Shielded Poisson Equation

In mathematics, the screened Poisson equation is a partial differential equation of the form:

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

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

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

Where ${\nabla }^{2}$ ${\ displaystyle {\ nabla} ^ {2}}$ ${\ displaystyle {\ nabla} ^ {2}}$ - Laplace operator , $\lambda$ ${\ displaystyle \ lambda}$ $\ lambda$ Is a constant $f$ ${\ displaystyle f}$ $f$ Is an arbitrary position function (known as a “source function”), and $u$ ${\ displaystyle u}$ $u$ Is the desired function. The screened Poisson equation is often used in physics , including the Yukawa theory of meson screening and screening of an electric field in a plasma .

When $\lambda$ ${\ displaystyle \ lambda}$ $\ lambda$ equal to zero, the equation turns into a Poisson equation . Therefore when $\lambda$ ${\ displaystyle \ lambda}$ $\ lambda$ very small, the solution approaches the solution of the unshielded Poisson equation, which is a superposition $1/r$ ${\ displaystyle 1 / r}$ ${\ displaystyle 1 / r}$ functions, statistically weighted source function $f$ ${\ displaystyle f}$ $f$ :

u(\mathbf {r} )_{(Poisson)}=\int d^{3}r'{\frac {f(\mathbf {r} ')}{4\pi |\mathbf {r} -\mathbf {r} '|}}.

{\ displaystyle u (\ mathbf {r}) _ {(Poisson)} = \ int d ^ {3} r '{\ frac {f (\ mathbf {r}')} {4 \ pi | \ mathbf {r } - \ mathbf {r} '|}}.}

{\ displaystyle u (\ mathbf {r}) _ {(Poisson)} = \ int d ^ {3} r '{\ frac {f (\ mathbf {r}')} {4 \ pi | \ mathbf {r } - \ mathbf {r} '|}}.}

On the other hand, when $\lambda$ ${\ displaystyle \ lambda}$ $\ lambda$ very big $u$ ${\ displaystyle u}$ $u$ approaching the value $f/\lambda ^{2}$ ${\ displaystyle f / \ lambda ^ {2}}$ ${\ displaystyle f / \ lambda ^ {2}}$ , which in turn approaches zero when $\lambda$ ${\ displaystyle \ lambda}$ $\ lambda$ goes to infinity. As we will see, the solution for averages $\lambda$ ${\ displaystyle \ lambda}$ $\ lambda$ behaves like a superposition of shielded (or fading) $1/r$ ${\ displaystyle 1 / r}$ ${\ displaystyle 1 / r}$ functions, and $\lambda$ ${\ displaystyle \ lambda}$ $\ lambda$ will be the power of shielding.

The screened Poisson equation can be solved for the general $f$ ${\ displaystyle f}$ $f$ using the Green function . Green function $G$ ${\ displaystyle G}$ $G$ defined as

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

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

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

Assuming that $u$ ${\ displaystyle u}$ $u$ and its derivatives are negligible at large $r$ ${\ displaystyle r}$ $r$ , we can perform the Fourier transform in spatial coordinates:

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

{\ displaystyle G (\ mathbf {k}) = \ int d ^ {3} r \; G (\ mathbf {r}) e ^ {i \ mathbf {k} \ cdot \ mathbf {r}}}

{\ displaystyle G (\ mathbf {k}) = \ int d ^ {3} r \; G (\ mathbf {r}) e ^ {i \ mathbf {k} \ cdot \ mathbf {r}}}

where the integral is taken over the entire space. Then it can be shown that

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

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

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

Therefore, the Green's function on $r$ ${\ displaystyle r}$ $r$ is given by the inverse Fourier transform:

G(\mathbf {r} )={\frac {1}{(2\pi )^{3}}}\;\int d^{3}\!k\;{\frac {e^{-i\mathbf {k} \cdot \mathbf {r} }}{k^{2}+\lambda ^{2}}}.

{\ displaystyle G (\ mathbf {r}) = {\ frac {1} {(2 \ pi) ^ {3}}};; \ int d ^ {3} \! k \; {\ frac {e ^ {-i \ mathbf {k} \ cdot \ mathbf {r}}} {k ^ {2} + \ lambda ^ {2}}}.

{\ displaystyle G (\ mathbf {r}) = {\ frac {1} {(2 \ pi) ^ {3}}};; \ int d ^ {3} \! k \; {\ frac {e ^ {-i \ mathbf {k} \ cdot \ mathbf {r}}} {k ^ {2} + \ lambda ^ {2}}}.

This integral can be estimated using spherical coordinates in $k$ ${\ displaystyle k}$ $k$ -space. Integration over angular coordinates is not difficult, and the integral is simplified - now you need to integrate only along one radial coordinate $k$ ${\ displaystyle k}$ $k$ :

G(\mathbf {r} )={\frac {1}{2\pi ^{2}r}}\;\int \limits _{0}^{\infty }dk\;{\frac {k\,\sin kr}{k^{2}+\lambda ^{2}}}.

{\ displaystyle G (\ mathbf {r}) = {\ frac {1} {2 \ pi ^ {2} r}} \; \ int \ limits _ {0} ^ {\ infty} dk \; {\ frac {k \, \ sin kr} {k ^ {2} + \ lambda ^ {2}}}.

{\ displaystyle G (\ mathbf {r}) = {\ frac {1} {2 \ pi ^ {2} r}} \; \ int \ limits _ {0} ^ {\ infty} dk \; {\ frac {k \, \ sin kr} {k ^ {2} + \ lambda ^ {2}}}.

This integral can be estimated by integration over the contour ( residue theory ). As a result, we get:

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

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

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

The final solution to the entire problem:

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

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

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

As stated above, this is a superposition of shielded $1/r$ ${\ displaystyle 1 / r}$ ${\ displaystyle 1 / r}$ functions statistically weighted by the source function $f$ ${\ displaystyle f}$ $f$ , and $\lambda$ ${\ displaystyle \ lambda}$ $\ lambda$ is the shielding factor. Shielded $1/r$ ${\ displaystyle 1 / r}$ ${\ displaystyle 1 / r}$ a function often appears in physics as a screened Coulomb potential, and the Yukawa potential is also known.

Shielded Poisson Equation

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