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

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

where $v(t)$ is the particle velocity, $\gamma$ is a drag coefficient, and $n(t)$ 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 $\langle v(t) v(0)\rangle$, the velocity autocorrelation function?

• 2. Complete the code hw5/free_hints.py to calculate and plot $\langle x^2\rangle$ as a function of time, assuming the particle at $t=0$ is at $x=0$. Units have been chosen so the $m=1$. What is the diffusion coefficient? Compare your results with the analytic solution for this problem, referenced in useful links that is here Links to an external site..