N !h3N

This is the starting point for the application, in the following section, of

statistical mechanics to systems of particles that follow classical equations

of motion. We observe that this equation is consistent with an equilibrium

probability density proportional to exp ’βH(p, r) in an isotropic phase space

p, r, divided into elementary units of area h with equal a priori statistical

weights. There is one single-particle quantum state per unit of 6D volume

h3 . The N ! means the following: If two identical particles (1) and (2)

exchange places in phase space, the two occupations p(1)p (2)r(1)r (2) and

p(2)p (1)r(2)r (1) should statistically be counted as one.

The quantum corrections to the classical partition function can be ex-

pressed in several ways. The e¬ect of quantum corrections on thermody-

namical quantities is best evaluated through the quantum corrections to the

Helmholtz free energy A. Another view is obtained by expressing quantum

corrections as corrections to the classical Hamiltonian. These can then be

used to generate modi¬ed equations of motion, although one should realize

that in this way we do not generate true quantum corrections to classical

dynamics, but only generate some kind of modi¬ed dynamics that happens

to produce proper quantum corrections to equilibrium phase-space distribu-

tions.

First look at the quantum correction to the free energy A = ’kB T ln Q.

Noting that

Q = Qcl (1 + fcor ), (17.105)

where · · · denotes a canonical ensemble average

dp dr fcor exp(’βH)

fcor = , (17.106)

dp dr exp(’βH)

we see, using ln(1 + x) ≈ x), that

A = Acl ’ kB T fcor . (17.107)

By partial integration the second derivative of V can be rewritten as:

∇2 V = β (∇j V )2 . (17.108)

j

2

The correction now reads

2 1

A = Acl + (∇j V )2 . (17.109)

24(kB T )2 mj

j

The averaged quantity is the sum of the squared forces on the particles.

478 Review of statistical mechanics

Helmholtz free energy /hν

1

Acl+corr

0.75

0.5

Aqu

0.25

Acl

0

’0.25

’0.5

0.25 0.5 0.75 1 1.25 1.5

Temperature kT/hν

Figure 17.5 The classical Helmholtz free energy of the harmonic oscillator (long-

dashed), the 2 -correction added to the classical free energy (short-dashed) and the

exact quantum free energy (solid).

The use of this 2 -Wigner correction is described in Section 3.5 on page

70. Quantum corrections for other thermodynamic functions follow from

A. As an example, we give the classical, quantum-corrected classical, and

exact quantum free energy for the harmonic oscillator in Fig. 17.5. The

improvement is substantial.

It is possible to include the 2 -term into the Hamiltonian as an extra

potential term:

V cor = ’kB T f cor . (17.110)

If done in this fashion, calculation of the force on particle i then requires

a double summation over particles j and k, i.e., a three-body interaction.19

The inclusion in a dynamics simulation would be cumbersome and time

consuming, while not even being dynamically correct. However, there are

several ways to devise e¬ective interparticle interactions that will lead to

the correct 2 -correction to the free energy when applied to equilibrium

simulations. An intuitively very appealing approach is to consider each

particle as a Gaussian distribution. The width of such a distribution can

be derived from Feynman™s path integrals (see Section 3.3 on page 44) and

leads to the Feynman“Hibbs potential, treated in Section 3.5 on page 70.

19 See, e.g., Huang (1987)

17.7 Pressure and virial 479

Next we consider the exchange term, i.e., the last line of (17.102). We

drop the ¬rst-order term in β, which is zero for an ideal gas and for high

temperatures; it may however reach values of the order 1 for condensed

phases. It is now most convenient to express the e¬ect in an extra correction

potential:

ln 1 “ e’mkB T rij / e’mkB T rij / .

2 2 2 2

V cor = ’kB T ≈ ±kB T

i<j i<j

(17.111)

The ¬rst form comes from the equations derived by Uhlenbeck and Gropper

(1932) (see also Huang, 1987); the second form is an approximation that is

invalid for very short distances. This is an interesting result, as it indicates

that fermions e¬ectively repel each other at short distance, while bosons

attract each other. This leads to a higher pressure for fermion gases and a

lower pressure for boson gases, as was already derived earlier (Eq. (17.68)

on page 469). The interparticle correction potential can be written in terms

of the de Broglie wave length Λ (see (17.69) on page 469):

rij rij

2 2

Vij = ’kB T ln 1 “ exp ’2π ≈ ±kB T exp ’2π

cor

.

Λ Λ

i<j

(17.112)

This exchange potential is not a true, but an e¬ective potential with the

e¬ect of correcting equilibrium ensembles to ¬rst-order for exchange e¬ects.

The e¬ective potential “acts” only between identical particles. Figure 17.6

shows the form and size of the exchange potential. When compared to the

interaction potential for a light atom (helium-4), it is clear that the e¬ects of

exchange are completely negligible for temperatures above 15 K. In “normal”

molecular systems at “normal” temperatures exchange plays no role at all.

17.7 Pressure and virial

There are two de¬nitions of pressure:20 one stems from (continuum) me-

chanics and equals the normal component of the force exerted on a surface

per unit area; the other stems from the ¬rst law of thermodynamics (16.24)

and equals minus the partial derivative of the internal energy with respect

to the volume at constant entropy, i.e., without exchange of heat. These

de¬nitions are equivalent for a system with homogeneous isotropic pressure:

if a surface with area S encloses a volume V and a force p dS acts on every

20 The author is indebted to Dr Peter Ahlstr¨m and Dr Henk Bekker for many discussions on

o

pressure in the course of preparing a review that remained unpublished. Some of the text on

continuum mechanics originates from Henk Bekker.

480 Review of statistical mechanics

potential energy /kBT

3

5 10 15 20 25 30 K

2

fermions

1

0