Statistical Mechanics - Physics at Oregon State University

110 CHAPTER 5. FERMIONS AND BOSONS
N(T, µ, V ) = (2S + 1)
nx,ny,nz
The condition on µ is µ 2 π 2
2M
e
2
2M ( π L )2 (n 2 x +n2 y +n2 z )−µ
kB T − 1
−1
(5.114)
V − 2
3 3. This guarantees that all terms are
positive, and hence each term is smaller than N
2S+1 . For large values of ni the
terms approach zero exponentially fast, and hence the series converges. That is
no surprise, since we know that the result is N. The convergence is uniform for
all values of T in [0, Tm] and V in [0, Vm]. This is true for all values of µ in the
range given above.
It is possible to give an upper-bound to these large terms by
e
2
2M ( π L )2 (n 2 x +n2y +n2z )−µ
kB T − 1
−1
with X of order 1 and for T < Tm if
< Xe −
2
2M ( π L )2 (n 2 x +n2y +n2z )−µ
kB Tm (5.115)
n 2 x + n 2 y + n 2 z > N(Tm). Therefore the
series converges uniformly for T Tm and µ 0, as stated above.
We can now write
lim N(T, µ, V ) = (2S + 1)
T →0
For any value of µ 2 π 2
2M
lim
T →0
nx,ny,nz
e
2
2M ( π L )2 (n 2 x +n2 y +n2 z )−µ
kB T − 1
−1
(5.116)
V − 2
3 3 the limit of each term is zero and hence the
limit of N is zero. If the chemical potential is specified, the system contains no
particles at zero temperature! This is the case for a system of photons, there is
no radiation at zero temperature!
In a gas of bosons, however, we often specify the number of particles < N >
and find µ from < N >= N(T, µ, V ). This tells us that µ is a function of T,V,
and < N > and that
< N >= lim
T →0 N(T, µ(T, < N >, V ), V ) (5.117)
We introduced the notation < N > to distinguish between the actual number
of particles in the system and the general function which yield the number of
particles when T,µ, and V are specified. Inserting this in the equation for the
limits gives:
N = (2S + 1)
lim
T →0
nx,ny,nz
e
2
2M ( π L )2 (n 2 x +n2y +n2z )−µ(T,N,V )
kB T − 1
−1
(5.118)