1st set of homework

28 Setembro 2018, 15:10 • Mykola Tasinkevych

1. Solve one dimensional, -\infty <x< +\infty,  Fokker-Planck equation for a free Brownian particle, i.e., external potential V = 0. Assume that at time t=0 the particle was at x=0. Plot the result as a function of x for several values of t. Calculate moments  <xⁿ>  and compare with the results of Langevin equation approach (<...> denotes the ensemble average).

2. Solve one dimensional, -\infty <x< +\infty,  Fokker-Planck equation for a Brownian particle in the external potential V(x) = mg x, m is the particle mass, and g is the gravitation acceleration. Assume that at time t=0 the particle was at x=x₀. Plot the result as a function of x for several values of t, choose  some value for x₀.

3. For a free one dimensional Brownian motion of a particle starting at time t = 0 from x = 0, determine the probability density P_x(t)  that the particle will reach for the first time a given point x at a moment of time in the interval (t, t + dt). 

[Remark:  P_x(t) dt is the probability that the particle for the first time will get from x=0 to the point x in time from the interval (t, t+dt)].