Diffusion equation

The diffusion equation describes how a quantity such as heat or particle concentration changes over time as it spreads down spatial gradients. Its simplest form makes the rate of change proportional to the Laplacian of the quantity.

Fick's laws of diffusion: Relations describing diffusive flux and the resulting change in concentration. Combining Fick's flux law with conservation of matter yields the diffusion equation.
Partial differential equation: An equation relating a function of several variables to its partial derivatives. The diffusion equation belongs to this broad class because it involves space and time derivatives.
Brownian motion: The irregular motion of particles suspended in a fluid, caused by molecular collisions. Its probability density evolves according to a diffusion equation.
Wave equation: A partial differential equation describing propagation of disturbances at finite speed. Unlike diffusion, it supports traveling waves rather than immediate smoothing.
Laplacian: A differential operator that sums second spatial derivatives of a function. The Laplacian measures local curvature that drives diffusion in the standard equation.
Continuity equation: A local conservation law relating changes in a density to the divergence of its flux. It supplies the conservation principle used to derive diffusion from a flux law.
Thermal conduction: The transfer of heat through matter by microscopic interactions. In uniform materials, temperature evolution follows the heat form of the diffusion equation.
Advection–diffusion equation: A transport equation combining bulk motion with diffusive spreading. It adds directional flow to the diffusion equation's gradient-driven spreading.
Diffusion coefficient: A parameter measuring how rapidly a substance or physical quantity diffuses. It sets the timescale and spatial rate of spreading in the equation.
Concentration gradient: The spatial rate and direction of change of a substance's concentration. Diffusive flux points down this gradient in Fick's first law.

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