Statistical Physics

10.3 Ideal Fermi Gases and Ideal Bose Gases 153Here, we have used the fact that the energy of the ith state, when it containsn particles, is nE i . The probability of having n particles in this state is1[Ξ e−β(nEi−nµ) = 1 − e −β(Ei−µ)] e −β(nEi−nµ) . (10.17)The expectation value of the number of particles in this state is〈n i 〉 =∞∑[ ne −nβ(Ei−µ) 1 − e −β(Ei−µ)]n=1= e−β(Ei−µ)1 − e −β(Ei−µ)1=e β(Ei−µ) − 1 . (10.18)If we consider the right-hand side of this equation as a function of E, weobtain the Bose distribution function1n(E) =e β(E−µ) − 1 . (10.19)We have already encountered this form, with µ = 0, in various situations,namely the harmonic oscillations of diatomic molecules, the oscillations ofa crystal lattice, and electromagnetic waves in a cavity. The behavior of theBose distribution function is shown in Fig. 10.2. The grand partition functionof the total system isΞ = ∏ iΞ i = ∏ i1. (10.20)1 − e−β(Ei−µ) Fig. 10.2. The Bose distribution function n(E) =1/[e β(E−µ) − 1]. The chemicalpotential µ must be negative; n(E) atk BT =10|µ| and 20|µ| are shown by the solidand dash-dotted lines, respectively