By: Robert Stresing
From: Univ. Oldenburg, Germany
At: Instituto de Investigação Interdisciplinar, A2-25
Complex systems can frequently be described in terms of scale-dependent quantities which obey stochastic differential equations. These quantities might be velocity increments in turbulent flows or log-returns of financial time series. In many cases, e.g. for turbulence, surface roughness, and seismic and financial data, the description is greatly simplified by the finding that the corresponding stochastic scale-to-scale process has the Markov property (although the signal itself, i.e. the velocity, surface height or asset price, does not follow a Markov process). We now show how this analysis can be extended in order to obtain the complete multi-point statistics of the corresponding system. We extend the stochastic scale-to-scale description of homogeneous isotropic turbulence by conditioning on the velocity value itself, and find that the corresponding process does also have the Markov property. The process is governed by a Fokker-Planck equation, which can be estimated from given experimental data. Because of the Markov property of the scale-to-scale process, the multi-point statistics can be expressed by three-point statistics of the velocity signal. Thus, we propose a stochastic three-point closure for the velocity field of homogeneous isotropic turbulence.
[see R. Stresing and J. Peinke, Towards a stochastic multi-point
description of turbulence, New J. Phys. 12, 103046 (2010)]