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Poisson's equation is an
elliptic partial differential equation Second-order linear Linearity is the property of a mathematical relationship (''function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be pr ...
of broad utility in
theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict List of natural phenomena, natural phenomena. This is in contrast to experimental ph ...
. 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
, which is also frequently seen in physics. The equation is named after French mathematician and physicist
Siméon Denis Poisson Baron Baron is a rank of nobility or title of honour, often hereditary, in various European countries, either current or historical. The female equivalent is baroness. Typically, the title denotes an aristocrat who ranks higher than a lord ...
.


Statement of the equation

Poisson's equation is \Delta\varphi = f where \Delta is the
Laplace operator In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
, and f and \varphi are
real Real may refer to: * Reality Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ...
or
complex The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London , mottoeng = Let all come who by merit deserve the most reward , established = , type = Public university, Public rese ...

complex
-valued
functions Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
on a
manifold In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

manifold
. Usually, f is given and \varphi is sought. When the manifold is
Euclidean space Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...
, the Laplace operator is often denoted as and so Poisson's equation is frequently written as \nabla^2 \varphi = f. In three-dimensional
Cartesian coordinate A Cartesian coordinate system (, ) in a plane Plane or planes may refer to: * Airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft A fixed-wing aircraft is a heavier-than-air flying machine Early fly ...

Cartesian coordinate
s, it takes the form \left( \frac + \frac + \frac \right)\varphi(x,y,z) = f(x,y,z). When f = 0 identically we obtain
Laplace's equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
. Poisson's equation may be solved using a
Green's function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
: \varphi(\mathbf) = - \iiint \frac\, \mathrm^3\! r', 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 In numerical mathematics, relaxation methods are iterative methods for solving simultaneous equations, systems of equations, including nonlinear systems. Relaxation methods were developed for solving large sparse matrix, sparse linear systems, whic ...
, an iterative algorithm.


Newtonian 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, \nabla\cdot\mathbf = -4\pi G\rho ~. Since the gravitational field is conservative (and
irrotational In vector calculus, a conservative vector field is a vector field that is the gradient of some function (mathematics), function. Conservative vector fields have the property that the line integral is path independent; the choice of any path between ...
), it can be expressed in terms of a scalar potential ''Φ'', \mathbf = -\nabla \phi ~. Substituting into Gauss's law \nabla\cdot(-\nabla \phi) = - 4\pi G \rho yields Poisson's equation for gravity, \nabla^2 \phi = 4\pi G \rho. If the mass density is zero, Poisson's equation reduces to Laplace's equation. The corresponding Green's function can be used to calculate the potential at distance from a central point mass (i.e., the
fundamental solution In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
). In three dimensions the potential is \phi(r) = \dfrac . which is equivalent to
Newton's law of universal gravitation Newton's law of universal gravitation is usually stated as that every particle In the Outline of physical science, physical sciences, a particle (or corpuscule in older texts) is a small wikt:local, localized physical body, object to which can ...
.


Electrostatics

One of the cornerstones of
electrostatics Electrostatics is a branch of physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related enti ...
is setting up and solving problems described by the Poisson equation. Solving the Poisson equation amounts to finding the
electric potential The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work Work may refer to: * Work (human activity), intentional activity people perform to support the ...

electric potential
for a given
charge Charge or charged may refer to: Arts, entertainment, and media Films * ''Charge, Zero Emissions/Maximum Speed'', a 2011 documentary Music * Charge (David Ford album), ''Charge'' (David Ford album) * Charge (Machel Montano album), ''Charge'' (Mac ...
distribution \rho_f. The mathematical details behind Poisson's equation in electrostatics are as follows (
SI
SI
units are used rather than
Gaussian units Gaussian units constitute a metric system of units of measurement, physical units. This system is the most common of the several electromagnetic unit systems based on Centimetre gram second system of units, cgs (centimetre–gram–second) units. ...
, which are also frequently used in
electromagnetism Electromagnetism is a branch of physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in ...

electromagnetism
). Starting with
Gauss's law In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. ...
for electricity (also one of
Maxwell's equations Maxwell's equations are a set of coupled partial differential equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), ...
) in differential form, one has \mathbf \cdot \mathbf = \rho_f where \mathbf \cdot is the
divergence operator
divergence operator
, D =
electric displacement field In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succ ...
, and ''ρf'' =
free charge In classical electromagnetism Classical electromagnetism or classical electrodynamics is a branch of theoretical physics that studies the interactions between electric charge Electric charge is the physical property of matter that causes it t ...
volume
density The density (more precisely, the volumetric mass density; also known as specific mass), of a substance is its per unit . The symbol most often used for density is ''ρ'' (the lower case Greek letter ), although the Latin letter ''D'' can also ...

density
(describing charges brought from outside). Assuming the medium is linear, isotropic, and homogeneous (see
polarization density In classical electromagnetism Classical electromagnetism or classical electrodynamics is a branch of theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and sys ...
), we have the
constitutive equation In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succ ...

constitutive equation
, \mathbf = \varepsilon \mathbf where is the
permittivity In electromagnetism Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric charge, electrically charged particles. The electromagnetic force is ...
of the medium and E is the
electric field An electric field (sometimes E-field) is the physical field that surrounds electrically-charged particle In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' ' ...

electric field
. Substituting this into Gauss's law and assuming is spatially constant in the region of interest yields \mathbf \cdot \mathbf = \frac ~. where \rho is a total volume charge density. In electrostatic, 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 \nabla \times \mathbf = 0, where is the curl operator. This equation means that we can write the electric field as the gradient of a scalar function (called the electric potential), since the curl of any gradient is zero. Thus we can write, \mathbf = -\nabla \varphi, where the minus sign is introduced so that is identified as the potential energy per unit charge. The derivation of Poisson's equation under these circumstances is straightforward. Substituting the potential gradient for the electric field, \nabla \cdot \mathbf = \nabla \cdot ( - \nabla \varphi ) = - ^2 \varphi = \frac, directly produces Poisson's equation for electrostatics, which is \nabla^2 \varphi = -\frac. Solving Poisson's equation for the potential requires knowing the charge density distribution. If the charge density is zero, then
Laplace's equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
results. If the charge density follows a
Boltzmann distribution In statistical mechanics In physics, statistical mechanics is a mathematical framework that applies Statistics, statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natu ...

Boltzmann distribution
, then the Poisson-Boltzmann equation results. The Poisson–Boltzmann equation plays a role in the development of the Debye–Hückel theory of dilute electrolyte solutions. Using Green's Function, the potential at distance from a central point charge (i.e., the Fundamental Solution) is: \varphi(r) = \frac . which is Coulomb's law of electrostatics. (For historic reasons, and unlike gravity's model above, the 4 \pi 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 In the physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior throu ...
is used. In this more general context, computing is no longer sufficient to calculate E, since E also depends on the
magnetic vector potential Magnetic vector potential, A, is the vector quantity in classical electromagnetism Classical electromagnetism or classical electrodynamics is a branch of theoretical physics Theoretical physics is a branch of physics that employs mathematica ...
A, which must be independently computed. See Maxwell's equation in potential formulation for more on 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 Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below. There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymous ...

Gaussian
charge density \rho_f(r) = \frac\,e^, where is the total charge, then the solution of Poisson's equation, ^2 \varphi = - , is given by \varphi(r) = \frac \frac \, \operatorname\left(\frac\right) where is the
error function In mathematics, the error function (also called the Gauss error function), often denoted by , is a complex function of a complex variable defined as: :\operatorname z = \frac\int_0^z e^\,dt. This integral is a special Special or specials may ...

error function
. This solution can be checked explicitly by evaluating . Note that, for much greater than , the erf function approaches unity and the potential approaches the
point charge A point particle (ideal particle or point-like particle, often spelled pointlike particle) is an idealization of particle In the Outline of physical science, physical sciences, a particle (or corpuscule in older texts) is a small wikt:local, ...
potential \varphi \approx \frac \frac , as one would expect. Furthermore, the error function approaches 1 extremely quickly as its argument increases; in practice for the relative error is smaller than one part in a thousand.


Surface reconstruction

Surface reconstruction is an
inverse problem An inverse problem in science is the process of calculating from a set of observations the causal factors that produced them: for example, calculating an image in X-ray computed tomography, sound source reconstruction, source reconstruction in aco ...

inverse problem
. The goal is to digitally reconstruct a smooth surface based on a large number of points ''pi'' (a
point cloud A point cloud image of a torus A point cloud is a set of data points in space. The points represent a 3D shape or object. Each point has its set of X, Y and Z coordinates. Point clouds are generally produced by 3D scanners or by photogrammetry so ...
) where each point also carries an estimate of the local
surface normal In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

surface normal
n''i''. Poisson's equation can be utilized to solve this problem with a technique called Poisson surface reconstruction. The goal of this technique is to reconstruct an
implicit function In mathematics, an implicit equation is a relation (mathematics), relation of the form , where is a function (mathematics), function of several variables (often a polynomial). For example, the implicit equation of the unit circle is . An implici ...
''f'' whose value is zero at the points ''pi'' and whose gradient at the points ''pi'' equals the normal vectors n''i''. The set of (''pi'', n''i'') is thus modeled as a continuous
vector Vector may refer to: Biology *Vector (epidemiology) In epidemiology Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and risk factor, determinants of health and disease conditions in defined pop ...
field V. The implicit function ''f'' is found by
integrating
integrating
the vector field V. Since not every vector field is the
gradient In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Prod ...

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 Curl or CURL may refer to: Science and technology * Curl (mathematics) In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the p ...
of V must be identically zero. In case this condition is difficult to impose, it is still possible to perform a
least-squares The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems (sets of equations in which there are more equations than unknowns) by minimizing the sum of the squares of the resid ...
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 rilinear interpolationon the set of points. The interpolation weights are then used to distribute the magnitude of the associated component of ''ni'' onto the nodes of the particular staggered grid cell containing ''pi''. 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. They suggest implementing this technique with an adaptive
octree An octree is a tree data structure In computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. ...

octree
.


Fluid dynamics

For the incompressible
Navier–Stokes equations In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. ...
, given by: \begin + \cdot\nabla &= -\nabla p + \nu\Delta + \\ \nabla\cdot &= 0 \end The equation for the pressure field p is an example of a nonlinear Poisson equation: \begin \Delta p &= -\rho \nabla\cdot(\cdot \nabla ) \\ &= -\rho\, \mathrm\big((\nabla ) (\nabla )\big). \end Notice that the above trace is not sign-definite.


See also

* Discrete Poisson equation *
Poisson–Boltzmann equation The Poisson–Boltzmann equation is a useful equation in many settings, whether it be to understand physiology, physiological interfaces, polymer science, electron interactions in a semiconductor, or more. It aims to describe the distribution of th ...
*
Helmholtz equation In mathematics, the eigenvalue In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their r ...
* Uniqueness theorem for Poisson's equation *
Weak formulation Weak formulations are important tools for the analysis of mathematical equation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which ...


References


Further reading

* * *


External links

* {{springer, title=Poisson equation, id=p/p073290
Poisson Equation
at EqWorld: The World of Mathematical Equations
Poisson's equation
on
PlanetMath PlanetMath is a free Free may refer to: Concept * Freedom, having the ability to act or change without constraint * Emancipate, to procure political rights, as for a disenfranchised group * Free will, control exercised by rational agents over ...
. Potential theory Partial differential equations Electrostatics Mathematical physics Equations of physics Electromagnetism