📢Decay rate always wins over diffusion

The sources establish that the behavior of a radioactive material in a substance is governed by a reaction-diffusion equation, $\partial_t n - D \nabla^2 n = -\lambda n$, where radioactive decay is incorporated as a negative source term or sink ($\kappa = -\lambda n$), reflecting the inherent decrease in the number density ($n$) of radioactive nuclei without production. This sink is defined by the decay constant ($\lambda$), which is related to the half-life ($\tau$) by $\lambda=1 / \tau$. Physically, the diffusivity ($D$) and the decay rate ($\lambda$) exert independent and accelerated control over the system; a higher diffusivity speeds up the spatial redistribution and peak flattening while conserving mass, whereas a higher decay rate dictates the system's temporal lifespan by acting as an accelerated mass sink, causing the concentration profile to shrink exponentially downwards. Crucially, in the combined scenario, the decay mechanism's rapid destruction of the material ensures that the overall mass vanishes before extensive diffusion can occur, confirming that the decay rate ultimately dominates the total existence time of the material within the domain.

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Decay rate always wins over diffusion | IllustraDemoviadean on Notion