Astroparticle Physics

B.1 Statistical Mechanics Review 395to zero ensures that the corresponding constraints are fulfilled.To find the set of distributions f which maximize(B.19), one substitutes f a (p i ) = ν ai /δ 3 p. Then the integralsare converted to sums, and further one has ∑ i ν ai =N a . In addition, the number of microstates can be obtainedfrom (B.13) for bosons and (B.14) for fermions. The derivativeof φ with respect to ν ai is∂φ= ln(γ ai ± ν ai ) − ln ν ai − βE ai∂ν ai− α Q Q a − α B B a − α L L a , (B.20)where γ ai = g a δ 3 p/(2π/L) 3 is the number of states availableto a particle of type a in the cell i, and for the derivationin this section the upper sign refers to bosons and the lowersign to fermions. Setting (B.20) equal to zero and solvingfor ν ai givesdiscretizing momentumintegralssolutionfor number of particlesν ai =γ aiexp [ α Q Q a + α B B a + α L L a + βE ai]∓ 1.(B.21)Re-expressing this in terms of the functions f a (p i ) =ν ai /δ 3 p givesf a (p) =g a (L/2π) 3exp [ α Q Q a + α B B a + α L L a + βE a] ∓ 1.(B.22)The temperature can be defined asdefinition of temperatureT = 1/β ,(B.23)and it can be shown that this has all of the desired propertiesof the usual thermodynamic temperature. Furthermore, thechemical potential for particle type a can be defined aschemical potentialµ a =−T(α Q Q a + α B B a + α L L a ), (B.24)which can be modified in the obvious way to include a differentset of conserved quantities. Note that, although theLagrange multipliers are specific to the system, i.e., are thesame for all particle types, the chemical potential depends onthe charge, baryon number, and lepton number of the particle.In a reaction where, say, a + b ↔ c + d, (B.24) impliesµ a + µ b = µ c + µ d .