Computational Physics

7.1 Basic theory 173

The product of the two functions peaks sharply at some valueE ̄and the system
will be found to have an energy very close to this value most of the time. This
suggests that there is in practice not much difference between the canonical and
the microcanonical system in which the energy is kept rigorously fixed atE ̄. This
is a manifestation of the so-called ensemble equivalence: because of the law of
large numbers, measurable physical quantities exhibit very small fluctuations –
hence fixing them to their average value leaves the system essentially unchanged.
For finite systems, the differences between the ensembles increase with decreasing
system size.
Using the definition of the entropy(7.5), we may write(7.11)as
Z(N,V,T)=

∑

E

e−β(E−TS)=

∑

E

e−βFE, (7.12)

whereFEis the free energyE−TSwithSevaluated in the microcanonical ensemble
with energyE, and we see that the sum is indeed dominated by the states for which
the free energy is minimal.
Again using the first law of thermodynamics, (7.7), we can derive the following
thermodynamic quantities from the free energy:

μ=

(

∂F

∂N

)

V,T

P=−

(

∂F

∂V

)

N,T

S=−

(

∂F

∂T

)

V,N

. (7.13)

If the pressurePis kept constant and not the volume, as in a cylinder closed
by a movable piston, we obtain an average over the isothermal-isobaric or(NPT)
ensemble:

〈A〉NPT=

1

N!Q

∫

dVe−βPV

∑

X

e−βH(X)A(X); (7.14a)

Q(N,P,T)=

∫

dVe−βPV

1

N!

∑

X

e−βH(X)=

∫

dVe−βPVZ(N,V,T),

(7.14b)

whereQ(N,P,T)is again called the partition function. We see thatQis related to
the canonical partition functionZin much the same way asZwas related to the
functionin the microcanonical ensemble – seeEq. (7.11).Qis related to the
Gibbs free energy or Gibbs potentialG:

G=−kBTlnQ(N,P,T). (7.15)

Gcan be expressed in terms of thermodynamic quantities as

G=E−TS+PV, (7.16)

and it assumes its mimimum value when the system has reached equilibrium under
the condition of fixed temperature and pressure. For magnetic systems, the role of