Diffusion equation - Wikipedia

Abstraction: Parabolic PDE governing density spread over time

Key points:

• The diffusion equation ∂ϕ/∂t = ∇·(D∇ϕ) is a parabolic PDE; when D is constant it reduces to the heat equation.
• Originally derived by Adolf Fick in 1855 from Fick's first law (flux proportional to density gradient) combined with the continuity equation.
• For anisotropic media, D is a symmetric positive-definite matrix rather than a scalar.
• Discretizing space yields the discrete Gaussian kernel; discretizing both space and time yields a random walk.
• Equivalent to the heat equation under constant diffusivity; generalized to include drift by the Fokker–Planck equation.
• Used in image filtering via anisotropic tensor diffusion with structure tensors built from image eigenvectors.

Connections: Adolf Fick · Diffusion Equation · Partial Differential Equations · Brownian Motion · Markov Processes

Source: http://en.wikipedia.org/wiki/Diffusion_equation