mathematics//differential equations//PDE//diffusion equation

How a quantity spreads in space: its time rate at a point is proportional to the **curvature** of its profile there.

How a quantity spreads in space: its time rate at a point is proportional to the curvature of its profile there.

∂c∂t=D ∂2c∂x2in one dimension,∂c∂t=D ∇2cin three.\frac{\partial c}{\partial t}=D\,\frac{\partial^2 c}{\partial x^2}\qquad\text{in one dimension,}\qquad \frac{\partial c}{\partial t}=D\,\nabla^2 c\qquad\text{in three.}∂t∂c​=D∂x2∂2c​in one dimension,∂t∂c​=D∇2cin three.

The second derivative says whether the profile bends upward or downward. Where it bends upward the point sits below the average of its neighbors, and the concentration rises; where it bends downward the point is above its neighbors and it falls. The relation between ccc and cxxc_{xx}cxx​ is like the one between a velocity vector and its acceleration: the second derivative is how the change of the slope changes at that point.

In three dimensions a point has neighbors in every direction, and the sum of the three second derivatives is the Laplacian, ∇2c=∇⋅∇c\nabla^2 c=\nabla\cdot\nabla c∇2c=∇⋅∇c: the divergence of the gradient. ∇\nabla∇, the del operator, is the vector of partial derivatives; applied once it gives the gradient, the direction of fastest increase; the divergence of that gradient measures how much the field at a point differs from its surroundings (gradient, divergence and curl). DDD is the diffusivity, the constant that turns curvature into a rate.

Diffusion smooths: bumps decay, and the sharper the bump the faster it decays. Written on a grid or a network, the same equation becomes c˙=−Lc\dot c=-Lcc˙=−Lc with the graph Laplacian, whose positive convention L=D−WL=D-WL=D−W carries the opposite sign of ∇2\nabla^2∇2; the decay rates of the spatial patterns are its eigenvalues (spectral gap).

It is the spatial term of the gains-minus-losses recipe: add D∇2cD\nabla^2 cD∇2c to an ODE for ccc and the model gains space; add a reaction such as an interaction term and it becomes a reaction-diffusion system.