Download Exercises in Statistical Mechanics

Exercises in Statistical Mechanics
Based on course by Doron Cohen, has to be proofed
Department of Physics, Ben-Gurion University, Beer-Sheva 84105, Israel
This exercises pool is intended for a graduate course in “statistical mechanics”. Some of the
problems are original, while other were assembled from various undocumented sources. In particular some problems originate from exams that were written by B. Horovitz (BGU), S. Fishman
(Technion), and D. Cohen (BGU).
====== [Exercise 8492]
Rate Equation with noise
Consider molecules A and B in a box. The molecules are subjected to weak electric field E that modifies the binding
(0)
energy of the B molecules such that εB = εB + αE, where α is a constant. In the lack of electric field the molecules
reach chemical equilibrium A↔B, such that their fractions are fA = hNA i/N and fB = hNA i/N . In item (c) it is
assumed that the system can be described by the rate equation
dNA
= kB NB − kA NA + A(t)
dt
Where A(t) is a stochastic term with zero average and correlation function hA(t)A(t0 )i = Cδ(t − t0 ). In item (e)
assume that only kB is affected by the weak electric field.
(a) Express hNA i, hNB i by N , and fA and fB , and αE.
(b) Express Var(NA ) by N , and hNA i and hNB i.
(c) Determine the constants kA and kB and D in the stochastic rate equation such that hNA i and Var(NA ) will
agree with the canonical expectation of items (a) and (b).
(d) Determine how kB is modified by the weak electric field. In other words find the coefficient in δkB ∝ E.
(e) Find the generalized susceptibility χ(ω) that relates the variation δNA to E.
(f) Find the power spectrum of δNA in steady state.