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| In [[mathematics]], '''Poisson's equation''' is a [[partial differential equation]] of elliptic type with broad utility in [[electrostatics]], [[mechanical engineering]] and [[theoretical physics]]. Commonly used to model diffusion, it is named after the [[France|French]] [[mathematician]], [[geometer]], and [[physicist]] [[Siméon Denis Poisson]].<ref>{{citation|title=Glossary of Geology|editor1-first=Julia A.|editor1-last=Jackson|editor2-first=James P.|editor2-last=Mehl|editor3-first=Klaus K. E.|editor3-last=Neuendorf|series=American Geological Institute|publisher=Springer|year=2005|isbn=9780922152766|page=503|url=http://books.google.com/books?id=SfnSesBc-RgC&pg=PA503}}.</ref>
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| ==Statement of the equation==
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| Poisson's equation is
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| :<math>\Delta\varphi=f</math>
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| where <math>\Delta</math> is the [[Laplace operator]], and ''f'' and ''φ'' are [[real number|real]] or [[complex number|complex]]-valued [[function (mathematics)|functions]] on a [[manifold]]. When the manifold is [[Euclidean space]], the Laplace operator is often denoted as ∇<sup>2</sup> and so Poisson's equation is frequently written as
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| :<math>\nabla^2 \varphi = f.</math>
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| In three-dimensional [[Cartesian coordinate]]s, it takes the form
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| :<math>
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| \left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} \right)\varphi(x,y,z) = f(x,y,z).
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| </math>
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| Poisson's equation may be solved using a [[Green's function]]; a general exposition of the Green's function for Poisson's equation is given in the article on the [[screened Poisson equation]]. There are various methods for numerical solution. The [[relaxation method]], an iterative algorithm, is one example.
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| ==Newtonian gravity==
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| {{main|gravitational field|Gauss' law for gravity}}
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| In the case of a gravitational field '''g''' due to an attracting massive object, of density ''ρ'', Gauss' law for gravity in differential form can be used to obtain the corresponding Poisson equation for gravity. Gauss' law for gravity is:
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| :<math>\nabla\cdot\bold{g} = -4\pi G\rho </math>,
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| and since the gravitational field is conservative, it can be expressed in terms of a scalar potential ''Φ'':
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| :<math>\bold{g} = -\nabla \Phi </math>,
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| substituting into Gauss' law
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| :<math>\nabla\cdot(-\nabla \Phi) = - 4\pi G \rho</math>
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| obtains '''Poisson's equation''' for gravity:
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| :<math>{\nabla}^2 \Phi = 4\pi G \rho.</math>
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| ==Electrostatics==
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| {{main|Electrostatics}}
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| One of the cornerstones of [[electrostatics]] is setting up and solving problems described by the Poisson equation. Finding φ for some given ''f'' is an important practical problem, since this is the usual way to find the [[electric potential]] for a given [[electric charge|charge]] distribution described by the density function.
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| The mathematical details behind Poisson's equation in electrostatics are as follows ([[SI]] units are used rather than [[Gaussian units]], which are also frequently used in [[electromagnetism]]).
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| Starting with [[Gauss' law]] for electricity (also one of [[Maxwell's equations]]) in differential form, we have:
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| :<math>\mathbf{\nabla} \cdot \mathbf{D} = \rho_f</math> | |
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| where <math>\mathbf{\nabla} \cdot</math> is the [[divergence|divergence operator]], '''D''' = [[electric displacement field]], and ''ρ<sub>f</sub>'' = [[free charge]] [[charge density|density]] (describing charges brought from outside). Assuming the medium is linear, isotropic, and homogeneous (see [[polarization density]]), we have the [[constitutive equation#Electromagnetism|constitutive equation]]:
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| :<math>\mathbf{D} = \varepsilon \mathbf{E}</math>
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| where ''ε'' = [[permittivity]] of the medium and '''E''' = [[electric field]]. Substituting this into Gauss' law and assuming ''ε'' is spatially constant in the region of interest obtains:
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| :<math>\mathbf{\nabla} \cdot \mathbf{E} = \frac{\rho_f}{\varepsilon}</math>
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| In the absence of a changing magnetic field, '''B''', [[Faraday's law of induction]] gives:
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| :<math>\nabla \times \mathbf{E} = -\dfrac{\partial \mathbf{B}} {\partial t} = 0</math>
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| where <math>\nabla \times</math> is the [[curl (mathematics)|curl operator]] and ''t'' is time. Since the [[Curl (mathematics)|curl]] of the electric field is zero, it is defined by a scalar electric potential field, <math>\varphi</math> (see [[Helmholtz decomposition]]).
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| :<math>\mathbf{E} = -\nabla \varphi</math>
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| The derivation of Poisson's equation under these circumstances is straightforward. Substituting the potential gradient for the electric field
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| :<math>\nabla \cdot \bold{E} = \nabla \cdot ( - \nabla \varphi ) = - {\nabla}^2 \varphi = \frac{\rho_f}{\varepsilon},</math>
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| directly obtains '''Poisson's equation''' for electrostatics, which is:
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| :<math>{\nabla}^2 \varphi = -\frac{\rho_f}{\varepsilon}.</math>
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| Solving Poisson's equation for the potential requires knowing the charge density distribution. If the charge density is zero, then [[Laplace's equation]] results. If the charge density follows a [[Boltzmann distribution]], then the [[Poisson-Boltzmann equation]] results. The Poisson–Boltzmann equation plays a role in the development of the [[Debye–Hückel equation|Debye–Hückel theory of dilute electrolyte solutions]].
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| The above discussion assumes that the magnetic field is not varying in time. The same Poisson equation arises even if it does vary in time, as long as the [[Coulomb gauge]] is used. In this more general context, computing ''φ'' is no longer sufficient to calculate '''E''', since '''E''' also depends on the [[magnetic vector potential]] '''A''', which must be independently computed. See [[Mathematical descriptions of the electromagnetic field#Maxwell's equation in potential formulation|Maxwell's equation in potential formulation]] for more on ''φ'' and '''A''' in Maxwell's equations and how Poisson's equation is obtained in this case.
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| === Potential of a Gaussian charge density ===
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| If there is a static spherically symmetric [[Gaussian distribution|Gaussian]] charge density
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| :<math> \rho_f(r) = \frac{Q}{\sigma^3\sqrt{2\pi}^3}\,e^{-r^2/(2\sigma^2)},</math>
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| where ''Q'' is the total charge, then the solution ''φ''(''r'') of Poisson's equation,
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| :<math>{\nabla}^2 \varphi = - { \rho_f \over \varepsilon } </math>,
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| is given by
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| :<math> \varphi(r) = { 1 \over 4 \pi \varepsilon } \frac{Q}{r}\,\mbox{erf}\left(\frac{r}{\sqrt{2}\sigma}\right)</math>
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| where erf(''x'') is the [[error function]].
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| This solution can be checked explicitly by evaluating <math>{\nabla}^2 \varphi</math>. Note that, for ''r'' much greater than ''σ'', the erf function approaches unity and the potential φ (''r'') approaches the [[electrical potential|point charge]] potential
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| :<math> \varphi \approx { 1 \over 4 \pi \varepsilon } {Q \over r} </math>,
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| as one would expect. Furthermore the erf function approaches 1 extremely quickly as its argument increases; in practice for r > 3''σ'' the relative error is smaller than one part in a thousand.
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| ==Surface Reconstruction==
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| Poisson's equation is also used to reconstruct a smooth 2D surface (in the sense of [[curve fitting]]) based on a large number of points ''p<sub>i</sub>'' (a [[point cloud]]) where each point also carries an estimate of the local [[surface normal]] '''n'''<sub>''i''</sub>.<ref>F. Calakli and G. Taubin, [http://mesh.brown.edu/ssd/pdf/Calakli-pg2011.pdf Smooth Signed Distance Surface Reconstruction], Pacific Graphics Vol 30-7, 2011</ref>
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| This technique reconstructs the [[implicit function]] ''f'' whose value is zero at the points ''p<sub>i</sub>'' and whose gradient at the points ''p<sub>i</sub>'' equals the normal vectors '''n'''<sub>''i''</sub>. The set of (''p<sub>i</sub>'', '''n'''<sub>''i''</sub>) is thus a sampling of a continuous [[Euclidean vector|vector]] field '''V'''. The implicit function ''f'' is found by [[Integral|integrating]] the vector field '''V'''. Since not every vector field is the [[gradient]] of a function, the problem may or may not have a solution: the necessary and sufficient condition for a smooth vector field '''V''' to be the gradient of a function ''f'' is that the [[Curl (mathematics)|curl]] of '''V''' must be identically zero. In case this condition is difficult to impose, it is still possible to perform a [[least-squares]] fit to minimize the difference between '''V''' and the gradient of ''f''.
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| ==See also==
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| * [[Discrete Poisson equation]]
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| * [[Poisson–Boltzmann equation]]
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| * [[Uniqueness theorem for Poisson's equation]]
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| ==References==
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| <references />
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| <div class="references-small">
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| * [http://eqworld.ipmnet.ru/en/solutions/lpde/lpde302.pdf Poisson Equation] at EqWorld: The World of Mathematical Equations.
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| * L.C. Evans, ''Partial Differential Equations'', American Mathematical Society, Providence, 1998. ISBN 0-8218-0772-2
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| * A. D. Polyanin, ''Handbook of Linear Partial Differential Equations for Engineers and Scientists'', Chapman & Hall/CRC Press, Boca Raton, 2002. ISBN 1-58488-299-9
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| </div>
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| ==External links==
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| *{{springer|title=Poisson equation|id=p/p073290}}
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| *[http://planetmath.org/encyclopedia/PoissonsEquation.html Poisson's equation] on [[PlanetMath]].
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| *[http://www.youtube.com/watch?v=sMJTWa-Z9Ho Poisson's Equation] Poisson's Equation video
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| [[Category:Potential theory]]
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| [[Category:Partial differential equations]]
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| [[Category:Electrostatics]]
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