10.3 Independent and Indistinguishable Particles
427
in the new space, the function we wish to maximize becomes unconstrained. After
finding the maximum, the new variables are then eliminated by re-imposing the
constraint relations.
As we mentioned, the n i are typically large numbers, and W even larger.
Since the logarithm of a number (i.e. the powers of the number with some base)
increase gently and monotonically with the original number, it is more convenient
to maximize the logarithm of W under the constraints rather than W . In addition,
since the n i are expected to be large near equilibrium, we can use Stirling’s
approximation, N ! ∼ N N e −N √
2πN for both N and the n i . The approximation
is good even for N as small as 10. We usually have N ’s many powers of 10 greater.
For the logarithm of factorials, we have
log (N!) ∼ N log N − N ,
dropping the last factor in Stirling’s formula because log
√
2πN = (1/2) log N +
(1/2) log 2π << N for large N .
So, we maximize the function W ≡ log W + α
n i − β
n i i by varying the
n i without constraint. At maximum, the slopes of this function along each of the
directions of the axes n i must vanish.
The slopes are first derivatives of W with respect to the n i and give
log
g i ± n i
n i
+ α − ββ i = 0 ,
with the upper sign for bosons, and the lower sign for fermions. Solving for n i gives
n i =
g i
e −α e ββ i ∓ 1
.
bosons
fermions
(10.6)
The total particle number and total energy constraints Eqs. (10.2) and (10.1) determine the Lagrange multipliers α and β in terms of the thermodynamic properties of
the system.
We have now found how the particles are distributed at or near equilibrium when
the total energy and total particle number are fixed.
Photons make a special case, in that there is no conservation law preserving the
number of photons. 9 If we drop the constraint
i n i = N , we will find that photons
distribute among the possible quantum states according to
n i =
g i
e ββ i − 1
photons .
(10.7)
9 Photons transfer no intrinsically conserved quantity, such as mass or charge. They are their own
antiparticle.
427
in the new space, the function we wish to maximize becomes unconstrained. After
finding the maximum, the new variables are then eliminated by re-imposing the
constraint relations.
As we mentioned, the n i are typically large numbers, and W even larger.
Since the logarithm of a number (i.e. the powers of the number with some base)
increase gently and monotonically with the original number, it is more convenient
to maximize the logarithm of W under the constraints rather than W . In addition,
since the n i are expected to be large near equilibrium, we can use Stirling’s
approximation, N ! ∼ N N e −N √
2πN for both N and the n i . The approximation
is good even for N as small as 10. We usually have N ’s many powers of 10 greater.
For the logarithm of factorials, we have
log (N!) ∼ N log N − N ,
dropping the last factor in Stirling’s formula because log
√
2πN = (1/2) log N +
(1/2) log 2π << N for large N .
So, we maximize the function W ≡ log W + α
n i − β
n i i by varying the
n i without constraint. At maximum, the slopes of this function along each of the
directions of the axes n i must vanish.
The slopes are first derivatives of W with respect to the n i and give
log
g i ± n i
n i
+ α − ββ i = 0 ,
with the upper sign for bosons, and the lower sign for fermions. Solving for n i gives
n i =
g i
e −α e ββ i ∓ 1
.
bosons
fermions
(10.6)
The total particle number and total energy constraints Eqs. (10.2) and (10.1) determine the Lagrange multipliers α and β in terms of the thermodynamic properties of
the system.
We have now found how the particles are distributed at or near equilibrium when
the total energy and total particle number are fixed.
Photons make a special case, in that there is no conservation law preserving the
number of photons. 9 If we drop the constraint
i n i = N , we will find that photons
distribute among the possible quantum states according to
n i =
g i
e ββ i − 1
photons .
(10.7)
9 Photons transfer no intrinsically conserved quantity, such as mass or charge. They are their own
antiparticle.