Rotational diffusion - Revision history

Carl McBride: Added an internal link.

2008-06-27T13:42:10Z

Added an internal link.

← Older revision		Revision as of 15:42, 27 June 2008
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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 diffusion, which maintains or restores the equilibrium statistical distribution of particles' position in space.		'''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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	==See also==		==See also==

	*[[~~diffusion~~]]		*[[Diffusion]]
	*[[~~rotational~~ relaxation]]		*[[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]]

Carl McBride: Style: expanded acronym

2008-06-20T12:24:42Z

Style: expanded acronym

← Older revision		Revision as of 14:24, 20 June 2008
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	units of inverse time.		units of inverse time.

	This [[partial differential equation]] ~~(PDE)~~ may be solved by expanding ''f(θ, φ, t)'' in [[spherical harmonics]], for which the mathematical identity holds:		This [[partial differential equation]] 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 ~~the PDE~~ may be written		Thus, the solution of this partial differential equation 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}}

Dduque: New page: '''Rotational diffusion''' is a process by which the equilibrium statistical distribution of the overall orientation of particles or molecules is maintained or restored. Rotational diffusi...

2008-06-20T10:13:20Z

New page: '''Rotational diffusion''' is a process by which the equilibrium statistical distribution of the overall orientation of particles or molecules is maintained or restored. Rotational diffusi...

New page

'''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
meaning of the θ and φ angles in spherical coordinates). The rotational diffusion equation is:
:<math>
\frac{1}{D_{\mathrm{rot}}} \frac{\partial f}{\partial t} = \nabla^{2} f =
\frac{1}{\sin\theta} \frac{\partial}{\partial \theta}\left( \sin\theta \frac{\partial f}{\partial \theta} \right) +
\frac{1}{\sin^{2} \theta} \frac{\partial^{2} f}{\partial \phi^{2}},
</math>
with the rotational diffusion coefficient <math>D_\mathrm{rot}</math>, which has
units of inverse time.

This [[partial differential equation]] (PDE) may be solved by expanding ''f(θ, φ, t)'' in [[spherical harmonics]], for which the mathematical identity holds:
:<math>
\frac{1}{\sin\theta} \frac{\partial}{\partial \theta}\left( \sin\theta \frac{\partial Y^{m}_{l}}{\partial \theta} \right) +
\frac{1}{\sin^{2} \theta} \frac{\partial^{2} Y^{m}_{l}}{\partial \phi^{2}} = -l(l+1) Y^{m}_{l}
</math>

Thus, the solution of the PDE may be written
:<math>
f(\theta, \phi, t) = \sum_{l=0}^{\infty} \sum_{m=-l}^{l} C_{lm} Y^{m}_{l}(\theta, \phi) e^{-t/\tau_{l}}
</math>

where ''C<sub>lm</sub>'' are constants determined by the initial distribution. The most relevant result is the one for relaxation times:

:<math>
\tau_{l} = \frac{1}{D_{\mathrm{rot}}l(l+1)}
</math>

==See also==

*[[diffusion]]
*[[rotational relaxation]]
*[http://en.wikipedia.org/wiki/Rotational_diffusion Rotational diffusion at wikipedia]

[[Category: Non-equilibrium thermodynamics]]