The diffusion equation is a

parabolic partial differential equation A parabolic partial differential equation is a type of partial differential equation (PDE). Parabolic PDEs are used to describe a wide variety of time-dependent phenomena, including heat conduction, particle diffusion, and pricing of derivat ...

. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion, resulting from the random movements and collisions of the particles (see Fick's laws of diffusion). In mathematics, it is related to Markov processes, such as

random walks In mathematics, a random walk is a random process that describes a path that consists of a succession of random steps on some mathematical space. An elementary example of a random walk is the random walk on the integer number line \mathbb Z ...

, and applied in many other fields, such as materials science,

information theory Information theory is the scientific study of the quantification, storage, and communication of information. The field was originally established by the works of Harry Nyquist and Ralph Hartley, in the 1920s, and Claude Shannon in the 1940s. ...

, and biophysics. The diffusion equation is a special case of the

convection–diffusion equation The convection–diffusion equation is a combination of the diffusion and convection (advection) equations, and describes physical phenomena where particles, energy, or other physical quantities are transferred inside a physical system due to two ...

, when bulk velocity is zero. It is equivalent to the heat equation under some circumstances.

Statement

The equation is usually written as: where is the

density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...

of the diffusing material at location and time and is the collective

diffusion coefficient Diffusivity, mass diffusivity or diffusion coefficient is a proportionality constant between the molar flux due to molecular diffusion and the gradient in the concentration of the species (or the driving force for diffusion). Diffusivity is enc ...

for density at location ; and represents the vector differential operator del. If the diffusion coefficient depends on the density then the equation is nonlinear, otherwise it is linear. The equation above applies when the diffusion coefficient is

isotropic Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence '' anisotropy''. ''Anisotropy'' is also used to describ ...

; in the case of anisotropic diffusion, is a symmetric

positive definite matrix In mathematics, a symmetric matrix M with real entries is positive-definite if the real number z^\textsfMz is positive for every nonzero real column vector z, where z^\textsf is the transpose of More generally, a Hermitian matrix (that is, a ...

, and the equation is written (for three dimensional diffusion) as: If is constant, then the equation reduces to the following linear differential equation: :

\frac = D\nabla^2\phi(\mathbf,t),

which is identical to the heat equation.

Historical origin

The particle diffusion equation was originally derived by

Adolf Fick Adolf Eugen Fick (3 September 1829 – 21 August 1901) was a German-born physician and physiologist. Early life and education Fick began his work in the formal study of mathematics and physics before realising an aptitude for medicine. He ...

in 1855.

Derivation

The diffusion equation can be trivially derived 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:

\frac+\nabla\cdot\mathbf=0,

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 states that the flux of the diffusing material in any part of the system is proportional to the local density gradient:

\mathbf=-D(\phi,\mathbf)\,\nabla\phi(\mathbf,t).

If drift must be taken into account, the

Fokker–Planck equation In statistical mechanics, the Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability density function of the velocity of a particle under the influence of drag forces and random forces, ...

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 In the areas of computer vision, image analysis and signal processing, the notion of scale-space representation is used for processing measurement data at multiple scales, and specifically enhance or suppress image features over different ranges o ...

, rather than the continuous

Gaussian kernel In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the base form f(x) = \exp (-x^2) and with parametric extension f(x) = a \exp\left( -\frac \right) for arbitrary real constants , and non-zero . It is ...

. In discretizing both time and space, one obtains the

random walk In mathematics, a random walk is a random process that describes a path that consists of a succession of random steps on some mathematical space. An elementary example of a random walk is the random walk on the integer number line \mathbb Z ...

Discretization (Image)

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:

\frac = \nabla\cdot \left (\phi,\mathbf)\right \nabla \phi(\mathbf,t) +  \Big D(\phi,\mathbf)\big(\nabla\nabla^T \phi(\mathbf,t)\big)\Big

where "tr" denotes the trace of the 2nd rank

tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other 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 In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...

with a varying kernel (stencil) of size 3 × 3 in 2D and 3 × 3 × 3 in 3D.

References

External links

Diffusion Calculator for Impurities & Dopants in Silicon

A tutorial on the theory behind and solution of the Diffusion Equation.

{{DEFAULTSORT:Diffusion Equation Diffusion Partial differential equations Parabolic partial differential equations Functions of space and time it:Leggi di Fick