1
$\begingroup$

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?

$\endgroup$
4
  • $\begingroup$ 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

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.