76 The hot universe
a closed system and does not use any concepts from equilibrium thermodynamics.
Therefore, it can also be applied to the expanding universe.
Let us assume that all possible states of some (complicated) closed system can
be completely characterized and enumerated by a (composite) discrete variable
α; differentαcorrespond to microscopically different states. If we know that the
system is in a certain stateα,the information about this system is complete and
its entropy should be zero. This follows from the general definition according to
which the entropy characterizes the missing information. If, on the other hand, we
know only the probabilityPαof finding the system in stateα,then the associated
(nonequilibrium) entropy is
S=−∑
αPαlnPα. (3.10)It takes its maximum value when all states are equally probable, that is,Pα= 1 /,
and is equal to
S=ln, (3.11)whereis the total number of possible microstates which the system can occupy.
Note that the last expression gives a finite result only if the total energy is bounded,
otherwise the number of possible states would be infinite.
Let us calculate the maximal possible entropy of anidealgas ofNbose particles
with total energyEplaced in a box of volumeV.It is clear that maximal entropy
occurs when there are no preferable directions and locations within the box. There-
fore, given the total energy and number of particles, each state of the system is
essentially given by the number of particles per mode of the one-particle energy
spectrum. Let us denote byNthe total number of particles, each having energy
in the interval betweenand+, and bygthe total number of different
possible microstates that a particle could occupy in theone-particlephase space.
The total number of all possible configurations (microstates) forNbose particles
is equal to the number of ways of redistributingNparticles amonggcells
(Figure 3.2):
G=
(N+g− 1 )!
(N)!(g− 1 )!. (3.12)
∆g − 1∆N
Fig. 3.2.