Diffusion equation

The diffusion equation is a partial differential equation. In physics, it describes the behavior of the collective motion of micro-particles in a material resulting from the random movement of each micro-particle. In mathematics, it is applicable in common to a subject relevant to the Markov process as well as in various other fields, such as the material sciences, information science, life science, social science, and so on. These subjects described by the diffusion equation are generally called Brown problems.

The equation is usually written as:

${\displaystyle {\frac {\partial \phi (\mathbf {r} ,t)}{\partial t}}=\nabla \cdot {\big [}D(\phi ,\mathbf {r} )\ \nabla \phi (\mathbf {r} ,t){\big ]},}$

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.

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