QT:{{”
The simplified diffusion equation you’re looking for, linking length (\(L\)) to diffusion (\(D\)) and time (\(t\)), is often represented as a characteristic length squared scaling with diffusivity times time, like \(L^{2}\propto Dt\), or more precisely, the Root Mean Square Displacement (\(\langle x^{2}\rangle \)) is \(2Dt\), showing that diffusion spreads particles proportionally to \(\sqrt{Dt}\), meaning distance (L) scales with \(\sqrt{Dt}\), so \(L\sim \sqrt{Dt}\) or \(L^{2}\sim Dt\). Here’s a breakdown: The Full Equation (1D): \(\frac{\partial u}{\partial t}=D\frac{\partial ^{2}u}{\partial x^{2}}\) (or \(\frac{\partial u}{\partial t}=\alpha \frac{\partial ^{2}u}{\partial x^{2}}\)), where \(u\) is concentration, \(t\) is time, and \(D\) (or \(\alpha \)) is the diffusion coefficient (diffusivity).Your Simplified Idea: \(L=Dt^{2}\) isn’t quite right dimensionally, as \(L\) (length) should relate to \(\sqrt{Dt}\) (length\(/\sqrt{time}\times \sqrt{time}=length\)).The Correct Scaling:Distance (L): A particle’s average distance diffused is proportional to \(\sqrt{Dt}\).Squared Distance (\(\langle x^{2}\rangle \)): The mean squared displacement (how far it spreads, averaged) is directly \(2Dt\).In short: For a characteristic length \(L\), \(L^{2}\propto Dt\). This shows how far things spread over time due to random motion (diffusion).
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