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

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

where $v (t)$ $v(t)$ is the particle velocity, $γ$ $\gamma$ 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 $⟨ v (t) v (0) ⟩$ $\langle v(t) v(0)\rangle$ , the velocity autocorrelation function?

2. Complete the code hw5/free_hints.py to calculate and plot $⟨ x^{2} ⟩$ $\langle x^2\rangle$ as a function of time, assuming the particle at $t = 0$ $t=0$ is at $x = 0$ $x=0$ . Units have been chosen so the $m = 1$ $m=1$ . 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..