### Poisson equation

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 French mathematician, geometer and physicist Siméon Denis Poisson.**
**

## Contents

## Statement of the equation

Poisson's equation is

- $\backslash Delta\backslash varphi=f$

where $\backslash Delta$ is the Laplace operator, and *f* and *φ* are real or complex-valued functions on a manifold. When the manifold is Euclidean space, the Laplace operator is often denoted as ∇^{2} and so Poisson's equation is frequently written as

- $\backslash nabla^2\; \backslash varphi\; =\; f.$

In three-dimensional Cartesian coordinates, it takes the form

- $$

\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).

For *f* = 0, the equation reduces to Laplace's equation.

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.

## Newtonian gravity

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:

- $\backslash nabla\backslash cdot\backslash bold\{g\}\; =\; -4\backslash pi\; G\backslash rho$,

and since the gravitational field is conservative, it can be expressed in terms of a scalar potential *Φ*:

- $\backslash bold\{g\}\; =\; -\backslash nabla\; \backslash Phi$,

substituting into Gauss' law

- $\backslash nabla\backslash cdot(-\backslash nabla\; \backslash Phi)\; =\; -\; 4\backslash pi\; G\; \backslash rho$

obtains **Poisson's equation** for gravity:

- $\{\backslash nabla\}^2\; \backslash Phi\; =\; 4\backslash pi\; G\; \backslash rho.$

## Electrostatics

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 charge distribution described by the density function.

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' law for electricity (also one of Maxwell's equations) in differential form, we have:

- $\backslash mathbf\{\backslash nabla\}\; \backslash cdot\; \backslash mathbf\{D\}\; =\; \backslash rho\_f$

where $\backslash mathbf\{\backslash nabla\}\; \backslash cdot$ is the divergence operator, **D** = electric displacement field, and *ρ _{f}* = free charge density (describing charges brought from outside). Assuming the medium is linear, isotropic, and homogeneous (see polarization density), we have the constitutive equation:

- $\backslash mathbf\{D\}\; =\; \backslash varepsilon\; \backslash mathbf\{E\}$

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:

- $\backslash mathbf\{\backslash nabla\}\; \backslash cdot\; \backslash mathbf\{E\}\; =\; \backslash frac\{\backslash rho\_f\}\{\backslash varepsilon\}$

In the absence of a changing magnetic field, **B**, Faraday's law of induction gives:

- $\backslash nabla\; \backslash times\; \backslash mathbf\{E\}\; =\; -\backslash dfrac\{\backslash partial\; \backslash mathbf\{B\}\}\; \{\backslash partial\; t\}\; =\; 0$

where $\backslash nabla\; \backslash times$ is the curl operator and *t* is time. Since the curl of the electric field is zero, it is defined by a scalar electric potential field, $\backslash varphi$ (see Helmholtz decomposition).

- $\backslash mathbf\{E\}\; =\; -\backslash nabla\; \backslash varphi$

The derivation of Poisson's equation under these circumstances is straightforward. Substituting the potential gradient for the electric field

- $\backslash nabla\; \backslash cdot\; \backslash bold\{E\}\; =\; \backslash nabla\; \backslash cdot\; (\; -\; \backslash nabla\; \backslash varphi\; )\; =\; -\; \{\backslash nabla\}^2\; \backslash varphi\; =\; \backslash frac\{\backslash rho\_f\}\{\backslash varepsilon\},$

directly obtains **Poisson's equation** for electrostatics, which is:

- $\{\backslash nabla\}^2\; \backslash varphi\; =\; -\backslash frac\{\backslash rho\_f\}\{\backslash varepsilon\}.$

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 theory of dilute electrolyte solutions.

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 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.

### Potential of a Gaussian charge density

If there is a static spherically symmetric Gaussian charge density

- $\backslash rho\_f(r)\; =\; \backslash frac\{Q\}\{\backslash sigma^3\backslash sqrt\{2\backslash pi\}^3\}\backslash ,e^\{-r^2/(2\backslash sigma^2)\},$

where *Q* is the total charge, then the solution *φ*(*r*) of Poisson's equation,

- $\{\backslash nabla\}^2\; \backslash varphi\; =\; -\; \{\; \backslash rho\_f\; \backslash over\; \backslash varepsilon\; \}$,

is given by

- $\backslash varphi(r)\; =\; \{\; 1\; \backslash over\; 4\; \backslash pi\; \backslash varepsilon\; \}\; \backslash frac\{Q\}\{r\}\backslash ,\backslash mbox\{erf\}\backslash left(\backslash frac\{r\}\{\backslash sqrt\{2\}\backslash sigma\}\backslash right)$

where erf(*x*) is the error function.

This solution can be checked explicitly by evaluating $\{\backslash nabla\}^2\; \backslash varphi$. Note that, for *r* much greater than *σ*, the erf function approaches unity and the potential φ (*r*) approaches the point charge potential

- $\backslash varphi\; \backslash approx\; \{\; 1\; \backslash over\; 4\; \backslash pi\; \backslash varepsilon\; \}\; \{Q\; \backslash over\; r\}$,

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.

## Surface Reconstruction

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 _{i}* (a point cloud) where each point also carries an estimate of the local surface normal

**n**

_{i}.

^{[1]}

This technique reconstructs the implicit function *f* whose value is zero at the points *p _{i}* and whose gradient at the points

*p*equals the normal vectors

_{i}**n**

_{i}. The set of (

*p*,

_{i}**n**

_{i}) is thus a sampling of a continuous vector ﬁeld

**V**. The implicit function

*f*is found by 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 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*.

## See also

## References

- Poisson Equation at EqWorld: The World of Mathematical Equations.
- L.C. Evans,
*Partial Differential Equations*, American Mathematical Society, Providence, 1998. ISBN 0-8218-0772-2 - 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

## External links

- Template:Springer
- PlanetMath.
- Poisson's Equation Poisson's Equation video