Diffusion equation. The  diffusion equation is a parabolic partial differential equation. 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, and applied in many other fields, such as materials science, information theory, and biophysics. The diffusion equation is a special case of the convection–diffusion equation 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 of the diffusing material at location  and time  and  is the collective diffusion coefficient 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; in the case of anisotropic diffusion,  is a symmetric positive definite matrix, and the equation is written (for three dimensional diffusion) as:

The diffusion equation has numerous analytic solutions. 

If  is constant, then the equation reduces to the following linear differential equation:
 
which is identical to the heat equation.

Historical origin 

The particle diffusion equation was originally derived by Adolf Fick 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:

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:

If drift must be taken into account, the Fokker–Planck equation 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, rather than the continuous Gaussian kernel. In discretizing both time and space, one obtains the random walk.

Discretization in image processing 
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:

where "tr" denotes the trace of the 2nd rank 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 with a varying kernel (stencil) of size 3 × 3 in 2D and 3 × 3 × 3 in 3D.

See also 
 Continuity equation
 Heat equation
 Self-similar solutions
 Reaction-diffusion equation
 Fokker–Planck equation
 Fick's laws of diffusion
 Maxwell–Stefan equation
 Radiative transfer equation and diffusion theory for photon transport in biological tissue
 Streamline diffusion
 Numerical solution of the convection–diffusion equation

References

Further reading 
  
 Carslaw, H. S. and Jaeger, J. C. (1959). Conduction of Heat in Solids Oxford: Clarendon Press
 Jacobs, M.H. (1935). Diffusion Processes Berlin/Heidelberg: Springer 
 Crank, J. (1956). The Mathematics of Diffusion Oxford: Clarendon Press
 Mathews, Jon; Walker, Robert L. (1970). Mathematical methods of physics (2nd ed.), New York: W. A. Benjamin, 
 Thambynayagam, R. K. M (2011). The Diffusion Handbook: Applied Solutions for Engineers.  McGraw-Hill
 Ghez, R. (1988). A Primer Of Diffusion Problems, Wiley
 Ghez, R. (2001). Diffusion Phenomena. Long Island, NY, USA:  Dover Publication Inc
 Pekalski, A. (1994). Diffusion Processes: Experiment, Theory, Simulations, Springer
 Bennett, T.D.  (2013). Transport by Advection and Diffusion. John Wiley & Sons
 Vogel, G. (2019). Adventure Diffusion  Springer
 Gillespie, D.T.; Seitaridou, E (2013). Simple Brownian Diffusion,Oxford University Press
 Nakicenovic, N.; Griübler, A.: (1991).   Diffusion of Technologies and Social Behavior; Springer
 Michaud, G.; Alecian, G.; Richer, G.: (2013). Atomic Diffusion in Stars, Springer
 Stroock, D. W.:, Varadhan, S.R.S.: (2006). Multidimensional diffusion processes, Springer
 Zhuoqun, W., Yin J., Li H., Zhao J., Jingxue Y., and  Huilai L. (2001). Nonlinear diffusion equations, World Scientific
 Shewmon, P. (1989). Diffusion in Solids, Wiley
 Banks, R.B. (2010). Growth and diffusion phenomena, Springer
 Roque-Malherbe, R.M.A. (2007). Adsorption and Diffusion in Nanoporous Materials, CRC Press
 Cunningham, R. (1980). Diffusion in gases and porous media, Plenum
 Pasquill,  F.,  Smith, F.B. (1983). Atmospheric diffusion, Horwood
 Ikeda, N., Watanabe, S. (1981). Stochastic Differential Equations and Diffusion Processes, Elsevier, Academic Press
Philibert, J.,  Laskar,  A.L., Bocquet,  J.L., Brebec, G., Monty, C. (1990). Diffusion in Materials, Springer Netherlands
Freedman, D., (1983). Brownian Motion and Diffusion, Springer-Verlag New York
Nagasawa, M., (1993). Schrödinger Equations and Diffusion Theory, Birkhäuser
Burgers, J.M., (1974). The Nonlinear Diffusion Equation: Asymptotic Solutions and Statistical Problems,Springer Netherlands  
Ito, S., (1992). Diffusion Equations, American Mathematical Society
Krylov, N. V. (1994). Introduction to the Theory of Diffusion Processes, American Mathematical Society
Knight, F.B., (1981). Essentials of Brownian Motion and Diffusion, American Mathematical Society
Ibe, O.C., (2013). Elements of random walk and diffusion processes, Wiley
Dattagupta, S. (2013). Diffusion: Formalism and Applications, CRC Press

External links 
 Diffusion Calculator for Impurities & Dopants in Silicon 
 A tutorial on the theory behind and solution of the Diffusion Equation.
 Classical and nanoscale diffusion (with figures and animations)

Diffusion
Partial differential equations
Parabolic partial differential equations
Functions of space and time

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