## Monday, April 21, 2008

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

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

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

with equal distribution function

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

and a powerlaw auto-correlation:

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

The d_n are positive step widths

$d_n\in&space;\left[&space;0,\infty&space;&space;\righ]$

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

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

but without any temporal correlations (white noise):

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

which means that

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

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

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

The velocity of the nth step is

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

For the auto-correlation function of the velocity we obtain

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

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

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

which yields for all m except 0:

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

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.