Editing Rotational diffusion
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Latest revision | Your text | ||
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units of inverse time. | units of inverse time. | ||
This [[partial differential equation]] | This [[partial differential equation]] (PDE) may be solved by expanding ''f(θ, φ, t)'' in [[spherical harmonics]], for which the mathematical identity holds: | ||
:<math> | :<math> | ||
\frac{1}{\sin\theta} \frac{\partial}{\partial \theta}\left( \sin\theta \frac{\partial Y^{m}_{l}}{\partial \theta} \right) + | \frac{1}{\sin\theta} \frac{\partial}{\partial \theta}\left( \sin\theta \frac{\partial Y^{m}_{l}}{\partial \theta} \right) + | ||
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</math> | </math> | ||
Thus, the solution of | Thus, the solution of the PDE may be written | ||
:<math> | :<math> | ||
f(\theta, \phi, t) = \sum_{l=0}^{\infty} \sum_{m=-l}^{l} C_{lm} Y^{m}_{l}(\theta, \phi) e^{-t/\tau_{l}} | f(\theta, \phi, t) = \sum_{l=0}^{\infty} \sum_{m=-l}^{l} C_{lm} Y^{m}_{l}(\theta, \phi) e^{-t/\tau_{l}} |