Thermal Physics: Problem Set 5

which is independent of ɛ !Problem 3: Fluctuations in a Fermi gasShow for a single orbital of a fermion system that〈(∆N) 2 〉 = 〈N〉(1 − 〈N〉), (3)if 〈N〉 is the average number of fermions in that orbital. Notice that the fluctuationvanishes for orbitals with energies deep within the Fermi sea, so that 〈N〉 = 1. Bydefinition, ∆N = N − 〈N〉.Problem 4: Fluctuations in a Bose gasIf 〈N〉 = ∑ n f(ɛ n, τ, µ)N n , is the average occupancy of a single orbital of a boson system,then using 〈(∆N) 2 〉 = τ∂〈N〉/∂µ show that〈(∆N) 2 〉 = 〈N〉(1 + 〈N〉). (4)Thus, if the occupancy is large, with 〈N〉 ≫ 1, the fractional fluctuations are of theorder of unity: 〈(∆N) 2 〉/〈N〉 ≈ 1, so that the actual fluctuations can be enormous.2