Screened Poisson equation

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In physics, the screened Poisson equation is a partial differential equation, which arises in (for example) Yukawa's theory of mesons and electric field screening in plasmas.

Statement of the equation[edit]


\left[ \Delta - \lambda^2 \right] u(\mathbf{r}) = - f(\mathbf{r})

Where \Delta 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.

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 n=3, is a superposition of 1/r functions weighted by the source function f:


u(\mathbf{r})_{(\text{Poisson})} = \iiint \mathrm{d}^3r' \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


\left[ \Delta - \lambda^2 \right] G(\mathbf{r}) = - \delta^3(\mathbf{r}).

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


G(\mathbf{k}) = \iiint \mathrm{d}^3r \; G(\mathbf{r}) e^{-i \mathbf{k} \cdot \mathbf{r}}

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


\left[ k^2 + \lambda^2 \right] G(\mathbf{k}) = 1.

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


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}.

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  k_r :


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}.

This may be evaluated using contour integration. The result is:


G(\mathbf{r}) = \frac{e^{- \lambda r}}{4\pi r}.

The solution to the full problem is then given by


u(\mathbf{r}) = \int \mathrm{d}^3r' G(\mathbf{r} - \mathbf{r}') f(\mathbf{r}')
= \int \mathrm{d}^3r' \frac{e^{- \lambda |\mathbf{r} - \mathbf{r}'|}}{4\pi |\mathbf{r} - \mathbf{r}'|} f(\mathbf{r}').

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:

 \left( \Delta_\perp -\frac{1}{\rho^2} \right)u(\mathbf{r}_\perp) = -f(\mathbf{r}_\perp)

with \Delta_\perp=\nabla\cdot\nabla_\perp and \nabla_\perp=\nabla-\frac{\mathbf{B}}{B}\cdot \nabla, with \mathbf{B} the magnetic field and \rho is the (ion) Larmor radius. The two-dimensional Fourier Transform of the associated Green's function is:

 G(\mathbf{k_\perp}) = \iint d^2 r~G(\mathbf{r}_\perp)e^{-i\mathbf{k}_\perp\cdot\mathbf{r}_\perp}.

The 2D screened Poisson equation yields:

 \left( k_\perp^2 +\frac{1}{\rho^2} \right)G(\mathbf{k}_\perp) = 1  .

The Green's function is therefore given by the inverse Fourier transform:


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}.

This integral can be calculated using polar coordinates in k-space:

 \mathbf{k}_\perp = (k_r\cos(\theta),k_r\sin(\theta))

The integration over the angular coordinate gives a Bessel function, and the integral reduces to one over the radial wavenumber  k_r :


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).

See also[edit]