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| {{Unreferenced|date=December 2009}}
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| In [[physics]], the '''screened Poisson equation''' is a [[partial differential equation]], which arises in (for example) [[Hideki Yukawa|Yukawa's]] theory of [[meson]]s and [[electric field screening]] in [[Plasma (physics)|plasma]]s.
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| ==Statement of the equation==
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| :<math>
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| \left[ \Delta - \lambda^2 \right] u(\mathbf{r}) = - f(\mathbf{r})
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| </math>
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| Where <math>\Delta</math> is the [[Laplace operator]], ''λ'' is a constant, ''f'' is an arbitrary function of position (known as the "source function") and ''u'' is the function to be determined.
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| In the homogenous 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.
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| ==Solutions==
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| ===Three dimensions===
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| 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'':
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| :<math> | |
| u(\mathbf{r})_{(\text{Poisson})} = \iiint \mathrm{d}^3r' \frac{f(\mathbf{r}')}{4\pi |\mathbf{r} - \mathbf{r}'|}.
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| </math>
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| 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.
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| 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
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| :<math> | |
| \left[ \Delta - \lambda^2 \right] G(\mathbf{r}) = - \delta^3(\mathbf{r}).
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| </math>
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| Assuming ''u'' and its derivatives vanish at large ''r'', we may perform a [[continuous Fourier transform]] in spatial coordinates:
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| :<math>
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| G(\mathbf{k}) = \iiint \mathrm{d}^3r \; G(\mathbf{r}) e^{-i \mathbf{k} \cdot \mathbf{r}}
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| </math>
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| where the integral is taken over all space. It is then straightforward to show that
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| :<math>
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| \left[ k^2 + \lambda^2 \right] G(\mathbf{k}) = 1.
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| </math>
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| The Green's function in ''r'' is therefore given by the inverse Fourier transform,
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| :<math>
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| G(\mathbf{r}) = \frac{1}{(2\pi)^3} \; \iiint \mathrm{d}^3\!k \; \frac{e^{i \mathbf{k} \cdot \mathbf{r}}}{k^2 + \lambda^2}.
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| </math>
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| 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>:
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| :<math>
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| 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}.
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| </math>
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| This may be evaluated using [[line integral|contour integration]]. The result is:
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| :<math>
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| G(\mathbf{r}) = \frac{e^{- \lambda r}}{4\pi r}.
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| </math>
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| The solution to the full problem is then given by
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| :<math>
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| u(\mathbf{r}) = \int \mathrm{d}^3r' G(\mathbf{r} - \mathbf{r}') f(\mathbf{r}')
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| = \int \mathrm{d}^3r' \frac{e^{- \lambda |\mathbf{r} - \mathbf{r}'|}}{4\pi |\mathbf{r} - \mathbf{r}'|} f(\mathbf{r}').
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| </math>
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| 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]]".
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| ===Two dimensions===
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| In two dimensions:
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| In the case of a magnetized plasma, the [[screened Poisson equation]] is quasi-2D:
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| : <math> \left( \Delta_\perp -\frac{1}{\rho^2} \right)u(\mathbf{r}_\perp) = -f(\mathbf{r}_\perp) </math>
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| 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]].
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| The two-dimensional [[Fourier Transform]] of the associated [[Green's function]] is:
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| : <math> G(\mathbf{k_\perp}) = \iint d^2 r~G(\mathbf{r}_\perp)e^{-i\mathbf{k}_\perp\cdot\mathbf{r}_\perp}. </math>
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| The 2D [[screened Poisson equation]] yields:
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| : <math> \left( k_\perp^2 +\frac{1}{\rho^2} \right)G(\mathbf{k}_\perp) = 1 </math>.
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| The [[Green's function]] is therefore given by the [[inverse Fourier transform]]:
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| :<math>
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| G(\mathbf{r}_\perp) = \frac{1}{4\pi^2} \; \iint \mathrm{d}^2\!k \; \frac{e^{i \mathbf{k}_\perp \cdot \mathbf{r}_\perp}}{k_\perp^2 + 1 / \rho^2}.
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| </math>
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| This integral can be calculated using [[polar coordinates]] in [[k-space]]:
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| : <math> \mathbf{k}_\perp = (k_r\cos(\theta),k_r\sin(\theta)) </math>
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| The integration over the angular coordinate gives a [[Bessel function]], and the integral reduces to one over the radial [[wavenumber]] <math> k_r </math>:
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| :<math>
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| 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).
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| </math>
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| ==See also==
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| * [[Yukawa interaction]]
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| {{DEFAULTSORT:Screened Poisson Equation}}
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| [[Category:Partial differential equations]]
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| [[Category:Plasma physics]]
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| [[Category:Electrostatics]]
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