How to calculate mean square angular displacement ? Do we need any periodic boundary conditions if at each step angle is updated using theta(t+dt) = theta(t) + eta(t) , where eta(t) is a gaussian noise . Please describe the procedure to calculate this quantity.
What this implies is that, when the angle varies as indicated, the quantities that have intetesting distributions are its exponentials, more precisely cosθ and sinθ. So just compute the corresponding time series for cosθ and sinθ.In that way the periodicity is automatically taken into account.
Stam Nicolis My question was How to calculate that angular mean square displacement ?
If you want to calculate it analytically, that would be or , depending on the question. In fact since the mapping means that the circle is sampled uniformly, it's easy to see that both quantities are equal and equal to 1/2 and they're given by the integral of cos2θ or sin2θ from 0 to 2π, dividing by the normalization, which is 2π.
These are the moments of the so-called Haar measure of the Lie group U(1).
What's important is to understand that it isn't that's the appropriate quantity, because it doesn't have the correct periodicity. So the angle θ is sampled uniformly, but it's the distribution of cosθ or sinθ that takes into account the periodicity properly.
Stam Nicolis I want to calculate the rotational diffusion constant from that angular mean square displacement. Is it possible to get ? If yes, could you please share the correct algorithm ?
Souvik Sadhukhan Yes: The way to do that is by relating the angular mean square displacement, expressed through , more precisely, . This is the relation that defines the diffusion coefficient (strictly speaking, in the approximation, where the sinθ(t) can be assumed to be drawn from a Gaussian distribution; otherwise it's more complicated) and is obtained by computing the probability distribution of the sinθ(t), from the knowledge of the probability distribution of the η(t) and the relation, dθ(t)/dt = η(t).
Indeed, if one sets u(t) = sinθ(t), then du/dt = η(t) cosθ(t) = η(t)(1-u(t)2)1/2 (there's a subtlety here about the sign, that deserves further study!), so it's possible to obtain the probability distribution for the u(t) that way.
A fuller discussion (that doesn't close the subject, by any means!) may be found here: Article Finite-dimensional colored fluctuation-dissipation theorem f...
You can do it using the solution P(theta, t) to the diffusion equation on the sphere. Calculate the integrals = int_{0}^{\pi} theta^k P(theta,t) dtheta using this solution and writing the Legendre polynomial as a linear combination of certain factorials times a cosine of particular frequencies related to the order of the Legendre polynomial. Then these integrals can be calculated by parts and you obtain the coefficients of a series that can be truncated retaining as many terms as you wish. You can see the details in the papers from Pavel Castro.
Adriano Valdes-Gomez Could you please share the links of the papers ?
Yes, sure.
Article Brownian motion of free particles on curved surfaces
Article Brownian motion meets Riemann curvature
and we have calculate those values numerically in two different ways in
Preprint A Geometrical Method for the Smoluchowski Equation on the Sphere
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