Rotational Diffusion - Basic Equations of Rotational Diffusion

Basic Equations of Rotational Diffusion

For rotational diffusion about a single axis, the mean-square angular deviation in time is

$\langle\theta^2\rangle = 2 D_r t \!$

where is the rotational diffusion coefficient (in units of radians2/s). The angular drift velocity in response to an external torque (assuming that the flow stays non-turbulent and that inertial effects can be neglected) is given by

$\Omega_d = \frac{\Gamma_\theta}{f_r}$

where is the frictional drag coefficient. The relationship between the rotational diffusion coefficient and the rotational frictional drag coefficient is given by the Einstein relation (or Einstein–Smoluchowski relation):

$D_r = \frac{k_B T}{f_r}$

where is the Boltzmann constant and is the absolute temperature. These relationships are in complete analogy to translational diffusion.

The rotational frictional drag coefficient for a sphere of radius is

$f_{r, \textrm{sphere}} = 8 \pi \eta R^3 \!$ where is the dynamic viscosity.

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