Statistical Mechanics - Physics at Oregon State University

92 CHAPTER 5. FERMIONS AND BOSONS
tnx,ny,nz e µm
kB T − 2 π 2
2ML2 mk (n2
B Tm x +n2y
+n2z
)
(5.12)
which means that the series for arbitrary values of T , µ, and L converges faster
than for the maximal values. Hence we can use the maximal values of T , µ ,
and L to describe the convergence of the series in general, and hence the series
converges uniformly on 0 T Tm , −∞ µ µm, and 0 L Lm.
Next we consider a domain for (T, µ, L) space with T ′ m T , L ′ m L , and
−∞ µ µm. The argument of the exponent in (5.7) is
ɛ(nx, ny, nz)
=
kBT
2π2 2MkBT L2 (n2x + n 2 y + n 2 z) = x 2 nx + y2 ny + z2 nz (5.13)
2π2 with xnx = nx 2MkBT L2 and similar for the y and z components. The size of
the steps therefore approaches zero when the temperature becomes very large or
when L becomes very large. Therefore, if we can replace the sum by an integral
with a certain error at a certain temperature or length, the error will be smaller
at all higher temperatures and lengths.
We now make the following observation. Suppose we choose a tolerance
ɛ. Because of the uniform convergence of the series we can find a value Nɛ
independent of T , µ , and L ( but with 0 T Tm , −∞ µ µm, and
0 L Lm ) such that:
−(2S
+ 1)kBT
N
N
nx=1 ny=1 nz=1
N
log(Znx,ny,nz (T, µ, V )) − S(T, µ, V )
< ɛ (5.14)
for all values of N with N > Nɛ. Next, use I(T, µ, V ) for the result of the
integral. We can find values of T ′ m and L ′ m in such a manner that
−(2S
+ 1)kBT
∞
∞
nx=1 ny=1 nz=1
∞
log(Znx,ny,nz(T, µ, V )) − I(T, µ, V )
< ɛ (5.15)
for all values of the temperature with T T ′ m and L L ′ m. The last equation
implies that the series converges, which we already knew, but does not require
uniform convergence.
We are now able to make the following statement. Consider a domain −∞
µ µm , T ′ m T Tm , and V ′ m V Vm. On this domain the series for
the grand partition function converges uniformly and the sum is close to the
value of the integral. We will choose both Tm and Vm very large. We will also
choose T ′ m very small. Since temperature and volume are multiplied together
(via the term T L 2 ), this means that we need to consider very large values of
the volume only. Hence for any value of µ and T in any interval −∞ µ µm,