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'''Rotational diffusion''' is a process by which the equilibrium statistical distribution of the overall orientation of particles or molecules is maintained or restored. Rotational diffusion is the counterpart of translational | '''Rotational diffusion''' is a process by which the equilibrium statistical distribution of the overall orientation of particles or molecules is maintained or restored. Rotational diffusion is the counterpart of translational diffusion, which maintains or restores the equilibrium statistical distribution of particles' position in space. | ||
If a molecule has an orientation along a unit vector '''n''', and ''f(θ, φ, t)'' represents the probability density distribution for the orientation of '''n''' at time ''t'' (with the usual | If a molecule has an orientation along a unit vector '''n''', and ''f(θ, φ, t)'' represents the probability density distribution for the orientation of '''n''' at time ''t'' (with the usual | ||
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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}} | ||
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==See also== | ==See also== | ||
*[[ | *[[diffusion]] | ||
*[[ | *[[rotational relaxation]] | ||
*[http://en.wikipedia.org/wiki/Rotational_diffusion Rotational diffusion at wikipedia] | *[http://en.wikipedia.org/wiki/Rotational_diffusion Rotational diffusion at wikipedia] | ||
[[Category: Non-equilibrium thermodynamics]] | [[Category: Non-equilibrium thermodynamics]] |