|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| The '''diffusion equation''' is a [[partial differential equation]] which describes density dynamics 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]].
| | Adrianne Le is the brand my parents gave me but you can cellphone me anything you like. Vermont has always been my home and [http://www.dailymail.co.uk/home/search.html?sel=site&searchPhrase=I+affection I affection] every [http://Search.Usa.gov/search?query=day+living day living] here. As a girl what I actually like is to play croquet but I can not make it my line of work really. Filing does have been my profession a few time and I'm starting pretty good financially. You can find my blog site here: http://circuspartypanama.com<br><br>Take a look at my webpage :: [http://circuspartypanama.com clash of clans hack free download] |
| | |
| ==Statement==
| |
| | |
| The equation is usually written as:
| |
| | |
| {{Equation box 1
| |
| |equation=<math>\frac{\partial\phi(\mathbf{r},t)}{\partial t} = \nabla \cdot \big[ D(\phi,\mathbf{r}) \ \nabla\phi(\mathbf{r},t) \big], </math>
| |
| |indent=:
| |
| |cellpadding
| |
| |border
| |
| |border colour = #0073CF
| |
| |background colour=#F5FFFA}}
| |
| | |
| 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.
| |
| | |
| More generally, when ''D'' is a symmetric [[positive definite matrix]], the equation describes [[Anisotropy|anisotropic]] diffusion, which is written (for three dimensional diffusion) as:
| |
| | |
| {{Equation box 1
| |
| |equation=<math>\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>
| |
| |indent=:
| |
| |cellpadding
| |
| |border
| |
| |border colour = #50C878
| |
| |background colour=#ECFCF4}}
| |
| | |
| If ''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>
| |
| | |
| also called the [[heat equation]].
| |
| | |
| == Historical origin == | |
| | |
| The [[Fick's law of diffusion|particle diffusion equation]] was originally derived by [[Adolf Fick]] in 1855.<ref>A. Fick, ''Ueber Diffusion'', Pogg. Ann. Phys. Chem. 170 (4. Reihe 94), 59-86 (1855).</ref>
| |
| | |
| == Derivation == | |
| | |
| The diffusion equation can be derived in a straightforward way 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>\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 assumes that the flux of the diffusing material in any part of the system is proportional to the local density gradient:
| |
| | |
| :<math>\mathbf{j}=-D(\phi)\,\nabla\phi(\mathbf{r},t)</math>. | |
| | |
| If [[drift (telecommunication)|drift]] must be taken into account, the [[Smoluchowski 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 (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:
| |
| | |
| <math> \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^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 ==
| |
| * [[Heat equation]]
| |
| * [[Fokker-Planck equation]]
| |
| * [[Fick's law of diffusion]]: [[Fick's law of diffusion#Second law|Fick's Second Law]]
| |
| * [[Maxwell-Stefan equation]]
| |
| * [[Radiative transfer equation and diffusion theory for photon transport in biological tissue]]
| |
| | |
| ==References==
| |
| <references/>
| |
| 2. Carslaw, H. S. and Jager, J. C. (1959). Conduction of Heat in Solids. Oxford: Clarendon Press
| |
| | |
| 3. Crank, J. (1956). The Mathematics of Diffusion. Oxford: Clarendon Press
| |
| | |
| 4. Thambynayagam, R. K. M (2011). The Diffusion Handbook: Applied Solutions for Engineers: McGraw-Hill
| |
| | |
| == External links ==
| |
| * [http://www.ee.byu.edu/cleanroom/DopConCalc.phtml Diffusion Calculator for Impurities & Dopants in Silicon]
| |
| * [http://dl.dropbox.com/u/46147408/tutorials/diffusion.pdf 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)]
| |
| | |
| {{DEFAULTSORT:Diffusion Equation}}
| |
| [[Category:Diffusion]]
| |
| [[Category:Partial differential equations]]
| |
| [[Category:Parabolic partial differential equations]]
| |
| | |
| [[it:Leggi di Fick]]
| |
Adrianne Le is the brand my parents gave me but you can cellphone me anything you like. Vermont has always been my home and I affection every day living here. As a girl what I actually like is to play croquet but I can not make it my line of work really. Filing does have been my profession a few time and I'm starting pretty good financially. You can find my blog site here: http://circuspartypanama.com
Take a look at my webpage :: clash of clans hack free download