Diffusion
Diffusion represents random local movement that tends to spread a population from regions of high density toward regions of lower density.
Diffusion equation
In one spatial dimension,
\[\frac{\partial u}{\partial t}=D\frac{\partial^2u}{\partial x^2},\]where \(u(x,t)\) is density and \(D>0\) is the diffusion coefficient.
Biological interpretation
A larger value of \(D\) represents faster spatial spreading. Diffusion does not imply directed movement toward a target; it represents net spreading generated by local random movement.
Flux
Fick's law expresses the diffusive flux as
\[J=-D\frac{\partial u}{\partial x}.\]Key idea. Diffusion smooths spatial differences by moving density down its spatial gradient.