Problem 10.7. A Brownian particle of mass m is attached to a harmonic spring with force constant, k, and is driven by an external force, F(r). The particle is constrained to move in one dimension. The Langevin equation is

where wok/m, y is the friction constant, and (r) is a Gaussian white noise with zero mean, (()) = 0. Here () denotes the average over values of the random force.

(a) Compute the equilibrium correlation function, (x(t)x(0)), starting from the Langevin equation above with F(t) = 0. Let (r denote the thermal average over the initial position and velocity of the Brownian particle. Assume that (x(0)v(0)) = 0 and (x(0)2)=kBT/mw (cf. Exercise (5.5)).

(b) The dynamic susceptibility for the Brownian oscillator is x(w) = (-mu + mw-iyw) (cf. Exercise (10.6)). Use the fluctuation-dissipation theorem, -1'

to compute the equilibrium correlation function. Do your results in

(a) and

(b) agree?

Transcribed Image Text:

m dx(t) dx(t) dt dt +7- +mwx(t) = {(t) + F(t),