Poisson's equation explained

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.^[1]

Statement of the equation

Poisson's equation is$$\backslash Delta\backslash varphi\; =\; f,$$where

\Delta is the Laplace operator, and f and \varphi are real or complex-valued functions on a manifold. Usually, f is given, and \varphi is sought. When the manifold is Euclidean space, the Laplace operator is often denoted as, and so Poisson's equation is frequently written as$$\backslash nabla^2\; \backslash varphi\; =\; f.$$

In three-dimensional Cartesian coordinates, it takes the form$$\backslash left(\backslash frac\; +\; \backslash frac\; +\; \backslash frac\; \backslash right)\backslash varphi(x,\; y,\; z)\; =\; f(x,\; y,\; z).$$

When

f = 0 identically, we obtain Laplace's equation.

Poisson's equation may be solved using a Green's function:$$\backslash varphi(\backslash mathbf)\; =\; -\; \backslash iiint\; \backslash frac\backslash ,\; \backslash mathrm^3\; r\text{'},$$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

See main article: Gravitational field and 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:$$\backslash nabla\backslash cdot\backslash mathbf\; =\; -4\backslash pi\; G\backslash rho.$$

Since the gravitational field is conservative (and irrotational), it can be expressed in terms of a scalar potential ϕ:$$\backslash mathbf\; =\; -\backslash nabla\; \backslash phi.$$

Substituting this into Gauss's law,$$\backslash nabla\backslash cdot(-\backslash nabla\; \backslash phi)\; =\; -\; 4\backslash pi\; G\; \backslash rho,$$yields Poisson's equation for gravity:$$\backslash nabla^2\; \backslash phi\; =\; 4\backslash pi\; G\; \backslash rho.$$

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 three dimensions the potential is$$\backslash phi(r)\; =\; \backslash frac,$$which is equivalent to Newton's law of universal gravitation.

Electrostatics

See main article: 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 for a given charge distribution

\rho _{f} .

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$$\backslash mathbf\; \backslash cdot\; \backslash mathbf\; =\; \backslash rho\_f,$$where

\nabla ⋅ is the divergence operator, D is the electric displacement field, and ρ_f 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$$\backslash mathbf\; =\; \backslash varepsilon\; \backslash mathbf,$$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$$\backslash mathbf\; \backslash cdot\; \backslash mathbf\; =\; \backslash frac.$$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$$\backslash nabla\; \backslash times\; \backslash mathbf\; =\; 0,$$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$$\backslash mathbf\; =\; -\backslash nabla\; \backslash varphi,$$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,$$\backslash nabla\; \backslash cdot\; \backslash mathbf\; =\; \backslash nabla\; \backslash cdot\; (-\backslash nabla\; \backslash varphi)\; =\; -\backslash nabla^2\; \backslash varphi\; =\; \backslash frac,$$directly produces Poisson's equation for electrostatics, which is$$\backslash nabla^2\; \backslash varphi\; =\; -\backslash frac.$$

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, 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$$\backslash varphi(r)\; =\; \backslash frac,$$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 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$$\backslash rho\_f(r)\; =\; \backslash frac\backslash ,e^,$$where is the total charge, then the solution of Poisson's equation$$\backslash nabla^2\; \backslash varphi\; =\; -\backslash frac$$is given by$$\backslash varphi(r)\; =\; \backslash frac\; \backslash frac\; \backslash operatorname\backslash left(\backslash frac\backslash right),$$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,$$\backslash varphi\; \backslash approx\; \backslash frac\; \backslash frac,$$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. The goal is to digitally reconstruct a smooth surface 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.^[2] Poisson's equation can be utilized to solve this problem with a technique called Poisson surface reconstruction.^[3]

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 (p_i, n_i) 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 (the grid is more finely divided) where there are more data points.^[3] 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

p is an example of a nonlinear Poisson equation:Notice that the above trace is not sign-definite.

See also

• Discrete Poisson equation
• Poisson–Boltzmann equation
• Helmholtz equation
• Uniqueness theorem for Poisson's equation
• Weak formulation
• Harmonic function
• Heat equation
• Potential theory

Further reading

• Book: Evans, Lawrence C. . Partial Differential Equations . American Mathematical Society . Providence (RI) . 1998 . 0-8218-0772-2 .
• Book: Mathews . Jon . Walker . Robert L. . 1970 . Mathematical Methods of Physics . 2nd . New York . W. A. Benjamin . 0-8053-7002-1 .
• Book: Polyanin, Andrei D. . Handbook of Linear Partial Differential Equations for Engineers and Scientists . Chapman & Hall/CRC Press . Boca Raton (FL) . 2002 . 1-58488-299-9 .

External links

• • Poisson Equation at EqWorld: The World of Mathematical Equations

Notes and References

1. Poisson . 1823 . Mémoire sur la théorie du magnétisme en mouvement . Memoir on the theory of magnetism in motion . Mémoires de l'Académie Royale des Sciences de l'Institut de France . 6 . 441–570 . fr . From p. 463: French: "Donc, d'après ce qui précède, nous aurons enfin:$$\backslash frac\; +\; \backslash frac\; +\; \backslash frac\; =\; 0,\; =\; -2k\backslash pi,\; =\; -4k\backslash pi,$$French: 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:$$\backslash frac\; +\; \backslash frac\; +\; \backslash frac\; =\; 0,\; =\; -2k\backslash pi,\; =\; -4k\backslash pi,$$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. 462) as$$V\; =\; \backslash iiint\backslash frac\backslash ,\; dx\text{'}\backslash ,dy\text{'}\backslash ,dz\text{'},$$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 (x', y', z') , k' is a given function of (x', y,' z') and in electrostatics, k' would be a measure of charge density, and \rho 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 (x, y, z) and k denotes the value of k' (the charge density) at M.
2. Fatih . Calakli . Gabriel . Taubin . Smooth Signed Distance Surface Reconstruction . Pacific Graphics . 2011 . 30 . 7 .
3. Book: Michael . Kazhdan . Matthew . Bolitho . Hugues . Hoppe . 2006 . Poisson surface reconstruction . Proceedings of the fourth Eurographics symposium on Geometry processing (SGP '06) . Eurographics Association, Aire-la-Ville, Switzerland . 61–70 . 3-905673-36-3 . https://dl.acm.org/doi/abs/10.5555/1281957.1281965 .