Diffusion equation

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The diffusion equation is a nonlinear partial differential equation, which describes the density fluctuations in a material undergoing diffusion. It is also used in population genetics to describe the 'diffusion' of alleles in a finite population.

The equation is usually written as:

[\frac=\nabla\cdot D(\phi)\nabla\phi(\vec,t)],

where [\phi] is the density of the diffusing material, [t] is time, [D] is the collective diffusion coefficient, and [\vec] the spatial coordinate. The collective diffusion coefficient depends on the density; if D is a constant, however, then the equation reduces to the following linear equation:

[\frac=D\nabla^2\phi(\vec,t)],

also called the heat equation.

Derivation

The diffusion equation can be derived in a straightforward way from the continuity equation, which states that a change in density in any part of the system is due to inflow and outflow of material into and out of that part of the system. Effectively, no material is created or destroyed:

[\frac+\nabla\cdot\vec=0],

where [\vec] is the flux of the diffusing material. The diffusion equation can be obtained easily from this when combined with the phenomenological Fick's first law, which assumes that the flux of the diffusing material in any part of the system is proportional to the local density gradient:

[\vec=-D(\phi)\nabla\phi(\vec,t)].

Diffusion equation

Derivation

See also