The diffusion equation is a partial differential equation which describes density dynamics
in a material undergoing diffusion. It is also used to describe processes exhibiting
diffusive-like behaviour, for instance the 'diffusion' of alleles in a population in
population genetics.
Statement The equation is usually written as:
where ϕ(r, t) is the density of the diffusing material at location r and time t and D(ϕ,
r) is the collective diffusion coefficient for density ϕ at location r; and ∇ represents
the vector differential operator del. If the diffusion coefficient depends on the density
then the equation is nonlinear, otherwise it is linear.
More generally, when D is a symmetric positive definite matrix, the equation describes anisotropic
diffusion, which is written as:
If D is constant, then the equation reduces to the following linear differential equation:
also called the heat equation. Historical origin
The particle diffusion equation was originally derived by Adolf Fick in 1855.
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:
, where j 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:
. If drift must be taken into account, the Smoluchowski
equation provides an appropriate generalization. Discretization
The diffusion equation is continuous in both space and time. One may discretize space,
time, or both space and time, which arise in application. Discretizing time alone just
corresponds to taking time slices of the continuous system, and no new phenomena arise. In discretizing
space alone, the Green's function becomes the discrete Gaussian kernel, rather than
the continuous Gaussian kernel. In discretizing both time and space, one obtains the random
walk. Discretization
The product rule is used to rewrite the anisotropic tensor diffusion equation, in standard discretization
schemes. Because direct discretization of the diffusion equation with only first order
spatial central differences leads to checkerboard artifacts. The rewritten diffusion equation
used in image filtering:
where "tr" denotes the trace of the 2nd rank tensor, and superscript "T" denotes transpose,
in which in image filtering D(ϕ, r) are symmetric matrices constructed from the eigenvectors
of the image structure tensors . The spatial derivatives can then be approximated by two
first order and a second order central finite differences. The resulting diffusion algorithm
can be written as an image convolution with a varying kernel of size 3 × 3 in 2D and
3 × 3 × 3 in 3D. See also
Continuity equation Heat equation
Fokker-Planck equation Fick's law of diffusion: Fick's Second Law
Maxwell-Stefan equation Radiative transfer equation and diffusion
theory for photon transport in biological tissue
References ^ A. Fick, Ueber Diffusion, Pogg. Ann. Phys.
Chem. 170, 59-86. 2. Carslaw, H. S. and Jager, J. C.. Conduction
of Heat in Solids. Oxford: Clarendon Press 3. Crank, J.. The Mathematics of Diffusion.
Oxford: Clarendon Press 4. Thambynayagam, R. K. M. The Diffusion Handbook:
Applied Solutions for Engineers: McGraw-Hill External links
Diffusion Calculator for Impurities & Dopants in Silicon
A tutorial on the theory behind and solution of the Diffusion Equation.
Classical and nanoscale diffusion