Statistical Physics, Second Revised and Enlarged Edition

Thegrand canonical ensemble 139

potential.Inequilibrium theelectrochemicalpotential(often calledthe Fermilevel
in this context, as in Chapter 8) of the electrons becomes the same throughout the
material.Inaninhomogeneous material,thisisachievedby small-charge transfers
between thedifferent regions, anidea thatisfundamentalto our understandingofthe
transistor and of a host of other devices.

1 3.2 THEGRANDCANONICAL ENSEMBLE

Inorder todiscuss the properties ofopen systems,itis usefultolookfurther at the
concept ofthegrandcanonicalensembleintroducedin section 12.2. Theideaisto
develop a technique for a system in which particle number is not fixed from the outset,
butisdeterminedbythe systembeing open to a ‘particlebath’witha specificchemical
potentialμ.
This may be visualized as, say, being interested in a litre of air in the middle of the
room. This system ofinterest, the particularlitreinthe room,has well-definedthermal
properties, even though its energyUanditsparticle numberNare not constants.
Rather,Uis controlled by the temperatureT,a property of the whole ensemble. As
discussedfor the canonicalensemblein section 12.2,Uhas a probabilitydistribution
withaverysharppeak. Wecansaythattemperatureisapotentialforenergy.Similarly,
Nhas a strongly peaked probability distribution determined by the chemical potential
μ,again a propertyofthewholeofthe ensemble (room). As statedin section 13.1,μ
is a potentialfor particle number.
This approach gives the right flavour, but is not enough to develop a quantitative
description ofan open system. Alittle mathematicalimaginationis required,andthis
we now discuss.

13 .2.1 Thenewmethod

Method1. Thebasic methodadoptedsofarinthisbookcentres aroundthefollowing
equation:

∂lnt

∂nnj

+α+βεεj=0(13.3)

This appeared in section 2.1.5 (compare (2.11), following(2.7)), and in section 5.4.
It is a statement of the Lagrange multiplier approach. Equation (13.3) is the recipe
forfindingthe most probabledistribution. Itisdesignedto pickout the maximum
value oftsubject to the two restrictive conditions

∑

nnj=Nand

∑

nnjεεj=U.It
is a method for dealing with any type of assembly of weakly interacting particles,
localizedorgaseous, solongas the number ofmicrostatest({nnj})for a particular
distribution is appropriatelyexpressed.

Method 2. But (13.3) can alsobe expressedin a way that relates to the canonical
ensemble, introduced in Chapter 12. It is also a recipe designed toguarantee that