Screened Poisson equation

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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[edit]

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[edit]

Three dimensions[edit]

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

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[edit]

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[edit]

References[edit]

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