Problem S5.8. A Brownian particle of mass m moves in one dimension in the presence of a harmonic potential V(x)=kx, where k is the force constant. The Langevin equations are given by m[dv(t)/dt] = v(t) - dv(x)/dx+(t) and dx(t)/dt = v(t), where is the friction coefficient and (t) is a Gaussian white noise force. The noise is delta-correlated, ((t)E()) = 86(tt), where g is the noise strength.

(a) Write the Fokker-Planck equation for the probability density, P(x, t), in the limit of large friction coefficient.

(b) Solve the Fokker-Planck equation and write the solution, P(x,t), for arbitrary times. Assume that at time = 0, the Brownian particle is at x=xo.

(c) Write an approximate expression for P(x, t) for very long times. ===