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Poisson's equation is an elliptic partial differential equation 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 natural phenomena. This is in contrast to experimental physics, which uses experim ...
. 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 Baron Siméon Denis Poisson FRS FRSE (; 21 June 1781 – 25 April 1840) was a French mathematician and physicist who worked on statistics, complex analysis, partial differential equations, the calculus of variations, analytical mechanics, electri ...
.

# Statement of the equation

Poisson's equation is $\Delta\varphi = f$ where $\Delta$ is the Laplace operator, and $f$ and $\varphi$ are
real Real may refer to: Currencies * Brazilian real (R\$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
or
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
-valued functions on a manifold. Usually, $f$ is given and $\varphi$ is sought. When the manifold is
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
, 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 is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
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. Poisson's equation may be solved using a
Green's function In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means that if \operatorname is the linear differenti ...
: $\varphi(\mathbf) = - \iiint \frac\, \mathrm^3\! r',$ 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, 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. A conservative vector field has the property that its line integral is path independent; the choice of any path between two points does not c ...
), it can be expressed in terms of a
scalar potential In mathematical physics, scalar potential, simply stated, describes the situation where the difference in the potential energies of an object in two different positions depends only on the positions, not upon the path taken by the object in trav ...
''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, a fundamental solution for a linear partial differential operator is a formulation in the language of distribution theory of the older idea of a Green's function (although unlike Green's functions, fundamental solutions do not a ...
). 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 attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distan ...
.

# Electrostatics

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 The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in ...
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'' (Machel Montano album) * ''Charge!!'', an album by The Aqu ...
distribution $\rho_f$. The mathematical details behind Poisson's equation in electrostatics are as follows ( SI units are used rather than
Gaussian units Gaussian units constitute a metric system of physical units. This system is the most common of the several electromagnetic unit systems based on cgs (centimetre–gram–second) units. It is also called the Gaussian unit system, Gaussian-cgs uni ...
, which are also frequently used in
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of ...
). Starting with
Gauss's law In physics and electromagnetism, Gauss's law, also known as Gauss's flux theorem, (or sometimes simply called Gauss's theorem) is a law relating the distribution of electric charge to the resulting electric field. In its integral form, it sta ...
for electricity (also one of
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. ...
) in differential form, one has $\mathbf \cdot \mathbf = \rho_f$ where $\mathbf \cdot$ is the divergence operator, D =
electric displacement field In physics, the electric displacement field (denoted by D) or electric induction is a vector field that appears in Maxwell's equations. It accounts for the effects of free and bound charge within materials. "D" stands for "displacement", as in ...
, and ''ρf'' =
free charge In classical electromagnetism, polarization density (or electric polarization, or simply polarization) is the vector field that expresses the density of permanent or induced electric dipole moments in a dielectric material. When a dielectric is ...
volume
density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
(describing charges brought from outside). Assuming the medium is linear, isotropic, and homogeneous (see
polarization density In classical electromagnetism, polarization density (or electric polarization, or simply polarization) is the vector field that expresses the density of permanent or induced electric dipole moments in a dielectric material. When a dielectric is ...
), we have the
constitutive equation In physics and engineering, a constitutive equation or constitutive relation is a relation between two physical quantities (especially kinetic quantities as related to kinematic quantities) that is specific to a material or substance, and app ...
, $\mathbf = \varepsilon \mathbf$ where is the
permittivity In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter ''ε'' (epsilon), is a measure of the electric polarizability of a dielectric. A material with high permittivity polarizes more in ...
of the medium and E is the 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 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 $\nabla \times \mathbf = 0,$ 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 The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in ...
), 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 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, $\nabla \cdot \mathbf = \nabla \cdot ( - \nabla \varphi ) = - ^2 \varphi = \frac,$ 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 results. If the charge density follows a
Boltzmann distribution In statistical mechanics and mathematics, a Boltzmann distribution (also called Gibbs distribution Translated by J.B. Sykes and M.J. Kearsley. See section 28) is a probability distribution or probability measure that gives the probability th ...
, 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 of gauge theories, gauge fixing (also called choosing a gauge) denotes a mathematical procedure for coping with redundant degrees of freedom in field variables. By definition, a gauge theory represents each physically distinct co ...
is used. In this more general context, computing is no longer sufficient to calculate E, since E also depends on the magnetic vector potential 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 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. 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 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, source reconstruction in acoustics, or calculating the ...
. The goal is to digitally reconstruct a smooth surface based on a large number of points ''pi'' (a
point cloud Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland * Poin ...
) where each point also carries an estimate of the local 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 of the form R(x_1, \dots, x_n) = 0, where is a function of several variables (often a polynomial). For example, the implicit equation of the unit circle is x^2 + y^2 - 1 = 0. An implicit func ...
''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 most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
field V. The implicit function ''f'' is found by integrating the vector field V. Since not every vector field is the
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
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 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 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 which each internal node has exactly eight children. Octrees are most often used to partition a three-dimensional space by recursively subdividing it into eight octants. Octrees are the three-dimensional ana ...
.

# Fluid dynamics

For the incompressible
Navier–Stokes equations In physics, the Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician Geo ...
, 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.

*
Discrete Poisson equation In mathematics, the discrete Poisson equation is the finite difference analog of the Poisson equation. In it, the discrete Laplace operator takes the place of the Laplace operator. The discrete Poisson equation is frequently used in numerical an ...
*
Poisson–Boltzmann equation The Poisson–Boltzmann equation is a useful equation in many settings, whether it be to understand physiological interfaces, polymer science, electron interactions in a semiconductor, or more. It aims to describe the distribution of the electric ...
* Helmholtz equation * Uniqueness theorem for Poisson's equation *
Weak formulation Weak formulations are important tools for the analysis of mathematical equations that permit the transfer of concepts of linear algebra to solve problems in other fields such as partial differential equations. In a weak formulation, equations or con ...