Problem 5.11. Consider a Brownian particle of mass m moving in one dimension in the presence of a constant force fo (such as a gravitational or electric field) in a fluid with force constant and in the presence of a delta-correlated random force (t) such that ((12)(1)) =88(121) and (()) = 0. Assume that the velocity and displacement of the Brownian particle at time t=0 are vo and xo, respectively.

(a) Compute the velocity correlation function (v(2)v(t)).

(b) Compute the variance ((x(t)-xo))