure can be locally de¬ned on an atomic basis. After that we can aver-

age over ensembles. So let us ¬rst look at the mechanical de¬nition in more

detail.

17.7 Pressure and virial 481

4

4

4

4

4

negative side 4 positive side

E normal

dS = n dS

©

4

dF 4

4

4

4

4

Figure 17.7 A force dF = σ · dS acts on the negative side of a surface element.

17.7.1 The mechanical pressure and its localization

In a continuous medium a quantity related to to the local pressure, called

stress, is given by a second-rank tensor σ(r), de¬ned through the following

relation (see Fig. 17.7): the force exerted by the material lying on the

positive side of a static surface element dS with normal n (which points

from negative to positive side), on the material lying on the negative side of

dS, is given by

dF = σ(r) · n dS = σ(r) · dS, (17.113)

dF± = σ±β dSβ . (17.114)

β

The stress tensor is often decomposed into a diagonal tensor, the normal

stress, and the shear stress „ which contains the o¬-diagonal elements. The

force acting on a body transfers momentum into that body, according to

Newton™s law. However, the stress tensor should be distinguished from the

momentum ¬‚ux tensor Π, because the actual transport of particles also con-

tributes to the momentum ¬‚ux. Also the sign di¬ers because the momentum

¬‚ux is de¬ned positive in the direction of the surface element (from inside

to outside).21 The momentum ¬‚ux tensor is de¬ned as

def

Π±β = ’σ±β + ρv± vβ , (17.115)

or

Π = ’σ + vJ , (17.116)

21 There is no sign consistency in the literature. We follow the convention of Landau and Lifschitz

(1987).

482 Review of statistical mechanics

where ρ is the mass density and vJ is the dyadic vector product of the

velocity v and the mass ¬‚ux J = ρv through the surface element.

It is this momentum ¬‚ux tensor that can be identi¬ed with the pressure

tensor, which is a generalization of the pressure. If Π is isotropic, Π±β =

p δ±β , the force on a surface element of an impenetrable wall, acting from

inside to outside, is normal to the surface and equals p dS.

Is the pressure tensor, as de¬ned in (17.116) unique? No, it is not. The

stress tensor itself is not a physical observable, but is observable only through

the action of the force resulting from a stress. From (17.113) we see that

the force F V acting on a closed volume V , as exerted by the surrounding

material, is given by the integral over the surface S of the volume

σ · dS.

FV = (17.117)

S

In di¬erential form this means that the force density f (r), i.e., the force

acting per unit volume, is given by the divergence of the stress tensor:

f (r) = ∇ · σ(r). (17.118)

Thus only the divergence of the stress tensor leads to observables, and we

are free to add any divergence-free tensor ¬eld σ 0 (r) to the stress tensor

without changing the physics. The same is true for the pressure tensor Π.

Without further proof we note that, although the local pressure tensor is

not unique, its integral over a ¬nite con¬ned system, is unique. The same

is true for a periodic system by cancelation of surface terms. Therefore the

average pressure over a con¬ned or periodic system is indeed unique.

Turning from continuum mechanics to systems of interacting particles,

we ask the question how the pressure tensor can be computed from particle

positions and forces. The particle ¬‚ux component vJ of the pressure tensor

is straightforward because we can count the number of particles passing over

a de¬ned surface area and know their velocities. For the stress tensor part

all we have is a set of internal forces F i , acting on particles at positions r i .22

From that we wish to construct a stress tensor such that

∇ · σ(r) = F i δ(r ’ r i ). (17.119)

i

Of course this construction cannot be unique. Let us ¬rst remark that a

solution where σ is localized on the interacting particles is not possible for

the simple reason that σ cannot vanish over a closed surface containing a

22 Here we restrict the pressure as resulting from internal forces, arising from interactions within

the system. If there are external forces, e.g., external electric or gravitational ¬elds, such forces

are separately added.

17.7 Pressure and virial 483

particle on which a force is acting, because the divergence inside the closed

surface is not zero. As shown by Scho¬eld and Henderson (1982), however, it

is possible to localize the stress tensor on arbitrary line contours C0i running

from a reference point r 0 to the particle positions r i :

σ±β (r) = ’ δ(r ’ r c )(dr c )β . (17.120)

F i,±

C0i

i

For each particle this function is localized on, and equally distributed over,

the contour C0i . Taking the divergence of σ we can show that (17.119) is

recovered:

∇ · σ(r) = ’ F i∇ · δ(r ’ r c ) dr c

C0i

i

F i [δ(r i ) ’ δ(r 0 )]