197 Appendix C Statistical Mechanics Results We wish to relate the plasma density to its temperature. In general where n$_{ε}$ = [exp $^{(}$( ε − µ ) /T ) ± 1] − 1 is the occupation number ( − taken for fermions, + taken for bosons), and g is the degeneracy. In an ultrarelativistic gas the single-particle energy is ε = cp . Therefore d $^{3}$p = 4 πp $^{2}$dp = 4 πε $^{2}$dε/c3 and dN = g n$_{ε}$dτ = g n$_{ε}$ dV d $^{3}$p (2 π$_{ℏ}$ ) $_{3}$, (C.1) N$_{F}$ = 4 πg$_{F}$ V (2 π$_{ℏ}$c ) 3 ∫ ∞ 0 dε ε 2 e ε/T − 1 (C.2) = 4 πg$_{F}$ V (2 π$_{ℏ}$c ) $_{3}$T $^{3}$2 ζ (3) (C.3) N$_{F}$ V = ρ$_{F}$ = g$_{F}$ ζ (3) π $^{2}$($_{ℏ}$ c ) $_{3}$T $^{3}$. (C.4) for a Fermi gas at zero chemical potential, µ = 0. Then the density is