Astroparticle Physics

B.3 Equations of State 399which relates the total energy U, temperature T ,entropyS,pressure P ,andvolumeV of the system. The differentialdU can also be written asdU =( ∂U∂S)VdS +( ) ∂UdV ,∂V S(B.36)where the subscripts indicate what is kept constant whencomputing the partial derivatives. Equating the coefficientsof dV in (B.35) and (B.36) gives the pressure,( ) ∂UP =− . (B.37)∂V SRecall that the entropy is simply the logarithm of the totalnumber of microstates Ω, and that to good approximationthis is given by the number of microstates of the equilibriumdistribution t[f(p)]. Thatis,pressureentropy andnumber of microstatesS = ln Ω ≈ ln t[f(p)] .(B.38)The important thing to notice here is that the entropy isentirely determined by the distribution f(p). Therefore, tokeep the entropy constant when computing (∂U/∂V ) S , onesimply needs to regard the distribution f(p) as remainingconstant when V is changed.The total energy U is∫U = Ef(p) d 3 p,(B.39)total energyand the pressure is therefore( ) ∫∂U ∂EP =− =−∂V∂V f(p) d3 p.S(B.40)The derivative of E with respect to the volume V = L 3 is∂E∂V = ∂E ∂p∂p ∂V = ∂E ∂p∂p ∂L/ ∂V∂L .(B.41)One has ∂V /∂L = 3L 2 and furthermore E = √ p 2 + m 2 ,so∂E∂p = 1 (p 2 + m 2) −1/2 p 2p =2E .(B.42)From (B.4) one gets that p ∼ L −1 , and therefore ∂p/∂L =−p/L. Substituting these into (B.41) gives