Diffusion equation
Encyclopedia
The diffusion equation is a partial differential equation
which describes density fluctuations in a material undergoing diffusion
. It is also used to describe processes exhibiting diffusive-like behaviour, for instance the 'diffusion' of alleles in a population in population genetics
.
The equation is usually written as:
where
is the density
of the diffusing material at location
and time 
and 
is the collective diffusion coefficient for density 
at location 
; the nabla symbol
represents the vector differential operator
del
acting on the space coordinates. If the diffusion coefficient depends on the density then the equation is nonlinear, otherwise it is linear. If
is constant, then the equation reduces to the following linear equation:
also called the heat equation
. More generally, when D is a symmetric positive definite matrix, the equation describes anisotropic
diffusion, which is written (for three dimensional diffusion) as:
, 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:
,
where
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 assumes that the flux of the diffusing material in any part of the system is proportional to the local density gradient:
.
If drift
must be taken into account, the Smoluchowski equation
provides an appropriate generalization.
was originally derived by Adolf Fick in 1855.
In discretizing space alone, the Green's function
becomes the discrete Gaussian kernel, rather than the continuous Gaussian kernel. In discretizing both time and space, one obtains the random walk
.
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 :
In which in image filtering
are symmetric matrices constructed from the eigenvectors of the image structure tensor
s . The spatial derivatives can then be approximated by two first order and a second order central finite difference
s. The resulting diffusion algorithm can be written as an image convolution
with a varying kernel (stencil) of size 3x3 in 2D and 3x3x3 in 3D.
Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
which describes density fluctuations in a material undergoing diffusion
Diffusion
Molecular diffusion, often called simply diffusion, is the thermal motion of all particles at temperatures above absolute zero. The rate of this movement is a function of temperature, viscosity of the fluid and the size of the particles...
. It is also used to describe processes exhibiting diffusive-like behaviour, for instance the 'diffusion' of alleles in a population in population genetics
Population genetics
Population genetics is the study of allele frequency distribution and change under the influence of the four main evolutionary processes: natural selection, genetic drift, mutation and gene flow. It also takes into account the factors of recombination, population subdivision and population...
.
The equation is usually written as:
where
is the densityDensity
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...
of the diffusing material at location
and time and is the collective diffusion coefficient for density at location ; the nabla symbolNabla symbol
right|thumb|256px|The [[harp]], the instrument after which the nabla symbol is namedNabla is the symbol \nabla . The name comes from the Greek word for a Hebrew harp, which had a similar shape. Related words also exist in Aramaic and Hebrew. The symbol was first used by William Rowan Hamilton in...
represents the vector differential operatorDifferential operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation, accepting a function and returning another .This article considers only linear operators,...
del
Del
In vector calculus, del is a vector differential operator, usually represented by the nabla symbol \nabla . When applied to a function defined on a one-dimensional domain, it denotes its standard derivative as defined in calculus...
acting on the space coordinates. If the diffusion coefficient depends on the density then the equation is nonlinear, otherwise it is linear. If
is constant, then the equation reduces to the following linear equation:also called the heat equation
Heat equation
The heat equation is an important partial differential equation which describes the distribution of heat in a given region over time...
. More generally, when D is a symmetric positive definite matrix, the equation describes anisotropic
Anisotropy
Anisotropy is the property of being directionally dependent, as opposed to isotropy, which implies identical properties in all directions. It can be defined as a difference, when measured along different axes, in a material's physical or mechanical properties An example of anisotropy is the light...
diffusion, which is written (for three dimensional diffusion) as:
Derivation
The diffusion equation can be derived in a straightforward way from the continuity equationContinuity equation
A continuity equation in physics is a differential equation that describes the transport of a conserved quantity. Since mass, energy, momentum, electric charge and other natural quantities are conserved under their respective appropriate conditions, a variety of physical phenomena may be described...
, 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:
,where
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 assumes that the flux of the diffusing material in any part of the system is proportional to the local density gradient:.If drift
Drift
- Film and literature :* Drift , a 2002 Doctor Who novel* Drift , a series of Japanese films written and directed by Futoshi Jinno* Drift, 2007 experimental short film by Max Hattler* Drift , a fictional character...
must be taken into account, the Smoluchowski equation
Smoluchowski equation
In physics, the diffusion equation with drift term is often called Smoluchowski equation .- The equation :Let w be a density, D a diffusion constant, ζ a friction coefficient,...
provides an appropriate generalization.
Historical origin
The particle diffusion equationFick's law of diffusion
Fick's laws of diffusion describe diffusion and can be used to solve for the diffusion coefficient, D. They were derived by Adolf Fick in the year 1855.- Fick's first law :...
was originally derived by Adolf Fick in 1855.
Discretization
The diffusion equation is continuous in both time and space. 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
Green's function
In mathematics, a Green's function is a type of function used to solve inhomogeneous differential equations subject to specific initial conditions or boundary conditions...
becomes the discrete Gaussian kernel, rather than the continuous Gaussian kernel. In discretizing both time and space, one obtains the random walk
Random walk
A random walk, sometimes denoted RW, is a mathematical formalisation of a trajectory that consists of taking successive random steps. For example, the path traced by a molecule as it travels in a liquid or a gas, the search path of a foraging animal, the price of a fluctuating stock and the...
.
Discretization (Image)
The product ruleProduct rule
In calculus, the product rule is a formula used to find the derivatives of products of two or more functions. It may be stated thus:'=f'\cdot g+f\cdot g' \,\! or in the Leibniz notation thus:...
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 :
In which in image filtering
are symmetric matrices constructed from the eigenvectors of the image structure tensorStructure tensor
In mathematics, the structure tensor, also referred to as the second-moment matrix, is a matrix derived from the gradient of a function. It summarizes the predominant directions of the gradient in a specified neighborhood of a point, and the degree to which those directions are coherent...
s . The spatial derivatives can then be approximated by two first order and a second order central finite difference
Finite difference
A finite difference is a mathematical expression of the form f − f. If a finite difference is divided by b − a, one gets a difference quotient...
s. The resulting diffusion algorithm can be written as an image convolution
Convolution
In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation...
with a varying kernel (stencil) of size 3x3 in 2D and 3x3x3 in 3D.
See also
- Heat equationHeat equationThe heat equation is an important partial differential equation which describes the distribution of heat in a given region over time...
- Fick's law of diffusionFick's law of diffusionFick's laws of diffusion describe diffusion and can be used to solve for the diffusion coefficient, D. They were derived by Adolf Fick in the year 1855.- Fick's first law :...
: Fick's Second Law - Radiative transfer equation and diffusion theory for photon transport in biological tissueRadiative transfer equation and diffusion theory for photon transport in biological tissuePhoton transport in biological tissue can be equivalently modeled numerically with Monte Carlo simulations or analytically by the radiative transfer equation . However, the RTE is difficult to solve without introducing approximations. A common approximation summarized here is the diffusion...