lecture notes on statistical mechanics - Scott Pratt - Michigan State ...

where i de<strong>notes</strong> a specific quantum state. For the Fermionic case this relation can be used to solvea problem considered by Euler, “How many ways can a integers be arranged to sum to E withoutusing the same number twice?”Iterative techniques similar to the ones mentioned here can also incorporate conservation ofmultiple charges, angular momentum, and in the case of QCD, can even include the constraint thatthe overall state is a coherent color singlet (Pratt and Ruppert, PRC68, 024904, 2003).2.8 Enforcing Canonical and Microcanonical Constraints through Integratingover Complex Chemical PotentialsFor systems that are interactive, or for systems where quantum degeneracy plays an important role,grand canonical partition functions are usually easier to calculate. However, knowing the grandcanonical partition function for all imaginary chemical potentials can lead one to the canonicalpartition function. First, consider the grand canonical partition function with an imaginary chemicalpotential, µ/T = iθ,Z GC (µ = iT θ, T ) = ∑ e −βE i+iθQ i. (2.65)iUsing the fact that,one can see that,δ Q,Qi = 1 ∫ 2πdθ e i(Q−Qi)θ , (2.66)2π 0∫1 2πdθ Z GC (µ = iT θ, T )e −iQθ2π 0= ∑ e −βE iδ Q,Qii(2.67)= Z C (Q, T ).A similar expression can be derived for the microcanonical ensemble,12π∫ ∞−∞dz Z C (T = i/z)e −izE = 12π= ∑ i∫ ∞−∞dz ∑ ie iz(ϵ i−E)δ(E − ϵ i ) = ρ(E).(2.68)As stated previously, the density of states plays the role of the partition function in the microcanonicalensemble.2.9 Problems1. Beginning with the expression for the pressure for a non-interacting gas of bosons,P VT= ln Z GC = ∑ ln ( 1 + e −β(ϵp−µ) + e −2β(ϵp−µ) + · · · ) , ∑ ∫V→ (2s + 1)(2π¯h) 3 ppd 3 p,show thatP =(2s + 1)(2π¯h) 3Here, the energy is relativistic, ϵ = √ p 2 + m 2 .∫d 3 p p2f(p), where f =e−β(ϵ−µ)3ϵ301 − e −β(ϵ−µ) .