C.M.'s Science Blog: [AD 11] Solution of the PL-Kurtosis puzzle

The question if powerlaw diffusion is compatible with positive kurtosis and, in particular, with an exponential step width distribution, could finally be settled by directly constructing a discrete time series with the requested properties:

We consider a simple stochastic process, which produces a time series of values x_k=x(t_k) for discrete time points t_k = k*dt:

$x_n=x_{n-1}+s_n\;d_n$

where the s_n are sign factors

$s_n \in \left\{ -1,+1 \right\}$

with equal distribution function

$P(s)=\frac{1}{2}\delta_{s,-1}+\frac{1}{2}\delta_{s,+1}$

and a powerlaw auto-correlation:

$C_{ss}(m)=\left< s_n\; s_{n+m} \right>=\delta_{m,0}\;\sigma_s^2 + (1-\delta_{m,0})\;m^{-p}$

The d_n are positive step widths

$d_n\in \left[ 0,\infty \righ]$

with an arbitrary distribution function of finite variance (Gaussian, Exponential, Poisson...)

$P(d>0) \;\;\mbox{arbitrary, with}\;\; \sigma_d^2=\left< (d\!-\!\overline{d})^2 \right> <\infty$

but without any temporal correlations (white noise):

$C_{dd}(m)=\left< (d_n\!-\!\overline{d})\;(d_{n+m}\!-\!\overline{d}) \right>=$
$=\left< d_n\;d_{n+m} \right> -\overline{d}^2 =\sigma_d^2\;\delta_{m,0}$

which means that

$\left< d_n\;d_{n+m} \right> = \overline{d}^2\;\;\mbox{for}\;\;m\neq 0$

We assume there are no cross-correlations between the sign-factors and the step widths:

$C_{sd}(m)=\left< s_n\;d_{n+m} \right>=0$

The velocity of the nth step is

$v_n=\frac{x_n-x_{n+1}}{\Delta t}=\frac{s_n\;d_n}{\Delta t}$

For the auto-correlation function of the velocity we obtain

$C_{vv}(m)=\left< v_n\;v_{n+m} \right>=\frac{1}{\Delta t^2}\left< s_n d_n\;s_{n+m}d_{n+m} \right>$

which can be factorized because the s and d and uncorrelated:

$C_{vv}(m)=\frac{1}{\Delta t^2}\left< s_n\;s_{n+m} \right>\left< d_n\;d_{n+m} \right>$

which yields for all m except 0:

$C_{vv}(m\! \neq \!0)=\frac{\overline{d}^2}{\Delta t^2}\;\;m^{-p}$

This means that the process has powerlaw VAC (and PSD and MSD) for arbitrary step widths distributions.

By using separate processes x_n and y_n of the above kind for the different spatial directions, the method can be generalized to multiple dimensions.

C.M.'s Science Blog

Monday, April 21, 2008

[AD 11] Solution of the PL-Kurtosis puzzle - The SD model

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