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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5.1. FERMIONS IN A BOX. 91<br />
Znx,ny,nz(T, µ, V ) = 1 + e µ−ɛ(nx,ny ,nz )<br />
k B T (5.8)<br />
We now introduce again λ = e µ<br />
k B T , which is always positive, and if we are<br />
allowed to replace the sum by an integral we obtain:<br />
1<br />
V Ω(T, µ, V ) = −(2S + 1)(2π)−3 <br />
kBT<br />
d 3 k log(1 + λe − 2 k 2<br />
2Mk B T ) (5.9)<br />
Formulas like these make it clear th<strong>at</strong> we really want to use kB for the Boltzmann<br />
constant, in order to avoid confusion with the norm of the wave vector, k.<br />
Free particles have a simple volume dependence!<br />
One remarkable thing happened. The right hand side does not depend on<br />
volume anymore! When we had a summ<strong>at</strong>ion over discrete k-vectors, the volume<br />
dependence was included in these vectors. But when we changed to a continuous<br />
summ<strong>at</strong>ion (or integr<strong>at</strong>ion), this volume dependence disappeared, and the only<br />
volume dependence is a simple pre-factor V , a linear behavior!<br />
Those m<strong>at</strong>hem<strong>at</strong>ical details!<br />
Wh<strong>at</strong> could go wrong here? There is an error associ<strong>at</strong>ed with the change<br />
from summ<strong>at</strong>ion to integr<strong>at</strong>ion, and in general we have:<br />
<br />
1<br />
Ω(T, µ, V ) = −(2S + 1)(2π)<br />
V −3 <br />
kBT<br />
d 3 k log(1 + λe − 2k 2<br />
<br />
2MkB T ) [1 + E(T, V, µ)]<br />
(5.10)<br />
where we know th<strong>at</strong> limV →∞ E(T, V, µ) = 0. But we would like to be able to<br />
differenti<strong>at</strong>e the result with respect to T and µ, and hence we need to require<br />
uniform convergence! But th<strong>at</strong> in turn also begs the question if the original<br />
series is uniformly convergent! In order to discuss the convergence properties<br />
we have to take two steps. First, we have to show th<strong>at</strong> the series converges<br />
uniformly. Second, we have to show th<strong>at</strong> the series in the thermodynamic limit<br />
converges uniformly to the integral. We will discuss the uniform convergence of<br />
the series first. The important thing to notice is th<strong>at</strong> both the integral and the<br />
series converge.<br />
For large values of the wave vector the terms in the series behave like:<br />
tnx,ny,nz = log(1 + λe − ɛ(nx,ny ,nz )<br />
k B T ) ≈ λe − ɛ(nx,ny ,nz )<br />
k B T (5.11)<br />
which approaches zero exponentially fast. Next we consider a domain in (T, µ, L)<br />
space with 0 T Tm , −∞ µ µm , and 0 L Lm. It is clear th<strong>at</strong> for<br />
large values of the indices we have