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