Poisson's equation

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 the corresponding electrostatic or gravitational 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 who published it in 1823.

Statement of the equation

Poisson's equation is
where is the Laplace operator, and and are real or complex-valued functions on a manifold. Usually, is given, and is sought. When the manifold is Euclidean space, the Laplace operator is often denoted as, and so Poisson's equation is frequently written as
In three-dimensional Cartesian coordinates, it takes the form
When identically, we obtain Laplace's equation.
Poisson's equation may be solved using a Green's function:
where the integral is over all of space. Note here is where we observe the field, where we are solving for. And integrating is equivalent to integrating all the "sources". In a sense that we integrate all the effects of sources at to get what's seen at . 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.

Applications in physics and engineering

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. Gauss's law for gravity is
Since the gravitational field is conservative, it can be expressed in terms of a scalar potential ϕ:
Substituting this into Gauss's law,
yields Poisson's equation for gravity:
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 . In three dimensions the potential is
which is equivalent to Newton's law of universal gravitation.

Electrostatics

Many problems in electrostatics are governed by the Poisson equation, which relates the electric potential
to the free charge density
, such as those found in conductors.
The mathematical details of Poisson's equation, commonly expressed in SI units, describe how the distribution of free charges generates the electrostatic potential in a given region.
Starting with Gauss's law for electricity in differential form, one has
where is the divergence operator, D is the electric displacement field, and ρ_f is the free-charge density.
Assuming the medium is linear, isotropic, and homogeneous, we have the constitutive equation
where is the permittivity of the medium, and E is the electric field.
Substituting this into Gauss's law and assuming that is spatially constant in the region of interest yields
In electrostatics, we assume that there is no magnetic field.
Then, we have that
where is the curl operator. This equation means that we can write the electric field as the gradient of a scalar function, since the curl of any gradient is zero. Thus we can write
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,
directly produces Poisson's equation for electrostatics, which is
Specifying the 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.
Using a Green's function, the potential at distance from a central point charge is
which is Coulomb's law of electrostatics.
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 class of cases, computing is no longer sufficient to calculate E, since E also depends on the vector potential">Euclidean vector">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 an appropriate Poisson's equation is obtained in this case.

Potential of a Gaussian charge density

If there is a static spherically symmetric Gaussian charge density
where is the total charge, then the solution of Poisson's equation
is given by
where is the error function. This solution can be checked explicitly by evaluating.
Note that for much greater than, approaches unity, and the potential approaches the point-charge potential,
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. The goal is to digitally reconstruct a smooth surface based on a large number of points p_i 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 f whose value is zero at the points p_i and whose gradient at the points p_i equals the normal vectors n_i. The set of is thus modeled as 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.
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_i onto the nodes of the particular staggered grid cell containing p_i. 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 where there are more data points. They suggest implementing this technique with an adaptive octree.

Fluid dynamics

For the incompressible Navier–Stokes equations, given by
The equation for the pressure field is an example of a nonlinear Poisson equation:
Notice that the above trace is not sign-definite.

Thermodynamics

Thermal conduction is modelled via the Heat equation. Stationary state heat conduction with a source term is modelled via the following Poisson equation:
where is the temperature, is the heat source term and is Thermal conductivity.