🎬the stationary solution to Poisson's equation driven by a Dirac delta function source

The two visualizations for the stationary diffusion problem, driven by the $\nabla^2 u \propto \delta^{(3)}(x)$ equation, collectively demonstrate how a point source creates a steady-state field governed by classical inverse power laws. The 3D concentration plot shows that the concentration potential, $u(r)$, adheres to the inverse distance law ( $u \propto 1 / r$ ), resulting in a sharp, conical peak at the origin. This potential establishes the necessary gradient to sustain flow. Simultaneously, the 2D animated flow field visualizes the current density, j, which follows the inverse square law ( |$\jmath| \propto 1 / r^2$ ). This faster decay in current magnitude ensures flux conservation, meaning the total amount of diffusing substance flowing outward through any spherical surface surrounding the source remains constant and equal to the production rate $P$.

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the stationary solution to Poisson's equation driven by a Dirac delta function source | Animated Resultsviadean on Notion