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, potentials. 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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Benzi, R., Biferale, L. & Sbragaglia, M. A Gibbs-Like Measure for Single-Time, Multi-Scale Energy Transfer in Stochastic Signals and Shell Model of Turbulence. Journal of Statistical Physics 114, 137–154 (2004). https://doi.org/10.1023/B:JOSS.0000003107.88526.bf
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DOI: https://doi.org/10.1023/B:JOSS.0000003107.88526.bf