1549380323-Statistical Mechanics Theory and Molecular Simulation

General formulation 415

11.4 General formulation for fermions and bosons

For systems of identical fermions or identical bosons, an exchange of particles does not
change the physical state. Therefore, the factorg({fnm}) is simply 1 for either particle
type. For fermions, the Pauli exclusion principle forbids two identical particles from
having the same set of quantum numbers. Note that the Slater determinant vanishes
if, for any two particlesiandj,ni=njandmi=mj. In the second quantization
formalism, this means that no two particles may occupy the same stateφn,m(x).
Consequently, the occupation numbers are restricted to be only 0or 1:

fnm= 0, 1 (Fermions). (11.4.1)

By contrast, since a permanent does not vanish ifni=njandmi=mj, the occupa-
tion numbersfnmfor a system of identical bosons have no such restriction and can,
therefore, take on any value between 0 andN:

fnm= 0, 1 , 2 ,...,N (Bosons). (11.4.2)

For either set of occupation numbers, the canonical partition function can be written
generally as

Q(N,V,T) =

∑

{fnm}

e−β

∑

m

∑

nfnmεn=

∑

{fnm}

∏

n

∏

m

e−βfnmεn. (11.4.3)

Note that the sum over occupation numbers in eqn. (11.4.3) must beperformed subject
to the restriction ∑

m

∑

n

fnm=N. (11.4.4)

This restriction makes performing the sum in eqn. (11.4.3) nontrivialwheng({fnm}) =

1. Evidently, the canonical ensemble is not the most convenient choice for deriving the
   thermodynamics of bosonic or fermionic ideal gases.
   Fortunately, since all ensembles are equivalent in the thermodynamic limit, we may
   choose from any of the other remaining ensembles. Of these, we willsee shortly that
   working in the grand canonical ensemble makes our task considerably easier. Recall
   that in the grand canonical ensemble,μ,V, andTare the control variables, and the
   partition function is given by

Z(μ,V,T) =

∑∞

N=0

ζNQ(N,V,T)

=

∑∞

N=0

eβμN

∑

{fnm}

∏

m

∏

n

e−βfnmεn. (11.4.5)

Note that the inner sum in eqn. (11.4.5) over occupation numbers is still subject to
the restriction

∑

m

∑

nfnm=N. However, in the grand canonical ensemble, there is a
final sum over all possible values ofN, and this sum allows us to lift the restriction on