View source for Poisson's equation

{{short description|Expression frequently encountered in mathematical physics, generalization of Laplace's equation}}
[[File:Simeon Poisson.jpg|thumb|Siméon Denis Poisson]]
'''Poisson's equation''' is an [[elliptic partial differential equation]] of broad utility in [[theoretical physics]]. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with the potential field known, one can then calculate electrostatic or gravitational (force) field. It is a generalization of [[Laplace's equation]], which is also frequently seen in physics.  The equation is named after French mathematician 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=https://books.google.com/books?id=SfnSesBc-RgC&pg=PA503 }}</ref><ref>{{cite journal |last1=Poisson |date=1823 |title=Mémoire sur la théorie du magnétisme en mouvement |trans-title=Memoir on the theory of magnetism in motion | url=https://www.biodiversitylibrary.org/item/55214#page/633/mode/1up |journal=Mémoires de l'Académie Royale des Sciences de l'Institut de France |volume=6 |pages=441–570 |language=fr }} From [https://www.biodiversitylibrary.org/item/55214#page/655/mode/1up p.&nbsp;463]: {{lang|fr|"Donc, d'après ce qui précède, nous aurons enfin:}}
<math display="block">\frac{\partial^2 V} {\partial x^2} + \frac{\partial^2 V} {\partial y^2} + \frac{\partial^2 V} {\partial z^2} = 0, = -2k\pi, = -4k\pi,</math>
{{lang|fr|selon que le point M sera situé en dehors, à la surface ou en dedans du volume que l'on considère."}} (Thus, according to what preceded, we will finally have:
<math display="block">\frac{\partial^2 V} {\partial x^2} + \frac{\partial^2 V} {\partial y^2} + \frac{\partial^2 V} {\partial z^2} = 0, = -2k\pi, = -4k\pi,</math>
depending on whether the point ''M'' is located outside, on the surface of, or inside the volume that one is considering.)  ''V'' is defined (p.&nbsp;462) as
<math display="block">V = \iiint\frac{k'}{\rho}\, dx'\,dy'\,dz',</math>
where, in the case of electrostatics, the integral is performed over the volume of the charged body, the coordinates of points that are inside or on the volume of the charged body are denoted by <math>(x', y', z')</math>, <math>k'</math> is a given function of <math>(x', y,' z')</math> and in electrostatics, <math>k'</math> would be a measure of charge density, and <math>\rho</math> is defined as the length of a radius extending from the point M to a point that lies inside or on the charged body. The coordinates of the point ''M'' are denoted by <math>(x, y, z)</math> and <math>k</math> denotes the value of <math>k'</math> (the charge density) at ''M''.</ref>

==Statement of the equation==

Poisson's equation is
<math display="block">\Delta\varphi = f,</math>
where <math>\Delta</math> is the [[Laplace operator]], and <math>f</math> and <math>\varphi</math> are [[real number|real]] or [[complex number|complex]]-valued [[function (mathematics)|functions]] on a [[manifold]]. Usually, <math>f</math> is given, and <math>\varphi</math> is sought. When the manifold is [[Euclidean space]], the Laplace operator is often denoted as {{math|∇<sup>2</sup>}}, and so Poisson's equation is frequently written as
<math display="block">\nabla^2 \varphi = f.</math>

In three-dimensional [[Cartesian coordinate]]s, it takes the form
<math display="block">\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).</math>

When <math>f = 0</math> identically, we obtain [[Laplace's equation]].

Poisson's equation may be solved using a [[Green's function]]:
<math display="block">\varphi(\mathbf{r}) = - \iiint \frac{f(\mathbf{r}')}{4\pi |\mathbf{r} - \mathbf{r}'|}\, \mathrm{d}^3 r',</math>
where the integral is over all of space. 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, such as the [[relaxation method]], an iterative algorithm.

==Newtonian gravity==
{{main|Gravitational field|Gauss's law for gravity}}

In the case of a gravitational field '''g''' due to an attracting massive object of density ''ρ'', Gauss's law for gravity in differential form can be used to obtain the corresponding Poisson equation for gravity:
<math display="block">\nabla\cdot\mathbf{g} = -4\pi G\rho.</math>

Since the gravitational field is conservative (and [[irrotational]]), it can be expressed in terms of a [[scalar potential]] ''ϕ'':
<math display="block">\mathbf{g} = -\nabla \phi.</math>

Substituting this into Gauss's law,
<math display="block">\nabla\cdot(-\nabla \phi) = - 4\pi G \rho,</math>
yields '''Poisson's equation''' for gravity:
<math display="block">\nabla^2 \phi  =  4\pi G \rho.</math>

If the mass density is zero, Poisson's equation reduces to Laplace's equation. The [[Green's function for the three-variable Laplace equation|corresponding Green's function]] can be used to calculate the potential at distance {{mvar|r}} from a central point mass {{mvar|m}} (i.e., the [[fundamental solution]]). In three dimensions the potential is
<math display="block">\phi(r) = \frac{-G m}{r},</math>
which is equivalent to [[Newton's law of universal gravitation]].

==Electrostatics==
{{main|Electrostatics}}
{{unsourced section|date=March 2023}}

One of the cornerstones of [[electrostatics]] is setting up and solving problems described by the Poisson equation. Solving the Poisson equation amounts to finding the [[electric potential]] {{mvar|φ}} for a given [[electric charge|charge]] distribution <math>\rho_f</math>.

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]]).

Starting with [[Gauss's law]] for electricity (also one of [[Maxwell's equations]]) in differential form, one has
<math display="block">\mathbf{\nabla} \cdot \mathbf{D} = \rho_f,</math>
where <math>\mathbf{\nabla} \cdot</math> is the [[divergence|divergence operator]], '''D''' is the [[electric displacement field]], and ''ρ<sub>f</sub>'' is the free-[[charge 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]]
<math display="block">\mathbf{D} = \varepsilon \mathbf{E},</math>
where {{mvar|ε}} is the [[permittivity]] of the medium, and '''E''' is the [[electric field]].

Substituting this into Gauss's law and assuming that {{mvar|ε}} is spatially constant in the region of interest yields
<math display="block">\mathbf{\nabla} \cdot \mathbf{E} = \frac{\rho_f}{\varepsilon}.</math>
In electrostatics, we assume that there is no magnetic field (the argument that follows also holds in the presence of a constant magnetic field). Then, we have that
<math display="block">\nabla \times \mathbf{E} = 0,</math>
where {{math|∇×}} is the [[Curl (mathematics)|curl operator]]. This equation means that we can write the electric field as the gradient of a scalar function {{mvar|φ}} (called the [[electric potential]]), since the curl of any gradient is zero. Thus we can write
<math display="block">\mathbf{E} = -\nabla \varphi,</math>
where the minus sign is introduced so that {{mvar|φ}} is identified as the [[electric potential energy]] per unit charge.

The derivation of Poisson's equation under these circumstances is straightforward. Substituting the potential gradient for the electric field,
<math display="block">\nabla \cdot \mathbf{E} = \nabla \cdot (-\nabla \varphi) = -\nabla^2 \varphi = \frac{\rho_f}{\varepsilon},</math>
directly produces '''Poisson's equation''' for electrostatics, which is
<math display="block">\nabla^2 \varphi = -\frac{\rho_f}{\varepsilon}.</math>

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]].

Using Green's function, the potential at distance {{mvar|r}} from a central point charge {{mvar|Q}} (i.e., the [[fundamental solution]]) is
<math display="block">\varphi(r) = \frac {Q}{4 \pi \varepsilon r},</math>
which is [[Coulomb's law]] of electrostatics. (For historic reasons, and unlike gravity's model above, the <math>4 \pi</math> factor appears here and not in Gauss's law.)

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 {{mvar|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 equations in potential formulation|Maxwell's equation in potential formulation]] for more on {{mvar|φ}} and '''A''' in Maxwell's equations and how Poisson's equation is obtained in this case.

=== Potential of a Gaussian charge density ===
If there is a static spherically symmetric [[Gaussian distribution|Gaussian]] charge density
<math display="block">\rho_f(r) = \frac{Q}{\sigma^3\sqrt{2\pi}^3}\,e^{-r^2/(2\sigma^2)},</math>
where {{mvar|Q}} is the total charge, then the solution {{math|''φ''(''r'')}} of Poisson's equation
<math display="block">\nabla^2 \varphi = -\frac{\rho_f}{\varepsilon}</math>
is given by
<math display="block">\varphi(r) = \frac{1}{4 \pi \varepsilon} \frac{Q}{r} \operatorname{erf}\left(\frac{r}{\sqrt{2}\sigma}\right),</math>
where {{math|erf(''x'')}} is the [[error function]].

This solution can be checked explicitly by evaluating {{math|∇<sup>2</sup>''φ''}}.

Note that for {{mvar|r}} much greater than {{mvar|σ}}, the erf function approaches unity, and the potential {{math|''φ''(''r'')}} approaches the [[electrical potential|point-charge]] potential,
<math display="block">\varphi \approx \frac{1}{4 \pi \varepsilon} \frac{Q}{r},</math>
as one would expect. Furthermore, the error function approaches 1 extremely quickly as its argument increases; in practice, for {{math|''r'' > 3''σ''}} the relative error is smaller than one part in a thousand.

==Surface reconstruction==

Surface reconstruction is an [[inverse problem]]. The goal is to digitally reconstruct a smooth surface 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>{{cite journal |first1=Fatih |last1=Calakli |first2=Gabriel |last2=Taubin |title=Smooth Signed Distance Surface Reconstruction |journal=Pacific Graphics |year=2011 |volume=30 |number=7 |url=http://mesh.brown.edu/ssd/pdf/Calakli-pg2011.pdf }}</ref> Poisson's equation can be utilized to solve this problem with a technique called Poisson surface reconstruction.<ref name="Kazhdan06">{{cite book |first1=Michael |last1=Kazhdan |first2=Matthew |last2=Bolitho |first3=Hugues |last3=Hoppe |year=2006 |chapter=Poisson surface reconstruction |title=Proceedings of the fourth Eurographics symposium on Geometry processing (SGP '06) |publisher=Eurographics Association, Aire-la-Ville, Switzerland |pages=61–70 |isbn=3-905673-36-3 |chapter-url=https://dl.acm.org/doi/abs/10.5555/1281957.1281965 }}</ref>

The goal of this technique is to reconstruct an [[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 modeled as 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''.

In order to effectively apply Poisson's equation to the problem of surface reconstruction, it is necessary to find a good discretization of the vector field '''V'''. The basic approach is to bound the data with a [[finite-difference]] grid. For a function valued at the nodes of such a grid, its gradient can be represented as valued on staggered grids, i.e. on grids whose nodes lie in between the nodes of the original grid. It is convenient to define three staggered grids, each shifted in one and only one direction corresponding to the components of the normal data. On each staggered grid we perform [[trilinear interpolation]] on the set of points. The interpolation weights are then used to distribute the magnitude of the associated component of ''n<sub>i</sub>'' onto the nodes of the particular staggered grid cell containing ''p<sub>i</sub>''. Kazhdan and coauthors give a more accurate method of discretization using an adaptive finite-difference grid, i.e. the cells of the grid are smaller (the grid is more finely divided) where there are more data points.<ref name="Kazhdan06"/> They suggest implementing this technique with an adaptive [[octree]].

== Fluid dynamics ==
For the incompressible [[Navier–Stokes equations]], given by
<math display="block">\begin{aligned}
 \frac{\partial\mathbf{v}}{\partial t} + (\mathbf{v} \cdot \nabla) \mathbf{v} &= -\frac{1}{\rho} \nabla p + \nu\Delta\mathbf{v} + \mathbf{g}, \\
 \nabla \cdot \mathbf{v} &= 0.
\end{aligned}</math>

The equation for the pressure field <math>p</math> is an example of a nonlinear Poisson equation:
<math display="block">\begin{aligned}
 \Delta p &= -\rho \nabla \cdot(\mathbf{v} \cdot \nabla \mathbf{v}) \\
          &= -\rho \operatorname{Tr}\big((\nabla\mathbf{v}) (\nabla\mathbf{v})\big).
\end{aligned}</math>
Notice that the above trace is not sign-definite.

==See also==
{{Portal|Mathematics|Physics}}
* [[Discrete Poisson equation]]
* [[Poisson–Boltzmann equation]]
* [[Helmholtz equation]]
* [[Uniqueness theorem for Poisson's equation]]
* [[Weak formulation#Example_2: Poisson's equation|Weak formulation]]

==References==
{{reflist}}

==Further reading==
* {{cite book |first=Lawrence C. |last=Evans |title=Partial Differential Equations |publisher=American Mathematical Society |location=Providence (RI) |year=1998 |isbn=0-8218-0772-2 }}
* {{cite book |last1=Mathews |first1=Jon |last2=Walker |first2=Robert L. |year=1970 |title=Mathematical Methods of Physics |edition=2nd |location=New York |publisher=W. A. Benjamin |isbn=0-8053-7002-1 }}
* {{cite book |first=Andrei D. |last=Polyanin |title=Handbook of Linear Partial Differential Equations for Engineers and Scientists |publisher=Chapman & Hall/CRC Press |location=Boca Raton (FL) |year=2002 |isbn=1-58488-299-9 }}

==External links==
* {{springer|title=Poisson equation|id=p/p073290}}
* [http://eqworld.ipmnet.ru/en/solutions/lpde/lpde302.pdf Poisson Equation] at EqWorld: The World of Mathematical Equations
* [http://planetmath.org/poissonsequation Poisson's equation] on [[PlanetMath]].

[[Category:Eponymous equations of physics]]
[[Category:Potential theory]]
[[Category:Partial differential equations]]
[[Category:Electrostatics]]
[[Category:Mathematical physics]]
[[Category:Electromagnetism]]