# Screened Poisson equation

In physics, the **screened Poisson equation** is a partial differential equation, which arises in (for example) the Klein–Gordon equation, electric field screening in plasmas, and nonlocal granular fluidity[1] in granular flow.

## Statement of the equation

The equation is

where 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 homogeneous case (f=0), the screened Poisson equation is the same as the time-independent Klein–Gordon equation. 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.

## Solutions

### Three dimensions

Without loss of generality, we will take *λ* to be non-negative. When *λ* is zero, the equation reduces to Poisson's equation. Therefore, when *λ* is very small, the solution approaches that of the unscreened Poisson equation, which, in dimension , is a superposition of 1/*r* functions weighted by the source function *f*:

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 δ^{3} is a delta function with unit mass concentrated at the origin of **R**^{3}.

Assuming *u* and its derivatives vanish at large *r*, we may perform a continuous Fourier transform in spatial coordinates:

where the integral is taken over all space. It is then straightforward to show that

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 result is:

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".

### Two dimensions

In two dimensions: In the case of a magnetized plasma, the screened Poisson equation is quasi-2D:

with and , with the magnetic field and 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 :

## See also

## References

- Kamrin, Ken; Koval, Georg (26 April 2012). "Nonlocal Constitutive Relation for Steady Granular Flow" (PDF).
*Physical Review Letters*.**108**(17): 178301. Bibcode:2012PhRvL.108q8301K. doi:10.1103/PhysRevLett.108.178301. PMID 22680912.