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:


where the s_n are sign factors

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

with equal distribution function


and a powerlaw auto-correlation:

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...)

but without any temporal correlations (white noise):

which means that

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

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

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

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.

No comments:

Post a Comment