Poisson's equation

This article deals with the Poisson equations from electrostatics and classical gravity theory. In thermodynamics , the Poisson equation relates to an adiabatic change in state .

The Poisson equation , named after the French mathematician and physicist Siméon Denis Poisson , is an elliptical partial differential equation of the second order that is used as part of boundary value problems in large parts of physics.

Mathematical formulation

The Poisson equation is general

${\displaystyle -\Delta u=f}$

Here designated

• ${\displaystyle \Delta }$the Laplace operator
• ${\displaystyle u}$ the solution you are looking for
• ${\displaystyle f}$a function. If the equation becomes Laplace's equation . ${\displaystyle f\equiv 0}$

To solve Poisson's equation, additional information must be given, e.g. B. in the form of a Dirichlet boundary condition :

${\displaystyle {\begin{cases}-\Delta u&=f&{\text{in}}&\Omega \\\quad u&=g&{\text{auf}}&\partial \Omega \end{cases}}}$

with open and restricted. ${\displaystyle \Omega \subset \mathbb {R} ^{n}}$

In this case, we construct a solution using the fundamental solution of Laplace's equation: ${\displaystyle \Phi }$

${\displaystyle \Phi (x):={\begin{cases}-{\dfrac {1}{2\pi }}\ln |x|&n=2\\{\dfrac {1}{(n-2)\omega _{n}}}\cdot {\dfrac {1}{|x|^{n-2}}}&n\geq 3\end{cases}}}$

It denotes the area of ​​the unit sphere in the -dimensional Euclidean space . ${\displaystyle \omega _{n}={\tfrac {2\pi ^{\frac {n}{2}}}{\Gamma ({\frac {n}{2}})}}}$ ${\displaystyle n}$

The convolution gives a solution to Poisson's equation. ${\displaystyle (\Phi *f)}$

In order to meet the boundary condition can be the Green's function use

${\displaystyle G(x,y):=\Phi (y-x)-\phi ^{x}(y)}$

${\displaystyle \phi ^{x}}$ is a correction function that

${\displaystyle {\begin{cases}\Delta \phi ^{x}=0&{\text{in}}\ \Omega \\\phi ^{x}=\Phi (y-x)&{\text{auf}}\ \partial \Omega \end{cases}}}$

Fulfills. It is generally dependent on and is easy to find only for simple areas. ${\displaystyle \Omega }$

If one knows , then a solution of the boundary value problem from above is given by ${\displaystyle G(x,y)}$

${\displaystyle u(x)=-\int _{\partial \Omega }g(y){\frac {\partial G(x,y)}{\partial n}}\mathrm {d} \sigma (y)+\int _{\Omega }f(y)G(x,y)\mathrm {d} y}$

where the surface dimension denotes. ${\displaystyle \sigma }$ ${\displaystyle \partial \Omega }$

The solution can also be found with the help of the platform method or a variation approach .

Applications in physics

The Poisson equation, for example, is satisfied by the electrostatic potential and the gravitational potential , each with symbols . The function is proportional to the electrical charge density or to the mass density ${\displaystyle \Delta \Phi (\mathbf {r} )=f(\mathbf {r} )}$ ${\displaystyle \Phi }$ ${\displaystyle f}$ ${\displaystyle \rho (\mathbf {r} ).}$

If it is known everywhere, the general solution of Poisson's equation, which approaches zero for large distances, is the integral ${\displaystyle f(\mathbf {r} )}$

${\displaystyle \Phi (\mathbf {r} )=-{\frac {1}{4\,\pi }}\int \mathrm {d} ^{3}r'\,{\frac {f(\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}}$.

In words: every charge at the location in the small area of ​​size contributes additively to the potential at the location with its electrostatic or gravitational potential: ${\displaystyle \mathrm {d} ^{3}r'\,f(\mathbf {r} ')}$ ${\displaystyle \mathbf {r} '}$ ${\displaystyle \mathrm {d} ^{3}r'}$ ${\displaystyle \Phi }$ ${\displaystyle \mathbf {r} }$

${\displaystyle -{\frac {\mathrm {d} ^{3}r'\,f(\mathbf {r} ')}{4\,\pi \,|\mathbf {r} -\mathbf {r} '|}}}$

Electrostatics

Since the electrostatic field is a conservative field , it can be expressed in terms of the gradient of a potential : ${\displaystyle \nabla \Phi }$ ${\displaystyle \Phi (\mathbf {r} )}$

${\displaystyle \mathbf {E} (\mathbf {r} )=-\nabla \Phi (\mathbf {r} ).}$

Applying the divergence results in

${\displaystyle \nabla \cdot \mathbf {E} (\mathbf {r} )=-\Delta \Phi (\mathbf {r} )}$

with the Laplace operator . ${\displaystyle \Delta }$

According to the first Maxwell equation , however, also applies

${\displaystyle \nabla \cdot \mathbf {E} (\mathbf {r} )={\frac {\rho (\mathbf {r} )}{\varepsilon }}}$

With

• the charge density ${\displaystyle \rho (\mathbf {r} )}$
• the permittivity . ${\displaystyle \varepsilon =\varepsilon _{\mathrm {r} }\cdot \varepsilon _{0}}$

It follows for the Poisson equation of the electric field

${\displaystyle \Delta \Phi (\mathbf {r} )=-{\frac {\rho (\mathbf {r} )}{\varepsilon }}}$

The special case for each location in the area under consideration is called the Laplace equation of electrostatics . ${\displaystyle \rho (\mathbf {r} )=0}$

Electrodynamics of stationary currents

The emitter of a silicon solar cell , which can be described as purely two-dimensional to a good approximation, is considered here as an example . The emitter is located in the xy-plane, the z-axis points into the base. The lateral surface current density in the emitter depends on the z-component of the (volume) current density of the base occurring at the emitter , which is indicated by the continuity equation in the form ${\displaystyle \mathbf {j} (x,y)}$ ${\displaystyle J_{z}(x,y,z=0)}$

${\displaystyle \nabla _{2}\cdot \mathbf {j} (x,y)=-J_{z}(x,y,z=0)}$

can be described (with the two-dimensional Nabla operator ). The surface current density depends on the local Ohm's law with the lateral electric field in the emitter together: ; here is the specific sheet resistance of the emitter assumed to be homogeneous . If you write (as discussed in the section on electrostatics) the electric field as a gradient of the electric potential, you get a Poisson equation for the potential distribution in the emitter in the form ${\displaystyle \nabla _{2}}$ ${\displaystyle \mathbf {j} (x,y)=R_{\Box }^{-1}\mathbf {E} (x,y)}$ ${\displaystyle R_{\Box }}$ ${\displaystyle \mathbf {E} (x,y)=-\nabla _{2}\Phi (x,y)}$

${\displaystyle \Delta _{2}\Phi (x,y)=R_{\Box }J_{z}(x,y,z=0).}$

Gravity

Just like the electrostatic field

${\displaystyle \mathbf {E} (\mathbf {r} )={\frac {1}{4\pi \varepsilon }}\int \rho _{\mathrm {el} }\left(\mathbf {r} '\right){\frac {\mathbf {r} -\mathbf {r} '}{\left|\mathbf {r} -\mathbf {r} '\right|^{3}}}\;dx'\,dy'\,dz'=-\nabla \Phi _{\mathrm {el} }(\mathbf {r} )}$,

the gravitational field  g is also a conservative field:

${\displaystyle \mathbf {g} (\mathbf {r} )=-G\int \rho _{\mathrm {m} }\left(\mathbf {r} '\right){\frac {\mathbf {r} -\mathbf {r} '}{\left|\mathbf {r} -\mathbf {r} '\right|^{3}}}\;dx'\,dy'\,dz'=-\nabla \Phi _{\mathrm {m} }(\mathbf {r} )}$.

It is

• G is the gravitational constant
• ${\displaystyle \rho _{\mathrm {m} }(\mathbf {r} ')}$ the mass density.

Since only the charges are replaced by masses and by , analogously to the first Maxwell equation applies ${\displaystyle {\frac {1}{4\pi \varepsilon }}}$ ${\displaystyle -G}$

${\displaystyle \nabla \cdot \mathbf {g} =-4\pi G\rho _{\mathrm {m} }(\mathbf {r} )}$.

This results in the Poisson equation for gravity

${\displaystyle \nabla \mathbf {g} =-\nabla \cdot (\nabla \Phi _{\mathrm {m} }(\mathbf {r} ))=-4\pi \cdot G\cdot \rho _{\mathrm {m} }(\mathbf {r} )\Leftrightarrow }$

${\displaystyle \Delta \Phi _{\mathrm {m} }(\mathbf {r} )=4\pi \cdot G\cdot \rho _{\mathrm {m} }(\mathbf {r} )}$.

literature

• Richard Courant , David Hilbert : Methods of Mathematical Physics. Volume 1. Springer, Berlin a. a. 1924 (= The Basic Teachings of Mathematical Sciences , 12). 4th edition, ibid. 1993, ISBN 3-540-56796-8 .
• Lawrence C. Evans : Partial Differential Equations. American Mathematical Society, Providence RI 1998, ISBN 0-8218-0772-2 (= Graduate studies in mathematics 19).

Individual evidence

1. ↑ Wolfgang Nolting: Basic course in theoretical physics . [Online excl. the] 8th [dr.]. 3. Electrodynamics. Springer, Berlin, ISBN 978-3-540-71252-7 .