In physics, the screened Poisson equation is a Poisson equation, which arises in (for example) the Klein–Gordon equation, electric field screening in plasmas, and nonlocal granular fluidity[1] in granular flow.
is the Laplace operator, λ is a constant that expresses the "screening", f is an arbitrary function of position (known as the "source function") and u is the function to be determined.
In the inhomogeneous case, the screened Poisson equation is very similar to the inhomogeneous Helmholtz equation, the only difference being the sign within the brackets.
is usually written as (SI units)
is the charge density produced by an external field in the absence of screening and
Without loss of generality, we will take λ to be non-negative.
Therefore, when λ is very small, the solution approaches that of the unscreened Poisson equation, which, in dimension
On the other hand, when λ is extremely large, u approaches the value f/λ2, which goes to zero as λ goes to infinity.
As we shall see, the solution for intermediate values of λ behaves as a superposition of screened (or damped) 1/r functions, with λ behaving as the strength of the screening.
The screened Poisson equation can be solved for general f using the method of Green's functions.
The Green's function G is defined by
where δ3 is a delta function with unit mass concentrated at the origin of R3.
Assuming u and its derivatives vanish at large r, we may perform a continuous Fourier transform in spatial coordinates:
The Green's function in r is therefore given by the inverse Fourier transform,
This integral may be evaluated using spherical coordinates in k-space.
The integration over the angular coordinates is straightforward, and the integral reduces to one over the radial wavenumber
This may be evaluated using contour integration.
The solution to the full problem is then given by
As stated above, this is a superposition of screened 1/r functions, weighted by the source function f and with λ acting as the strength of the screening.
The screened 1/r function is often encountered in physics as a screened Coulomb potential, also called a "Yukawa potential".
In two dimensions: In the case of a magnetized plasma, the screened Poisson equation is quasi-2D:
is the (ion) Larmor radius.
The two-dimensional Fourier Transform of the associated Green's function is:
The 2D screened Poisson equation yields:
The Green's function is therefore given by the inverse Fourier transform:
This integral can be calculated using polar coordinates in k-space:
The integration over the angular coordinate gives a Bessel function, and the integral reduces to one over the radial wavenumber
The Green's functions in both 2D and 3D are identical to the probability density function of the multivariate Laplace distribution for two and three dimensions respectively.
The homogeneous case, studied in the context of differential geometry, involving Einstein warped product manifolds, explores cases where the warped function satisfies the homogeneous version of the screened Poisson equation.
Under specific conditions, the manifold size, Ricci curvature, and screening parameter are interconnected via a quadratic relationship.