I would like to calculate the non-gaussian parameter from MSD. I think I am doing mistake in calculating it?
3 < r(t)**4 >
NGP ((alpha)(t))= ----------------------- - 1
5 < r(t)**2 >**2
In some articles it is delta r(t). I am bit confused. Someone please help me out in calculating it by explain the terms in it?
Thank you
You cannot! The MSD is almost by definition a second moment of the probability distribution -- when it is Gaussian, this is the only parameter of the distribution and thus you find it. But with a non-Gaussian probability you have at least one more (or more than one) additional independent parameters that determine the 4th moment -- so you cannot determine both from just one measurement of MSD.
People sometimes "cheat" and make a guess about what is the distribution (e.g. Gaussian with non-quadratic correction, or Poisson, or some other explicit function) and then calculate your NGP analytically, and then try to fit two parameters to one measurement. But to be even partially respectable, you have to have independent physical reasons for your guess... For instance -- if all you measure is MSD, then why do you even think you have a non-Gaussian distribution ??
r(t) and delta r(t) are (I infer) the displacement. Simple discussions of diffusion lead to a Gaussian displacement distribution whose MSD increases linearly with time. These descriptions do not describe diffusion in glassy liquids or complex fluids such as polymer solutions, for which the diffusion is in general non-Gaussian. As a mathematical point, for a Gaussian distribution the object you are discussing is zero. However, if it is non-zero, then all the other even higher relatives of NGP are also non-zero. The time dependence of the displacement gives a clue. If the MSD increases non-linearly with time (e.g. subdiffusion) then it is mathematically impossible for the displacement distribution function P(Delta x, t) to be a central limit theorem Gaussian. (Alas, there are other ways of reaching a Gaussian P(Delta x, t), so life is more complicated.) I refer you to my recent paper in Soft Matter 11, 580-586 (2015) for an extended discussion on this matter.
Thank you Eugene M Terentjev for your time and explanation.
And also George Phillies for spending your time in answering my query and suggesting me your paper.
Thank you
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