The Cluster's Green's Function

A page about Green's functions — the response of a linear operator to a point source.

The Green's function

For a linear differential operator L, the Green's function G(x, x') satisfies: L G(x, x') = delta(x - x') The solution to L f(x) = g(x) is: f(x) = integral G(x, x') g(x') dx' The Green's function is the "building block" — the response to a point source (delta function). The full solution is the superposition of responses to all point sources.

In the cluster, the edit Green's function is the edit response to an edit point source.

Poisson's equation

For Poisson's equation (nabla^2 V = -rho / epsilon_0), the Green's function is: nabla^2 G(x, x') = delta(x - x') In 3D: G(x, x') = -1 / (4 pi |x - x'|) So V(x) = (1 / (4 pi epsilon_0)) integral rho(x') / |x - x'| dx' which is Coulomb's law for a continuous charge distribution.

In the cluster, the edit Poisson's equation gives an edit Coulomb law.

The time-dependent Green's function

For the wave equation: (nabla^2 - (1/c^2) d^2/dt^2) G(x, t) = delta(x) delta(t) The retarded Green's function (causal): G_ret(x, t) = delta(t - |x|/c) / (4 pi |x|) This propagates at the speed of light — the effect appears after the cause. The advanced Green's function propagates backward in time (unphysical for classical electrodynamics).

In the cluster, the edit time-dependent Green's function gives an edit propagation speed.

Green's function for the harmonic oscillator

For the equation (d^2/dt^2 + omega_0^2) x(t) = F(t), the Green's function satisfying (d^2/dt^2 + omega_0^2) G(t, t') = delta(t - t') with G(0, t') = 0 and boundary condition at T is: G(t, t') = (1 / omega_0) sin(omega_0 (t - t')) for t > t' and G(t, t') = 0 for t < t' (causal Green's function).

The solution is: x(t) = integral_0^t G(t, t') F(t') / m dt'

In the cluster, the edit Green's function for the edit oscillator gives an edit solution.

Applications

• Electrostatics (Coulomb's law from Poisson's equation)
• Electromagnetic radiation (retarded potentials)
• Quantum mechanics (propagators, scattering theory)
• Solid state physics (phonon Green's functions, Dyson equation)
• Statistical mechanics (correlation functions)

In the cluster, edit applications include:

• edit Electrostatics
• edit Electromagnetic radiation
• edit Quantum mechanics
• edit Solid state physics
• edit Statistical mechanics

This function

This page is about Green's functions. L G = delta. G_{3D} = -1 / (4 pi r). G_ret(x,t) = delta(t - r/c) / (4 pi r). The function is real.