Statistical Mechanics - Physics at Oregon State University

4.1. INTRODUCTION. 73
and hence in the distribution functions the terms ±1 can be ignored. In that
case both distribution functions reduce to the same form
fMB(ɛ; T, µ) = e µ−ɛ
k B T (4.21)
which is the Maxwell-Boltzmann function. Quantum effects do not play a role
anymore, since in this limit there is no distinction between Fermions and Bosons.
This is therefore the classical limit and an ideal gas is also called a Boltzmann
gas.
Use of distribution functions.
Once we have chosen the appropriate distribution function for our system,
it is easy to calculate thermodynamic variables. A general property Q of a
independent gas as a function of T, µ, and V follows from the values of Qo(V )
for all orbitals from
Q(T, V, µ) =
f(ɛo(V ); T, µ)Qo(V ) (4.22)
orb
Two important examples are the number of particles and the internal energy
N(T, V, µ) =
f(ɛo(V ); T, µ) (4.23)
orb
U(T, V, µ) =
f(ɛo(V ); T, µ)ɛo(V ) (4.24)
orb
These two equation are sufficient to derive all of thermodynamics, although
we later will see that there is a better way of doing that. From the first equation
we can find by inversion µ(T, V, N) and then by integration F (T, V, N). The
second equation can be written as a function of T, V, N too when we substitute
µ. But in many cases we do not need that, and the strength of this approach is
really when we can use equation 4.22 directly. Note: in order for these sums to
converge, we always need µ < ɛo. In other words we need µ < ɛmin.
What have we gained by introducing this formulation of independent particles?
The formulas we need still look very similar to those we used before. The
important difference, however, is the number of terms involved. If the number
of orbitals No is finite, the number of N particle states in the partition function
is No N
N0
for bosons and N for fermions. Since in practice No ≫ N the
numbers are very similar for bosons and fermions. This indicates that the sum
over many particle states in the general formalism is much more complicated
than the sum over orbitals in the independent particle systems! In many cases
the sum over orbitals reduces to an integral over a single variable, and these
integrals are easy to analyze. Only for very small systems (e.g. 3 orbitals, 2
particles) is the complexity of the general approach similar to the independent
particle formalism. But keep in mind that using distribution functions implies
independent particles!