a) A particle moving in one dimension interact with environment in such a way that its equation of motion for the momentum $p$ is

$\dot{p}(t) = -\gamma p(t) + \xi(t)$

with $\gamma$ a positive damping factor and $\xi(t)$ a gaussian white noise:

$<\xi(t)>=0$

$<\xi(t)\xi(t^\prime)>=\sigma\delta(t-t^\prime)$

Determine the coefficient $\sigma$ such that asymptotically ($t\rightarrow\infty$) the process $p(t)$ has the correct Boltzmann statistical properties.

b) Once the point a) has been solved determine the correlation function for all times $t$ and $t^\prime$

$C(t,t^\prime)=<p(t) p(t^\prime)>$

c) using the result of the point b) show that the coordinate mean square displacement from initial position asymptotically shows a diffusive behaviour:

$<|x(t)-x(0)|>\rightarrow 2 D t $

and determine the diffusion coefficient $D$.

d) In the derivation of the Fokker-Planck equation as done in the lectures, starting from the conditional probability definition, obtain the final partial differential equation for the probability distribution of the process.