Editing Screened Poisson equation

{{Refimprove|date=July 2017}}
In [[physics]], the '''screened Poisson equation''' is a [[Poisson equation]], which arises in (for example) the [[Klein–Gordon equation]], [[electric field screening]] in [[Plasma (physics)|plasma]]s, and nonlocal granular fluidity<ref>{{cite journal| last1=Kamrin | first1=Ken| last2=Koval| first2=Georg | title=Nonlocal Constitutive Relation for Steady Granular Flow| journal=Physical Review Letters| date=26 April 2012| volume=108| issue=17| doi=10.1103/PhysRevLett.108.178301| bibcode = 2012PhRvL.108q8301K| pmid=22680912| page=178301| url=https://dspace.mit.edu/bitstream/1721.1/71598/1/Kamrin-2012-Nonlocal%20Constitutive%20Relation%20for.pdf| hdl=1721.1/71598|hdl-access=free}}</ref> in [[granular flow]].

==Statement of the equation==

The equation is
<math display="block">
\left[ \Delta - \lambda^2 \right] u(\mathbf{r}) = - f(\mathbf{r}),
</math>

where <math>\Delta</math> 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 [[Helmholtz equation#Inhomogeneous Helmholtz equation|inhomogeneous Helmholtz equation]], the only difference being the sign within the brackets.

=== Electrostatics ===
In [[electric-field screening]], screened Poisson equation for the electric potential <math>\phi(\mathbf r)</math> is usually written as (SI units)

<math display="block">
\left[ \Delta - k_0^2 \right] \phi(\mathbf{r}) = - \frac{\rho_{\rm ext}(\mathbf{r})}{\epsilon_0},
</math>

where <math>k_0^{-1}</math> is the screening length, <math>\rho_{\rm ext}(\mathbf r)</math> is the charge density produced by an external field in the absence of screening and <math>\epsilon_0</math> is the [[vacuum permittivity]]. This equation can be derived in several screening models like [[Thomas–Fermi screening]] in [[solid-state physics]] and [[Debye length|Debye screening]] in [[Plasma (physics)|plasmas]].

==Solutions==

===Three dimensions===

Without loss of generality, we will take ''λ'' to be non-negative. When ''λ'' is [[0 (number)|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 <math>n=3</math>, is a superposition of 1/''r'' functions weighted by the source function ''f'':

<math display="block">
u(\mathbf{r})_{(\text{Poisson})} = \iiint \mathrm{d}^3\mathbf{r}' \frac{f(\mathbf{r}')}{4\pi |\mathbf{r} - \mathbf{r}'|}.
</math>

On the other hand, when ''λ'' is extremely large, ''u'' approaches the value ''f''/''λ''<sup>2</sup>, 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 function]]s. The Green's function ''G'' is defined by

<math display="block">
\left[ \Delta - \lambda^2 \right] G(\mathbf{r}) = - \delta^3(\mathbf{r}),
</math>
where δ<sup>3</sup> is a [[Dirac delta function|delta function]] with unit mass concentrated at the origin of '''R'''<sup>3</sup>.

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

<math display="block">
\tilde{G}(\mathbf{k}) = \iiint \mathrm{d}^3\mathbf{r} \; G(\mathbf{r}) e^{-i \mathbf{k} \cdot \mathbf{r}}
</math>

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

<math display="block">
\left[ k^2 + \lambda^2 \right] \tilde{G}(\mathbf{k}) = 1.
</math>

The Green's function in ''r'' is therefore given by the inverse Fourier transform,

<math display="block">
G(\mathbf{r}) = \frac{1}{(2\pi)^3} \; \iiint \mathrm{d}^3\!\mathbf{k} \; \frac{e^{i \mathbf{k} \cdot \mathbf{r}}}{k^2 + \lambda^2}.
</math>

This integral may be evaluated using [[Spherical coordinate system|spherical coordinates]] in ''k''-space. The integration over the angular coordinates is straightforward, and the integral reduces to one over the radial [[wavenumber]] <math> k_r </math>:

<math display="block">
G(\mathbf{r}) = \frac{1}{2\pi^2 r} \; \int_0^\infty \mathrm{d}k_r \; \frac{k_r \, \sin k_r r }{k_r^2 + \lambda^2}.
</math>

This may be evaluated using [[line integral|contour integration]]. The result is:

<math display="block">
G(\mathbf{r}) = \frac{e^{- \lambda r}}{4\pi r}.
</math>

The solution to the full problem is then given by

<math display="block">
u(\mathbf{r}) = \int \mathrm{d}^3\mathbf{r}' G(\mathbf{r} - \mathbf{r}') f(\mathbf{r}')
= \int \mathrm{d}^3\mathbf{r}' \frac{e^{- \lambda |\mathbf{r} - \mathbf{r}'|}}{4\pi |\mathbf{r} - \mathbf{r}'|} f(\mathbf{r}').
</math>

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:
<math display="block"> \left( \Delta_\perp - \frac{1}{\rho^2} \right)u(\mathbf{r}_\perp) = -f(\mathbf{r}_\perp) </math>
with <math>\Delta_\perp=\nabla\cdot\nabla_\perp</math> and <math>\nabla_\perp=\nabla-\frac{\mathbf{B}}{B}\cdot \nabla</math>, with <math>\mathbf{B}</math> the magnetic field and <math>\rho</math> is the (ion) [[Larmor radius]].
The two-dimensional [[Fourier Transform]] of the associated [[Green's function]] is:
<math display="block"> \tilde{G}(\mathbf{k_\perp}) = \iint d^2 \mathbf{r} ~G(\mathbf{r}_\perp) e^{-i\mathbf{k}_\perp\cdot\mathbf{r}_\perp}. </math>
The 2D screened Poisson equation yields:
<math display="block"> \left( k_\perp^2 +\frac{1}{\rho^2} \right) \tilde{G}(\mathbf{k}_\perp) = 1 . </math>
The [[Green's function]] is therefore given by the [[inverse Fourier transform]]:
<math display="block">
G(\mathbf{r}_\perp) = \frac{1}{4\pi^2} \; \iint \mathrm{d}^2\!\mathbf{k} \; \frac{e^{i \mathbf{k}_\perp \cdot \mathbf{r}_\perp}}{k_\perp^2 + 1 / \rho^2}.
</math>
This integral can be calculated using [[polar coordinates]] in [[frequency domain|k-space]]:
<math display="block"> \mathbf{k}_\perp = (k_r\cos(\theta), k_r\sin(\theta)) </math>
The integration over the angular coordinate gives a [[Bessel function]], and the integral reduces to one over the radial [[wavenumber]] <math> k_r </math>:
<math display="block">
G(\mathbf{r}_\perp) = \frac{1}{2\pi} \; \int_0^\infty \mathrm{d}k_r \; \frac{k_r \, J_0(k_r r_\perp)}{k_r^2 + 1 / \rho^2} = \frac{1}{2\pi} K_0(r_\perp \, / \, \rho).
</math>

==Connection to the Laplace distribution==
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.

==Application in differential geometry==
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 dimension, Ricci curvature, and screening parameter are interconnected via a quadratic relationship.<ref>{{cite journal |last1=Pigazzini |first1=Alexander |last2=Lussardi |first2=Luca |last3=Toda |first3=Magdalena |author3-link=Magdalena Toda |last4=DeBenedictis |first4=Andrew |date=29 July 2024 |title=Einstein warped-product manifolds and the screened Poisson equation |url=https://bookstore.ams.org/view?ProductCode=CONM/821 |journal=Contemporary Mathematics series of the American Mathematical Society (AMS) - Book entitled: "Recent Advances in Differential Geometry and Related Areas" (2025)}}</ref>

==See also==
* [[Yukawa interaction]]

==References==
{{Reflist}}

{{DEFAULTSORT:Screened Poisson Equation}}
[[Category:Partial differential equations]]
[[Category:Plasma physics equations]]
[[Category:Electrostatics]]