tion of thermodynamic entropy by Clausius, with Boltzmann and Gibbs as

the main innovators near the end of the nineteenth century. The concepts

of atoms and molecules already existed but there was no notion of quantum

theory. The link to thermodynamics was properly made, including the in-

terpretation of entropy in terms of probability distributions over ensembles

of particle con¬gurations, but the quantitative counting of the number of

possibilities required an unknown elementary volume in phase space that

could only later be identi¬ed with Planck™s constant h. The indistinguisha-

bility of particles of the same kind, which had to be introduced in order to

avoid the Gibbs™ paradox,1 got a ¬rm logical basis only after the invention of

quantum theory. The observed distribution of black-body radiation could

not be explained by statistical mechanics of the time; discrepancies of this

kind have been catalysts for the development of quantum mechanics in the

beginning of the twentieth century. Finally, only after the completion of

basic quantum mechanics around 1930 could quantum statistical mechan-

ics “ in principle “ make the proper link between microscopic properties at

the atomic level and macroscopic thermodynamics. The classical statistical

mechanics of Gibbs is an approximation to quantum statistics.

In this review we shall reverse history and start with quantum statistics,

proceeding to classical statistical mechanics as an approximation to quantum

1 In a con¬guration of N particles there are N ! ways to order the particles. If each of these

ways are counted as separate realizations of the con¬guration, a puzzling paradox arises: ther-

modynamic quantities that involve entropy appear not to be proportional to the size of the

system. This paradox does not arise if the number of realizations is divided by N !. In quantum

statistics one does not count con¬gurations, but quantum eigenstates which incorporate the

indistinguishability of particles of the same kind in a natural way.

453

454 Review of statistical mechanics

statistics. This will enable us to see the limitations of classical computational

approaches and develop appropriate quantum corrections where necessary.

Consider an equilibrium system of particles, like nuclei and electrons,

possibly already pre-organized in a set of molecules (atoms, ions). Sup-

pose that we know all the rules by which the particles interact, i.e., the

Hamiltonian describing the interactions. Then we can proceed to solve the

time-independent Schr¨dinger equation to obtain a set of wave functions

o

and corresponding energies, or “ by classical approximation “ a set of con-

¬gurations in phase space, i.e., the multidimensional space of coordinates

and momenta with their corresponding energies. The aim of statistical me-

chanics is to provide a recipe for the proper averaging over these detailed

solutions in order to obtain thermodynamic quantities. Hopefully the av-

eraging is such that the (very di¬cult and often impossible) computation

of the detailed quantities can be avoided. We must be prepared to handle

thermodynamic quantities such as temperature and entropy which cannot

be obtained as simple averages over microscopic variables.

17.2 Ensembles and the postulates of statistical mechanics

The basic idea, originating from Gibbs,2 is to consider a hypothetical ensem-

ble of a large number of replicas of the system, with the same thermodynamic

conditions but di¬erent microscopic details. Properties are then obtained

by averaging over the ensemble, taken in the limit of an in¬nite number of

replicates. The ensemble is supposed to contain all possible states of the

system and be representative for the single system considered over a long

time. This latter assumption is the ergodic postulate. Whether a realistic

system is in practice ergodic (i.e., are all microscopic possibilities indeed

realized in the course of time?) is a matter of time scale: often at low tem-

peratures internal processes may become so slow that not all possibilities are

realized in the experimental observation time, and the system is not ergodic

and in fact not in complete equilibrium. Examples are metastable crystal

modi¬cations, glassy states, polymer condensates and computer simulations

that provide incomplete sampling or insu¬cient time scales.

Let us try to set up an appropriate ensemble. Suppose that we can de-

scribe discrete states of the system, numbered by i = 1, 2, . . . with energies

Ei . In quantum mechanics, these states are the solutions of the Schr¨dinger

o

2 Josiah Willard Gibbs (1839“1903) studied mathematics and engineering in Yale, Paris, Berlin

and Heidelberg, and was professor at Yale University. His major work on statistical mechanics

dates from 1876 and later; his collected works are available (Gibbs, 1957). See also http://www-

gap.dcs.st-and.ac.uk/∼history/Mathematicians/Gibbs.html.

17.2 Ensembles and the postulates of statistical mechanics 455

equation for the whole system.3 Note that energy levels may be degenerate,

i.e., many states can have the same energy; i numbers the states and not

the distinct energy levels. In classical mechanics a state may be a point

in phase space, discretized by subdividing phase space into elementary vol-

umes. Now envisage an ensemble of N replicas. Let Ni = wi N copies be in

state i with energy Ei . Under the ergodicity assumption, the fraction wi is

the probability that the (single) system is to be found in state i. Note that

wi = 1. (17.1)

i

The number of ways W the ensemble can be made up with the restriction

of given Ni ™s equals

N!

W= (17.2)

Πi Ni !

because we can order the systems in N ! ways, but should not count per-

mutations among Ni as distinct. Using the Stirling approximation for the

factorial,4 we ¬nd that

ln W = ’N wi ln wi . (17.3)

i

If the assumption is made (and this is the second postulate of statistical

mechanics) that all possible ways to realize the ensemble are equally prob-

able, the set of probabilities {wi } that maximizes W is the most probable

distribution. It can be shown that in the limit of large N , the number of

ways the most probable distribution can be realized approaches the total

number of ways that any distribution can be realized, i.e., the most proba-

ble distribution dominates all others.5 Therefore our task is to ¬nd the set

of probabilities {wi } that maximizes the function

H=’ wi ln wi . (17.4)

i

This function is equivalent to Shannon™s de¬nition of information or un-

certainty over a discrete probability distribution (Shannon, 1948).6 It is

3 The states as de¬ned here are microscopic states unrelated to the thermodynamic states de¬ned

in Chapter 16. √

N ! ≈ N N e’N 2πN {1 + (O)(1/N )}. For the present application and with N ’ ∞, the

4

approximation N ! ≈ N N e’N su¬ces.

5 More precisely: the logarithm of the number of realizations of the maximum distribution

approaches the logarithm of the number of all realizations in the limit of N ’ ∞.

6 The relation with information theory has led Jaynes (1957a, 1957b) to propose a new foundation

for statistical mechanics: from the viewpoint of an observer the most unbiased guess he can

make about the distribution {wi } is the one that maximizes the uncertainty H under the

constraints of whatever knowledge we have about the system. Any other distribution would

456 Review of statistical mechanics

also closely related (but with opposite sign) to the H-function de¬ned by

Boltzmann for the classical distribution function for a system of particles in

coordinate and velocity space. We shall see that this function is proportional

to the entropy of thermodynamics.

17.2.1 Conditional maximization of H

The distribution with maximal H depends on further conditions we may

impose on the system and the ensemble. Several cases can be considered,

but for now we shall concentrate on the N, V, T or canonical ensemble. Here

the particle number and volume of the system and the expectation of the

energy, i.e., the ensemble-averaged energy i wi Ei , are ¬xed. The systems

are allowed to interact weakly and exchange energy. Hence the energy per

system is not constant, but the systems in the ensemble belong to the same

equilibrium conditions. Thus H is maximized under the conditions

wi = 1, (17.5)

i

wi Ei = U. (17.6)

i