17.4 Other ensembles

Thus far we have considered the canonical ensemble with constant N and

V , and given expectation U of the energy over the ensemble. It appeared

that the latter requirement implied the existence of a constant β, identi¬ed

with the inverse temperature. Thus the canonical ensemble is also called

the N, V, T ensemble.

Although the canonical ensemble is often the most useful one, it is by no

means the only possible ensemble. For example, we can constrain not only N

and V , but also the energy E for each system and obtain the microcanonical

ensemble; the ensemble then consists of systems in di¬erent microstates

460 Review of statistical mechanics

(wave functions) with the same degenerate energy.9 Instead of constraining

the volume for each system, we can prescribe the ensemble average of the

volume, which introduces another constant that appears to be related to the

pressure. This is the N, p, T ensemble. Finally, if we do not ¬x the particle

number for each system, but only ¬x the ensemble average of the particle

number, a constant appears that is related to the chemical potential. This

produces the grand-canonical or μ, V, T ensemble if the volume is ¬xed, or

the μ, p, T ensemble if the ensemble-averaged volume is ¬xed.

The recipe is always the same: Let wi be the probability that the system

is in state i (numbering each of all possible states, given the freedom we give

the various parameters), and maximize ’ i wi ln wi under the conditions

we impose on the ensemble. Each condition introduces one Lagrange multi-

plier, which can be identi¬ed with a thermodynamic quantity. This can be

summarized as follows:

• The N, V, E or microcanonical ensemble. The system will have a degener-

acy ©, being the number of states with energy E (or within a very small,

¬xed energy interval). ’ i wi ln wi is maximized under the only condi-

tion that i wi = 1, which implies that all probabilities wi are equal and

equal to 1/©; it follows that S = kB ln ©. Knowledge of the “partition

function” © (and hence S) as a function of V and E = U then generates

all thermodynamic quantities. For example, T = ‚S/‚E.

• The N, V, T or canonical ensemble. See above. The partition function

is Q(N, V, β) = i exp(’βEi ) and thermodynamic functions follow from

β = (kB T )’1 and A = ’kB T ln Q.

• The N, p, T or isobaric-isothermal ensemble. Here the particle number of

the system and the ensemble-averaged energy and volume are ¬xed. The

wi are now a function of volume (a continuous variable), and we look for

the probability wi (V ) dV that the system is in state i with volume between

V and V +dV . Thus10 H = ’ dV i wi (V ) ln wi (V ) is maximized under

the conditions

dV wi (V ) = 1, (17.26)

i

9 Neither experimentally, nor in simulations is it possible to generate an exact microcanonical

ensemble. The spacing between energy levels becomes very small for macroscopic systems and

complete thermal isolation including radiation exchange is virtually impossible; algorithms

usually do not conserve energy exactly. But the microcanonical ensemble can be de¬ned as

having an energy in a small interval (E, E + ”E).

10 This is a somewhat sloppy extension of the H-function of (17.4) with an integral. The H-

function becomes in¬nite when a continuous variable is introduced, because the number of

choices in the continuous variable is in¬nite. The way out is to discretize the variable V in

small intervals ”V . The equation for H then contains ln[wi (V ) ”V ]. But for maximization

the introduction of ”V is immaterial.

17.4 Other ensembles 461

dV wi (V )Ei (V ) = U, (17.27)

i

dV V wi (V ) = V. (17.28)

i

The Lagrange optimization yields

ln wi (V ) ∝ ’βEi (V ) ’ γV, (17.29)

or

1 ’βEi (V )’γV

wi (V ) = e , (17.30)

”

e’βEi (V )’γV

”= dV (17.31)

i

” is the isothermal-isobaric partition function. Identifying ’kB ln wi (V )

with the entropy S, we ¬nd that

S

= ln ” + βU + γV. (17.32)

kB

Hence, β = (kB T )’1 , γ = βp, and the thermodynamic functions follow

from

G = U ’ T S + pV = ’kB T ln ”. (17.33)

• The μ, V, T or grand-canonical ensemble.11 Here the ensemble averages of

E and N are ¬xed, and a Lagrange multiplier δ is introduced, related to

the condition N = NA n, where NA is Avogadro™s number and n is the

(average) number of moles in the system.12 The microstates now involve

every possible particle number N and all quantum states for every N .

The probabilities and partition function are then given by

1 ’βEN ,i +δN

wN,i = e , (17.34)

Ξ

e’βEN ,i .

eδN

Ξ= (17.35)

i

N

Working out the expression for the entropy S = ’kB ln wN,i and com-

paring with the thermodynamic relation

T S = U + pV ’ nμ, (17.36)

11 Often just called “the grand ensemble.”

12 For a multi-component system, there is a given average number of particles and a Lagrange

multiplier for each species. Many textbooks do not introduce Avogadro™s number here, with

the consequence that the thermodynamic potential is de¬ned per particle and not per mole as

is usual in chemistry.

462 Review of statistical mechanics

one ¬nds the identi¬cations β = (kB T )’1 , δ = μ/RT and

pV = kB T ln Ξ(μ, V, T ). (17.37)

This equation relates the grand partition function to thermodynamics.

Note that

∞

μN

Ξ= QN exp . (17.38)

RT

N =0

If we de¬ne the absolute activity » as:13

μ

def

» = exp(

, (17.39)

RT

then the grand partition function can be written as

∞

»N QN .

Ξ= (17.40)

N =0

Partial derivatives yield further thermodynamic functions:

‚ ln Ξ

» = N = NA n, (17.41)

‚»

‚ ln Ξ p

= , (17.42)

‚V kB T

‚ ln Ξ

= ’U. (17.43)

‚β

This ends our review of the most important ensembles. In simulations one

strives for realization of one of these ensembles, although integration errors

and deviations from pure Hamiltonian behavior may cause distributions that

are not exactly equal to those of a pure ensemble. If that is the case, one

may in general still trust observed averages, but observed ¬‚uctuations may

deviate signi¬cantly from those predicted by theory.

17.4.1 Ensemble and size dependency

One may wonder if and if so, why, the di¬erent ensembles yield the same