Editing Diffusion equation

{{short description|Equation that describes density changes of a material that is diffusing in a medium}}

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 process|Markov processes]], such as [[Random walk|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:
<math display="block">\frac{\partial\phi(\mathbf{r},t)}{\partial t} = \nabla \cdot \big[ D(\phi,\mathbf{r}) \ \nabla\phi(\mathbf{r},t) \big],</math>
where {{math|''ϕ''('''r''', ''t'')}} is the [[density]] of the diffusing material at location {{math|'''r'''}} and time {{mvar|t}} and {{math|''D''(''ϕ'', '''r''')}} is the collective [[diffusion coefficient]] for density {{mvar|ϕ}} at location {{math|'''r'''}}; and {{math|∇}} 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 [[Isotropy|isotropic]]; in the case of anisotropic diffusion, {{mvar|D}} is a symmetric [[positive definite matrix]], and the equation is written (for three dimensional diffusion) as:
<math display="block">\frac{\partial\phi(\mathbf{r},t)}{\partial t} = \sum_{i=1}^3\sum_{j=1}^3 \frac{\partial}{\partial x_i}\left[D_{ij}(\phi,\mathbf{r})\frac{\partial \phi(\mathbf{r},t)}{\partial x_j}\right]</math>
The diffusion equation has numerous analytic solutions.<ref>{{Cite journal |last1=Barna |first1=I.F. |last2=Mátyás |first2=L. |year=2022 |title=Advanced Analytic Self-Similar Solutions of Regular and Irregular Diffusion Equations |journal=Mathematics |volume=10 |issue=18 |pages=3281 |doi=10.3390/math10183281|doi-access=free |arxiv=2204.04895 }}</ref> 

If {{mvar|D}} is constant, then the equation reduces to the following [[linear differential equation]]:
: <math>\frac{\partial\phi(\mathbf{r},t)}{\partial t} = D\nabla^2\phi(\mathbf{r},t), </math>
which is identical to the [[heat equation]].

== Historical origin ==

The [[Fick's law of diffusion|particle diffusion equation]] was originally derived by [[Adolf Fick]] in 1855.<ref name="Fick1855">{{cite journal |last1=Fick |first1=Adolf |title=Ueber Diffusion |journal=Annalen der Physik und Chemie |volume=170 |issue=1 |year=1855 |pages=59–86 |issn=0003-3804 |doi=10.1002/andp.18551700105|bibcode=1855AnP...170...59F |doi-access=free }}</ref>

== 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:
<math display="block">\frac{\partial\phi}{\partial t}+\nabla\cdot\mathbf{j}=0,</math>
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 law|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:
<math display="block">\mathbf{j}=-D(\phi,\mathbf{r})\,\nabla\phi(\mathbf{r},t).</math>

If drift must be taken into account, the [[Fokker–Planck equation]] provides an appropriate generalization.

== Discretization ==
{{see also|Discrete Gaussian kernel}}

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:
<math display="block"> \frac{\partial\phi(\mathbf{r},t)}{\partial t} = \nabla\cdot \left[D(\phi,\mathbf{r})\right] \nabla \phi(\mathbf{r},t) + {\rm tr} \Big[ D(\phi,\mathbf{r})\big(\nabla\nabla^\text{T} \phi(\mathbf{r},t)\big)\Big] </math>
where "tr" denotes the [[Trace (linear algebra)|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 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 3 × 3 in 2D and 3 × 3 × 3 in 3D.

== See also ==
* [[Continuity equation]]
* [[Heat equation]]
* [[Self-similar solution|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 ==
{{reflist|30em}}

== Further reading ==
* {{Cite journal |last1=Mehrer |first1=H. |last2=Stolwijk |first2=A |date=2009 |title=Heroes and Highlights in the History of Diffusion |url=https://www.diffusion-fundamentals.org/journal/11/index11.html |journal=Diffusion Fundamentals |volume=11 |issue= |pages=1–32 |doi=10.62721/diffusion-fundamentals.11.453 |jstor=}} 
* 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, {{ISBN|0-8053-7002-1}}
* 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
*[[J. M. Burgers|Burgers, J.M.]], (1974). T''he 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 ==
* [http://www.ee.byu.edu/cleanroom/DopConCalc.phtml Diffusion Calculator for Impurities & Dopants in Silicon] {{Webarchive|url=https://web.archive.org/web/20090502033108/http://ee.byu.edu/cleanroom/DopConCalc.phtml |date=2009-05-02 }}
* [https://www.dropbox.com/s/s0470m594llwcc7/diffusion.pdf?raw=1 A tutorial on the theory behind and solution of the Diffusion Equation.]
* [http://dragon.unideb.hu/~zerdelyi/Diffusion-on-the-nanoscale/index.html Classical and nanoscale diffusion (with figures and animations)]

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{{DEFAULTSORT:Diffusion Equation}}
[[Category:Diffusion]]
[[Category:Partial differential equations]]
[[Category:Parabolic partial differential equations]]
[[Category:Functions of space and time]]

[[it:Leggi di Fick]]