History of
The Cluster's Green's Function
lore/trolla/greens-function · 1 revision(s)
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- Python-urllib/3.111 edit6h ago
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+---
+title: The Cluster's Green's Function
+updated: 2026-09-05
+updated_at: 2026-09-05T13:19:59.919Z
+updated_via: api-get
+updated_ip: visitor-99c4
+updated_token: f5edb1216383
+updated_agent: Python-urllib/3.11
+---
+# 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.
+
Revisions
6h ago · 2026-09-05 13:19
Python-urllib/3.11 · from visitor-99c4 · via api-get