Gibbsian Hypothesis in Turbulence

Details Gibbsian Hypothesis in Turbulence Gibbsian Hypothesis in Turbulence

2002-05-14

arxiv.org/abs/cond-mat/0205286v2

Physics

Condensed Matter

Statistical Mechanics

19 pages, 9 figures, greatly expanded content, accepted to appear in J. Stat. Phys

Scientific paper

We show that Kolmogorov multipliers in turbulence cannot be statistically independent of others at adjacent scales (or even a finite range apart) by numerical simulation of a shell model and by theory. As the simplest generalization of independent distributions, we suppose that the steady-state statistics of multipliers in the shell model are given by a translation-invariant Gibbs measure with a short-range potential, when expressed in terms of suitable ``spin'' variables: real-valued spins that are logarithms of multipliers and XY-spins defined by local dynamical phases. Numerical evidence is presented in favor of the hypothesis for the shell model, in particular novel scaling laws and derivative relations predicted by the existence of a thermodynamic limit. The Gibbs measure appears to be in a high-temperature, unique-phase regime with ``paramagnetic'' spin order.

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