Diffusion Mathematics

In no particular order, here are some common mathematical concepts often found in diffusion modeling. To simulate and quantify simple diffusion (e.g. 2D Brownian Motion) only requires a working knowledge of basic statistics and trigonometry. Modeling complex forms of diffusion requires the use of vector calculus and differential geometry. There is a thin boundary between simple and complex diffusion and the math becomes dense very quickly. For example, using just a few lines of code, one can easily simulate stochastic diffusion on a flat 2D surface:

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% MATLAB CODE %
Ndots = 100; Nsteps = Ndots;
D = .5;                     % Diffusion Rate
dm = 2;                     % dimensions
tau = 1;                    % time step
k = sqrt(dm*D*tau);         % stdev of step-size distribution
xyl = ones(2,Ndots); xyds = xyl;

for t = 1:Nsteps
   xyds = (k * randn(2,Ndots));
   for j = 1:Ndots
     xyl(:,j) = xyl(:,j)+xyds(:,j);
   end
   gscatter(xyl(1,:),xyl(2,:));
end

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On the other hand, simulating stochastic diffusion on a ruffled 2D membrane is often done using parallel-computing systems at your local university's supercomputer center. However, with an understanding of the mathematical concepts that underlie complex diffusion, we can understand diffusion properties without a powerful computing cluster. I am a neuroscientist, not a mathematician or physicist, and based on my own personal trials and tribulations in modeling diffusion as a non-math expert, here are some concepts I needed to brush up on before getting serious about diffusion modeling...

Common Mathematical Concepts in Diffusion

Laplace operator

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In mathematics the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space. It is usually denoted by the symbols ∇·∇, ∇2 or ∆. The Laplacian ∆f(p) of a function f at a point p, up to a constant depending on the dimension, is the rate at which the average value of f over spheres centered at p, deviates from f(p) as the radius of the sphere grows.