Astroparticle Physics

392 B Results from Statistical Physics: Thermodynamics of the Early Universenumber of microstatesA given distribution f(p) could result from a numberof distinct N-particle wave functions, i.e., from a number ofdifferent microstates. All available microstates are equallylikely, but the overwhelming majority of them will correspondto a single specific f(p), the equilibrium distribution.This is what one needs to find.To find this equilibrium momentum distribution, onemust determine the number of microstates t for a givenf(p). To do this, one considers the momentum space to bedivided into cells of size δ 3 p. The number of particles in theith cell isν i = f(p i )δ 3 p.(B.6)The number of possible one-particle momentum states in thecell is δ 3 p divided by the number of states per unit volumetotal number in momentum space, (2π/L) 3 . The total number of one-of one-particle states particle states in δ 3 p is therefore 1δ 3 pγ i = g(2π/L) 3 ,(B.7)where g represents the number of internal (e.g., spin) de-grees of freedom for the particle. For an electron with spin1/2, for example, one has g = 2.It is assumed that the element δ 3 p is large compared tothe volume of momentum space per available state, which is(2π/L) 3 , but small compared to the typical momenta of theparticles. Within this approximation, the set of numbers ν ifor all i contains the same information as f(p).For a system of bosons, there is no restriction on thenumber of particles that can have the same momentum.Therefore, each of the γ i states can have from zero up to ν iparticles. The number of ways of distributing the ν i particlesamong the γ i states is a standard problem of combinatorics(see, e.g., [47]). One obtainsnumber of degreesof freedomsystem of bosons(ν i + γ i − 1)!ν i !(γ i − 1)!(B.8)total number of microstatespossible arrangements. The total number of microstates forthe distribution is therefore1 In many references the number of particles is called n i and thenumber of states g i . Unfortunately, these letters need to be usedwith different meanings later in this appendix, so here ν i and γ iwill be used instead.