I've been investigating a 2-dimensional random walk, where the direction of a particle changes from $\theta$ to $\theta + \Delta \theta$ every time step. The particle moves a distance $v\Delta t$ in this direction at every time step. The random variable $\Delta\theta$ I'm using is uniformly distributed along the interval $[-\eta,\eta]$ where $0<\eta\leq 2\pi$.
I'm going to use the heat equation to try and model the diffusion of such particles. To do this, I need to know how the diffusion constant $D$ depends on $\eta$. I've done some research, but everything I've found only gives a value for $D=\frac{v^2}{4}$ in the $\eta = 2\pi$ case.
I guessed that $D(\eta) = \frac{2\pi v^2}{\eta 4}$. This gives the correct expression in the $\eta = 2\pi$ case. I tried to verify this by using numerics.
I started with a large number of particles (5000) localised at $x=y=0$ I then measured how they spread through time. At each time step, I measured their full-width half-maximum, and used curve fitting to find a value for $D$. I repeated this process many times, for many different values of $\eta$.
The black line is $D = \frac{v^2}{4}$.
My ansatz is not correct. Can anyone show me how $D$ would depend on $\eta$?