Statistical Mechanics - Physics at Oregon State University
Statistical Mechanics - Physics at Oregon State University
Statistical Mechanics - Physics at Oregon State University
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70 CHAPTER 4. STATISTICS OF INDEPENDENT PARTICLES.<br />
tial µ are specified. The volume V of the gas determines the orbital energies<br />
ɛo(V ). We will now divide this gas in a large number of subsystems, each of<br />
which contains exactly one orbital. If all these subsystems are in thermal and<br />
diffusive equilibrium with the reservoir, they are in equilibrium with each other.<br />
Therefore, the properties of the total system can be obtained by adding the<br />
corresponding properties of all subsystems and the total system is really the<br />
sum of the subsystems! The properties of each subsystem, or each orbital o,<br />
follow from<br />
Zo(T, µ, V ) =<br />
?<br />
n=0<br />
e n(µ−ɛo)<br />
k B T (4.6)<br />
because the energy of a st<strong>at</strong>e of the subsystem with n particles in the orbital<br />
o is simple nɛo. Quantum effects are still allowed to play a role here, and they<br />
determine the upper limit of the summ<strong>at</strong>ion. The Pauli exclusion principle for<br />
fermions tells us th<strong>at</strong> we can have <strong>at</strong> most one particle in each orbital and hence<br />
for fermions there are only two terms in the sum. For bosons, on the other hand,<br />
there is no such a restriction, and the upper limit is ∞.<br />
Fermions.<br />
We will first consider the case of Fermions. The grand partition function for<br />
an orbital is<br />
Z(T, µ, V ) = 1 + e µ−ɛo<br />
k B T (4.7)<br />
Once we have the partition function for each orbital, we can calcul<strong>at</strong>e the average<br />
number of particles < no > in th<strong>at</strong> orbital. In terms of the probabilities Pn of<br />
finding n particles in this orbital we have < no >= 0P0 + 1P1. Hence we find<br />
< n o >=<br />
e µ−ɛo<br />
k B T<br />
1 + e µ−ɛo<br />
k B T<br />
=<br />
1<br />
e ɛo−µ<br />
k B T + 1<br />
(4.8)<br />
Hence the average number of particles in a subsystem depends on T and µ and<br />
ɛo(V ). The only quantity of the orbital we need to know to determine this<br />
average is the energy! No other aspects of the orbital play a role. In general,<br />
the function of the energy ɛ which yields the average number of particles in an<br />
orbital with energy ɛ is called a distribution function and we have in the case<br />
of Fermions:<br />
fF D(ɛ; T, µ) =<br />
1<br />
e ɛ−µ<br />
k B T + 1<br />
(4.9)<br />
This function is called the Fermi-Dirac distribution function. The shape of this<br />
function is well known, and is given in figure (4.1).<br />
The Fermi-Dirac distribution function has the following general properties: