Information Theory Analysis of Cascading Process in a Synthetic Model of Fluid Turbulence

Massimo Materassi,Rossana De Marco,Nathan Smith,Giuseppe Consolini

doi:10.3390/e16031272

Massimo Materassi, Rossana De Marco + Show 2 more

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https://doi.org/10.3390/e16031272

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Abstract

The use of transfer entropy has proven to be helpful in detecting which is the verse of dynamical driving in the interaction of two processes, X and Y . In this paper, we present a different normalization for the transfer entropy, which is capable of better detecting the information transfer direction. This new normalized transfer entropy is applied to the detection of the verse of energy flux transfer in a synthetic model of fluid turbulence, namely the Gledzer–Ohkitana–Yamada shell model. Indeed, this is a fully well-known model able to model the fully developed turbulence in the Fourier space, which is characterized by an energy cascade towards the small scales (large wavenumbers k), so that the application of the information-theory analysis to its outcome tests the reliability of the analysis tool rather than exploring the model physics. As a result, the presence of a direct cascade along the scales in the shell model and the locality of the interactions in the space of wavenumbers come out as expected, indicating the validity of this data analysis tool. In this context, the use of a normalized version of transfer entropy, able to account for the difference of the intrinsic randomness of the interacting processes, appears to perform better, being able to discriminate the wrong conclusions to which the “traditional” transfer entropy would drive.

Highlights

This paper is about the use of quantities, referred to as information dynamical quantities (IDQ), derived from the Shannon information [1] to determine cross-predictability relationships in the study of a dynamical system
A dynamical “driving” of the large on the small scales is expected, which is verified here through simulations: mutual information and transfer entropy analysis are applied to the synthetic time series of the Fourier amplitudes that are interacting
Before applying the calculation of the quantities, ∆TY →X (τ ; t) and ∆KY →X (τ ; t), to the turbulence model considered in Section 3, it is worth underlining again the dependence of all these IDQs on the delay, τ : the peaks of the IDQs on the τ axis indicate those delays after which the process, X, shares more information with the process, Y, i.e., the characteristic time scales of their cross-predictability, due to their interaction

Summary

Introduction

This paper is about the use of quantities, referred to as information dynamical quantities (IDQ), derived from the Shannon information [1] to determine cross-predictability relationships in the study of a dynamical system. The relationship between TE and the mathematical structure of the system (1) has been investigated in [16], while a more exhaustive study on the application of these information theoretical tools to systems with local dynamics is presented in [21] and the references therein This “physical sense” of the IDQs will be the subject of our future studies.

Normalized Mutual Information and Transfer Entropy

Turbulent Cascades and Information Theory

Conclusions