Computational Physics

172 Classical equilibrium statistical mechanics

chemical potentialμand pressurePare given as derivatives of the entropy with
respect to the system parameters:

T=

(

∂S

∂E

)− 1

N,V

μ=−T

(

∂S

∂N

)

E,V

P=T

(

∂S

∂V

)

E,N

(7.6)

as can be readily seen from the first law of thermodynamics:^3

dE=TdS−PdV+μdN. (7.7)
In experimental situations, it is often the temperature that is kept constant and
not the energy (for the latter to be constant, the system must be insulated thermally
and mechanically). In order to achieve constant temperature, the system under
consideration is coupled to a heat bath, a much larger system with which it can
exchange heat. It turns out that a time average for the system under consideration
is equal to a weighted average over states with fixed volume and particle number
(the energy is no longer restricted); the weighting factor is the so-calledBoltzmann
factorexp[−H(X)/(kBT)]. Writingβ= 1 /(kBT),wehave

〈A〉NVT=

1

N!Z

∑

X

A(X)e−βH(X); (7.8a)

Z(N,V,T)=

1

N!

∑

X

e−βH(X). (7.8b)

The factorZensures proper normalisation. It is called thepartition functionand
it is related to the free energyF:

F=−kBTlnZ(N,V,T) (7.9)

which, in terms of thermodynamic quantities, is given by

F=E−TS. (7.10)

In equilibrium, the free energy assumes its minimum under the constraint of fixed
volume and particle number. The average in(7.8)is called thecanonical ensemble
averageor(NVT)ensemble average. Note that the partition function can be written
as a sum over sets of states with fixed energy:

Z(N,V,T)=

∑

E

e−βE(N,V,E), (7.11)

where(N,V,E)is the number of states with energyEas defined already in the
microcanonical ensemble. The number of states(N,V,E)is a rapidly increasing
function ofEand the Boltzmann distribution is a rapidly decreasing function ofE.

(^3) Often, the first law is stated without including changes in particle number dN.