Potential theory

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a> and <a href="/facts/Mathematical_physics/sLKg8T1G">mathematical physics</a>, potential theory is the study of <a href="/facts/Harmonic_function/H6pg4dMT">harmonic functions</a>.
The term "potential theory" was coined in 19th-century <a href="/facts/Physics/a7aH2y08">physics</a> when it was realized that the two fundamental <a href="/facts/Force/533Q1uzq">forces</a> of nature known at the time, namely gravity and the electrostatic force, could be modeled using functions called the <a href="/facts/Gravitational_potential/x4ITT4gY">gravitational potential</a> and <a href="/facts/Electrostatic_potential/Nmqm6hm5">electrostatic potential</a>, both of which satisfy <a href="/facts/Poisson%2527s_equation/Pr8zkCXB">Poisson's equation</a>—or in the vacuum, <a href="/facts/Laplace%2527s_equation/8n1YkmFS">Laplace's equation</a>.
There is considerable overlap between potential theory and the theory of Poisson's equation to the extent that it is impossible to draw a distinction between these two fields. The difference is more one of emphasis than subject matter and rests on the following distinction: potential theory focuses on the properties of the functions as opposed to the properties of the equation. For example, a result about the <a href="/facts/Mathematical_singularity/Y5sMQY4I">singularities</a> of harmonic functions would be said to belong to potential theory whilst a result on how the solution depends on the boundary data would be said to belong to the theory of Poisson's equation. This is not a hard and fast distinction, and in practice there is considerable overlap between the two fields, with methods and results from one being used in the other.
Modern potential theory is also intimately connected with probability and the theory of <a href="/facts/Markov_chain/5oP5hntk">Markov chains</a>. In the continuous case, this is closely related to analytic theory. In the finite state space case, this connection can be introduced by introducing an <a href="/facts/Electrical_network/zt94kTW0">electrical network</a> on the state space, with resistance between points inversely proportional to transition probabilities and densities proportional to potentials. Even in the finite case, the analogue I-K of the Laplacian in potential theory has its own maximum principle, uniqueness principle, balance principle, and others.

Potential theory open-in-new

Potential theory