Astroparticle Physics

B.1 Statistical Mechanics Review 393t BE [f(p)] = ∏ i(ν i + γ i − 1)!ν i !(γ i − 1)!≈ ∏ i(ν i + γ i )!ν i !γ i !,(B.9)where the product extends over all cells in momentum space.The subscript BE in (B.9) stands for Bose–Einstein sincethis relation holds for a collection of identical bosons.For fermions, the antisymmetric nature of the total wavefunction implies that it can contain a given one-particle stateat most only once. Therefore, each of the γ i states in theith cell in momentum space can be occupied either once ornot at all. This implies γ i ≥ ν i . The number of possiblearrangements of ν i particles in the γ i states where each stateis occupied zero or one time is another standard problem ofcombinatorics, for which one findsγ i !ν i !(γ i − ν i )! .(B.10)The total number of combinations for all cells is thust FD [f(p)] = ∏ iγ i !ν i !(γ i − ν i )! ,(B.11)where FD stands for Fermi–Dirac.As the number of microstates t[f(p)] is astronomicallylarge, it is more convenient to work with its logarithm, andfurthermore one can use Stirling’s approximation,system of fermionstotal number of microstateslogarithm of the numberof microstatesln N! ≈N ln N − N,(B.12)valid for large N. Thisgivesln t BE [f(p)] ≈(B.13)≈ ∑ [(ν i + γ i ) ln(ν i + γ i ) − ν i ln ν i − γ i ln γ i ]iandln t FD [f(p)] ≈(B.14)≈ ∑ [γ i ln γ i − ν i ln ν i − (γ i − ν i ) ln(γ i − ν i )]ifor bosons and fermions, respectively.The next step is to find the distribution f(p) which maximizesln t[f(p)]. Before doing this, however, the problemshould be generalized to allow for more than one type ofparticle. As long as a particle’s mass is small compared tomore than one typeof particle