A Gibbs-like measure for single-time, multi-scale energy transfer in stochastic signals and Shell Model of turbulence

Roberto Benzi; Luca Biferale; Mauro Sbragaglia

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arXiv · nlin/0211045

A Gibbs-like measure for single-time, multi-scale energy transfer in stochastic signals and Shell Model of turbulence

Roberto Benzi·Luca Biferale·Mauro Sbragaglia

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Abstract

A Gibbs-like approach for simultaneous multi-scale correlation functions in random, time-dependent, multiplicative processes for the turbulent energy cascade is investigated. We study the optimal log-normal Gibbs-like distribution able to describe the subtle effects induced by non-trivial time dependency on both single-scale (structure functions) and multi-scale correlation functions. We provide analytical expression for the general multi-scale correlation functions in terms of the two-point correlations between multipliers and we show that the log-normal distribution is already accurate enough to reproduce quantitatively many of the observed behavior. The main result is that non-trivial time effects renormalize the Gibbs-like effective potential necessary to describe single-time statistics. We also present a generalization of this approach to more general, non log-normal, potential. In the latter case one obtains a formal expansion of both structure functions and multi-scale correlations in terms of cumulants of all orders.

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Roberto Benzi, Luca Biferale, Mauro Sbragaglia. 2002-11-27. A Gibbs-like measure for single-time, multi-scale energy transfer in stochastic signals and Shell Model of turbulence. https://arxiv.org/abs/nlin/0211045

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