3.10. Langevin equation and Brownian motion Consider a particle of mass m in motion in a fluid (for simplicity we consider a onedimensional motion), subjected to a friction force proportional to the velocity and a random force η(t) due to the randomfluctuations of the fluid density. Denoting by x(t)

and v(t) the position and the velocity of the particle at time t, its equations of motion of the particle dv(t)
dt = −γ
m v(t)+ 1 m η(t), dx(t)
dt = v(t), where γ >0 is the friction coefficient. Assume that η(t) is random variable, with zero mean and delta-correlated η(t)η = 0, η(t1)η(t2)η = 2γ kBT δ(t1 −t2)
where kB is the Boltzmann constant, T is the temperature and the average η is with respect to the probability distribution of the stochastic variable η(t).

a. Let x0 and v0 the position and velocity of the particle at t = 0. Integrating the equations of motion and taking the average with respect to η, show that the correlation function of the velocity is v(t2)v(t1)η = v20 − kBT m e−(γ /m)(t1+t2) + kBT m e−(γ /m)(t2−t1).
with t2 > t1.

b. Compute the variance of the displacement and show that (x(t)−x0)2η = m2 γ
v20 − kBT m 1−e−(γ /m)t
2 + 2kBT γ
t − m γ
1−e−(γ /m)t
.

c. Assuming that the particle is in thermal equilibrium, we can now average over all possible initial velocities v0. Let us denote this thermal average by T. By the equipartion theorem we have v20 T = kBT/m. Show that, for t m/γ , the thermal average of the variance of the displacement becomes (x(t)−x0)2ηT  (2kBT/γ )t.