dom that we allow for the system and therefore their entropies are di¬erent

as well. It is quite clear that the entropy of a given system is larger in a

canonical than in a microcanonical ensemble, and larger still in a grand en-

semble, because there are more microstates allowed. This would seemingly

13 The name “absolute activity” is logical if we compare μ = RT ln » with the de¬nition of activity

a (e.g., (16.66)): μ = μ0 + RT ln a.

17.5 Fermi“Dirac, Bose“Einstein and Boltzmann statistics 463

lead to di¬erent values for the entropy, as well as for other thermodynamic

functions.

The point is that, although the entropies are not strictly the same, they

tend to the same value when the system is macroscopic and contains a large

number of particles. Each of the thermodynamic variables that is not ¬xed

per system has a probability distribution over the ensemble that tends to a

delta function in the limit of an in¬nite system, with the same value for each

kind of ensemble. The ensembles do di¬er, however, both in the values of

averages and in ¬‚uctuations, for systems of ¬nite size with a ¬nite number

of particles. In numerical simulations in particular one deals with systems

of ¬nite size, and one should be aware of (and correct for) the ¬nite-size

e¬ects of the various ensembles.

Let us, just for demonstration, consider the ¬nite-size e¬ects in a very

simple example: a system of N non-interacting spins, each of which can

be either “up” or “down”. In a magnetic ¬eld the total energy will be

proportional to i mi . Compare the microcanonical ensemble with exact

energy E = 0, requiring 1 N spins up and 1 N spins down, with the canonical

2 2

ensemble at such high temperature that all 2N possible con¬gurations are

equally likely (the Boltzmann factor for any con¬guration equals 1). The

entropy in units of kB is given by

N!

= ln N ! ’ 2 ln( 1 N )!,

microcanonical : S = ln (17.44)

2

1 2

[( 2 N )!]

canonical : S = N ln 2. (17.45)

We see that for large N , in the limit where the Stirling approximation

ln N ! ≈ N ln N ’ N is valid, the two entropies are equal. For smaller N

this is not the case, as Fig. 17.1 shows. Plotting the “observed” entropy

versus N ’1 ln N allows extrapolation to in¬nite N .

17.5 Fermi“Dirac, Bose“Einstein and Boltzmann statistics

In this section ¬rst a more general formulation for the canonical partition

function will be given in terms of the trace of an exponential matrix in

Hilbert space. The reader may wish to review parts of Chapter 14 as an in-

troduction. Then we shall look at a system of non-interacting particles (i.e.,

an ideal gas) where the symmetry properties of the total wave function ap-

pear to play a role. For fermions this leads to Fermi“Dirac (FD) statistics,

while bosons obey Bose“Einstein (BE) statistics. In the limit of low den-

sity or high temperature both kinds of statistics merge into the Boltzmann

approximation.

464 Review of statistical mechanics

Smicro

1000 500 300 200 100 N

0.70

2000

ln 2

Scan

0.69

0.68

0.67

0.66

0.01 0.02 0.03 0.04 0.05

1

- ln N

N

Figure 17.1 Di¬erence between canonical and microcanonical ensemble entropies

for ¬nite systems. Case: N non-interacting spins in a magnetic ¬eld at high tem-

perature. The canonical entropy per mole in units of R (or per spin in units of kB )

equals ln 2, independent of system size (solid line); the microcanonical (at E = 0)

molar entropy depends on N and tends to ln 2 for large N . Extrapolation is nearly

linear if S is plotted against N ’1 ln N : the dashed line is a linear connection be-

tween the data points at N = 1000 and N = 2000.

17.5.1 Canonical partition function as trace of matrix

Consider a system of N particles in a box of volume V . For simplicity we take

a cubic box and assume the system to be in¬nite and periodic with the box

as repeating unit cell. These restrictions are convenient but not essential:

the system may contain various types of di¬erent particles, or have another

shape. Although not rigorously proved, it is assumed that e¬ects due to

particular choices of boundary conditions vanish for large system sizes since

these e¬ects scale with the surface area and hence the e¬ect per particle

scales as N ’1/3 . The wave function Ψ(r 1 , r 2 , . . .) of the N -particle system

can be constructed as a linear combination of properly (anti)symmetrized

products of single-particle functions. The N -particle wave function must

17.5 Fermi“Dirac, Bose“Einstein and Boltzmann statistics 465

change sign if two identical fermions are interchanged, and remain the same

if two identical bosons are interchanged (see Chapter 2, page 37, for details).

In equilibrium we look for stationary quantum states involving all parti-

cles. There will be stationary solutions with wave functions Ψi and energies

Ei and the canonical partition function is given by

Q= exp(’βEi ). (17.46)

i

Consider a Hilbert space spanned by all the stationary solutions Ψi of the

Hamiltonian (see Chapter 14 for details of vectors and transformations in

Hilbert spaces). Then the matrix H is diagonal and Ei = Hii . Thus we can

also write

Q = tr exp(’βH). (17.47)

This equality is quite general and also valid on any complete basis set on

which the Hamiltonian is not diagonal. This is easily seen by applying a

unitary transformation U that diagonalizes H, so that U † HU is diagonal,

and realizing that (see (14.30) on page 386)

exp(’βU † HU ) = U † exp(’βH)U, (17.48)

and that, because the trace of a product is invariant for cyclic exchange of

the elements in the product,

tr U † AU = tr U U † A = tr A. (17.49)

Solving Q would now require the computation of all diagonal elements

of the Hamiltonian (on a complete basis set). This seems simpler than

solving the Schr¨dinger equation for the whole system, but is still in practice

o

impossible for all but the simplest systems.

17.5.2 Ideal gas: FD and BE distributions

In order to get insight into the e¬ect of the symmetry requirements of the

wave function on the partition function we shall now turn to a system which

is solvable: the ideal gas. In the ideal gas there are no interactions between

particles. We shall also, for convenience, but without loss of generality, as-

sume that the system contains identical particles, which are either fermions

or bosons. Let the single-particle wave functions be given by φk (r) with

energy µk (Fig. 17.2). The φk form an orthonormal set of functions, and the

466 Review of statistical mechanics

. k