I am trying to do a computer simulation as part of a research project of the diffusion of a few particles undergoing brownian motion along a two-dimensional canvas. I am having some holes in my understanding of diffusion dynamics, which makes me blocked on this task.

I understand that there exists the relationship $$ \tau = \frac{L^2}{4D}, $$ where $D$ is a diffusion coefficient, $\tau$ is a time change and $L$ is the distance traveled.

Then I suppose that in two-dimensions this scales like $L_r = \sqrt{L_x^2 + L_y^2}$, where $$ (L_x, L_y) = (2\sqrt{\tau D}, 2\sqrt{\tau D}) \implies L_r = \sqrt{8 \tau D} $$

I am able at each step to generate a random walk of the particles by sampling discretely from $-1$ to $1$ using a normal distribution, append this value over the $x$ coordinate of a particle and repeat the sampling for the $y$ coordinate, but how and where does diffusion come to play into this? More specifically, what is the role of the diffusion coefficient and how can I implement it in my particle diffusion simulation?

  • $\begingroup$ So to be clear, you're modeling diffusion not by solving a diffusion PDE (e.g., $\partial_tu=\partial_x(D\partial_xu)$), but by simulating individual particle walks? $\endgroup$
    – Kyle Kanos
    Commented Sep 24, 2022 at 3:16
  • 1
    $\begingroup$ Do you sample discretely from $\{-1,1\}$ from a (symmetric) Binomial distribution ? Or from $(-\infty,+\infty)$ from a normal distribution ? In any case let's call this sample $\varepsilon$. It has mean zero and variance one. You should do that, both, in $x$-direction and in $y$-direction: $\varepsilon_x,\varepsilon_y$. To bring in your physical parameters increment every direction by $\sqrt{2\tau D}L$ times the respective $\varepsilon$. That's it. $\endgroup$
    – Kurt G.
    Commented Sep 24, 2022 at 6:12
  • $\begingroup$ @KurtG. Awesome, thanks, I did not think of just multiplying by the $\sqrt{2 \tau D} L$, but can you elaborate on what is $L$ in this scenario? So for instance, given a particle's position $(x_i, y_i)$ at $i^{\text{th}}$ iteration, would $L$ be the distance between $x_i$ and $x_{i-1}$, and same for $y_i, y_{i-1}$ respectively? $\endgroup$
    – hexaquark
    Commented Sep 24, 2022 at 19:27
  • $\begingroup$ @KyleKanos Yes that is correct, simulate diffusion (Brownian motion) for a few particles, given a diffusion coefficient $D$. $\endgroup$
    – hexaquark
    Commented Sep 24, 2022 at 19:56


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