Problem S5.7. Consider a Brownian rotor with moment of inertia, I, constrained to rotate through angle, 0, about the z axis. The Langevin equations of motion for the rotor are I(dw/dt)=-Tw+ (t) and (de/dt) =w, where w is the angular velocity of the rotor, I is the friction coefficient, and (t) is a Gaussian white noise torque. The torque is delta-correlated, ((r)E(t)) = Go(tt), where G is the noise strength.

(a) For the case of large friction coefficient, I, write the Fokker-Planck equation for the probability density, P(0, 1), to find the Brownian rotor in the interval 00+ do at time, t.

(b) Solve the Fokker-Planck equation assuming that at time = 0 the rotor is at 0 = 80.

(c) Compute the probability current at time 1.