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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94 CHAPTER 5. FERMIONS AND BOSONS<br />
with<br />
Ω(0, µ, V ) = −(2S + 1) <br />
lim<br />
T →0<br />
nx,ny,nz<br />
kBT log(Znx,ny,nz(T, µ, V )) (5.17)<br />
Znx,ny,nz(T, µ, V ) = 1 + e µ−ɛ(nx,ny ,nz )<br />
k B T (5.18)<br />
The limit of the argument is easy, since µ(0) = ɛF (N) > 0. Therefore the<br />
argument is µ − ɛ(nx, ny, nz) if µ > ɛ(nx, ny, nz), because in th<strong>at</strong> case we have<br />
and<br />
Znx,ny,nz(T, µ, V ) ≈ e µ−ɛ(nx,ny ,nz )<br />
k B T (5.19)<br />
kBT log(Znx,ny,nz (T, µ, V )) ≈ kBT µ − ɛ(nx, ny, nz)<br />
kBT<br />
(5.20)<br />
The argument is 0 if µ < ɛ(nx, ny, nz), because now the function Znx,ny,nz(T, µ, V )<br />
approaches 1 rapidly. As a result we have<br />
Ω(0, µ, V )<br />
V<br />
2S + 1<br />
= −<br />
(2π) 3<br />
<br />
d 3 k(µ − ɛ( k))Θ(µ − ɛ( k)) + error (5.21)<br />
with Θ(x) being the well-known step function: Θ(x) = 0 for x < 0 and 1 for<br />
x > 0. In this case the error also goes to zero for V → ∞, only in a different<br />
algebraic manner. This shows th<strong>at</strong> we can use<br />
Ω(T, µ, V )<br />
V<br />
2S + 1<br />
= −<br />
(2π) 3<br />
<br />
d 3 µ−<br />
kkBT log(1 + e<br />
2k 2<br />
2M<br />
kB T ) (5.22)<br />
for all values of T on [0, Tm] and also in the limit T → 0 with an error less than<br />
a small number which only depends on V and not on T. The limit for small<br />
temper<strong>at</strong>ures in the last general form of the integral is the same step function<br />
we need for the integral th<strong>at</strong> is equal to the sum for T = 0. Therefore, we<br />
have shown th<strong>at</strong> in the limit T → 0 the series also converges to an integral,<br />
and th<strong>at</strong> this is the same integral we obtain by taking the limit T → 0 of the<br />
general form. Hence we can use the general form of the integral for all our<br />
calcul<strong>at</strong>ions and derive low temper<strong>at</strong>ure series expansions, for example. This<br />
situ<strong>at</strong>ion will be very different for bosons, where these two forms are not the<br />
same, and Bose-Einstein condens<strong>at</strong>ion occurs!<br />
Why bother?<br />
So we went through all this m<strong>at</strong>h stuff and found th<strong>at</strong> there was nothing<br />
special. Why bother? Why would there be anything special? The answer is very<br />
simple and important. Every time a convergence fails, we have a singularity and