Physical Foundations of Cosmology

3.3 Rudiments of thermodynamics 77

The total number of states for the whole system, therefore, is

({N})=

∏



G. (3.13)

Substituting (3.13) into (3.11), we find that the maximal possible entropy of the
system with thegiven energy spectrum{N}is

S({N})=

∑



lnG. (3.14)

Let us assume thatN andg are much larger than unity. Using Stirling’s
formula,

lnN!=

∑N

n= 1

lnn≈

∫N

1

lnxdx+

1

2

lnN=

(

N+

1

2

)

lnN−N, (3.15)

we find from (3.12) and (3.14) that, to leading order,

S({N})≡S({n})=

∑



[(n+ 1 )ln( 1 +n)−nlnn]g, (3.16)

wheren≡N/gare called occupation numbers. They characterize the aver-
age number of particles per microstate of asingleparticle. The entropy depends
on the energy spectrum{n}and we want to maximize it subject to the given total
energy

E({n})=

∑



N=

∑



ng, (3.17)

and total number of particles

N({n})=

∑



N=

∑



ng. (3.18)

To extremize (3.16) with the two extra constraints (3.17) and (3.18), we apply the
method of Lagrange multipliers. The variation of expression

S({n})+λ 1 E({n})+λ 2 N({n})

with respect tonvanishes for

n=

1

exp(−λ 1 −λ 2 )− 1

. (3.19)

Given spectrum (3.19), the Lagrange multipliersλ 1 and λ 2 are the parame-
ters which allow us to satisfy the constraints. They can be expressed in terms
ofEandN,or, instead, in terms of temperatureT≡− 1 /λ 1 and chemical potential