Brownian motion of a free particle

The Langevin equation for a free particle is a model for the stochastic motion of a particle in a liquid or a gas for short times, and is


m {\dot v} = -\gamma v + n(t)
m˙v=γv+n(t)

 

where v(t)v(t) is the particle velocity, \gammaγ is a drag coefficient, and n(t)n(t) is random noise. This is analogous to a massless particle in a harmonic potential, that's also a homework problem.

  • 1. Using this analogy, what is \langle v(t) v(0)\ranglev(t)v(0), the velocity autocorrelation function?