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 in granular flow.
Statement of the equation The equation is
[ Δ − λ 2 ] u ( r ) = − f ( r ) , {\displaystyle \left[\Delta -\lambda ^{2}\right]u(\mathbf {r} )=-f(\mathbf {r} ),}
where Δ {\displaystyle \Delta } 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.
Electrostatics In electric-field screening, screened Poisson equation for the electric potential ϕ ( r ) {\displaystyle \phi (\mathbf {r} )} is usually written as (SI units)
[ Δ − k 0 2 ] ϕ ( r ) = − ρ e x t ( r ) ϵ 0 , {\displaystyle \left[\Delta -k_{0}^{2}\right]\phi (\mathbf {r} )=-{\frac {\rho _{\rm {ext}}(\mathbf {r} )}{\epsilon _{0}}},}
where k 0 − 1 {\displaystyle k_{0}^{-1}} is the screening length, ρ e x t ( r ) {\displaystyle \rho _{\rm {ext}}(\mathbf {r} )} is the charge density produced by an external field in the absence of screening and ϵ 0 {\displaystyle \epsilon _{0}} is the vacuum permittivity. This equation can be derived in several screening models like Thomas–Fermi screening in solid-state physics and Debye screening in plasmas.
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 n = 3 {\displaystyle n=3} , is a superposition of 1/r functions weighted by the source function f:
u ( r ) ( Poisson ) = ∭ d 3 r ′ f ( r ′ ) 4 π | r − r ′ | . {\displaystyle u(\mathbf {r} )_{({\text{Poisson}})}=\iiint \mathrm {d} ^{3}\mathbf {r} '{\frac {f(\mathbf {r} ')}{4\pi |\mathbf {r} -\mathbf {r} '|}}.}
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
[ Δ − λ 2 ] G ( r ) = − δ 3 ( r ) , {\displaystyle \left[\Delta -\lambda ^{2}\right]G(\mathbf {r} )=-\delta ^{3}(\mathbf {r} ),}
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:
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