Physics 262: Statistical Physics Fall 2008 Problem Set 2 due: Wed Oct 1 Reading: Parthria, Chapter 1; Lifshitz+Pitaevski, Chapters 1,2 1. The classical canonical ensemble is defined by the phase space density ρ = 1 Z c exp( - βH ) (1) where the Hamiltonian H is a function of the particle co-ordinates and momenta, and Z c = Z d 3 N pd 3 N q h 3 N N ! exp( - βH ) (2) By foliating the integral over equal energy surfaces, we argued in class that Z c = Z ∞ 0 dE Δ exp ( - β ( E - TS ( E ))) ≈ f exp - β ( E - T S ) (3) where S is the entropy, E is the average energy, and f is given by f = √ 2 πk B T 2 C V Δ (4) (a) Establish that in the thermodynamic limit lim N →∞ ln f N = 0 (5) (b) Using the above identity, show that in the thermodynamic limit ln Z c = - β ( E - TS ) (6) holds exactly as N → ∞ , with corrections which are smaller by a factor of ln N/N . This is the reason the value of f is unimportant. Also, I have dropped the bars over E , because the fluctuations of E are negligible, and E = E with probability 1 in the thermodynamic limit. Eq. (6) is a key result which we will use frequently in this course.