[Paper Review] Gibbsian Hypothesis in Turbulence

Gregory L. Eyink, Shiyi Chen+1 more|arXiv (Cornell University)|May 14, 2002

Statistical Mechanics and Entropy2 references4 citations

TL;DR

This paper challenges the long-standing assumption of statistical independence among Kolmogorov multipliers in turbulence by demonstrating through theory and numerical simulations of a shell model that multipliers are instead correlated across scales. It proposes a Gibbsian statistical mechanics framework with short-range interactions in logarithmic amplitude and phase variables, showing that the system exhibits a unique, high-temperature Gibbs measure with paramagnetic spin order, resolving inconsistencies in prior models and enabling new scaling laws.

ABSTRACT

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.

Motivation & Objective

• To challenge the assumption of statistical independence among Kolmogorov multipliers in turbulence, which underpins classical K41 theory.
• To investigate whether the steady-state statistics of multipliers in shell models can be described by a Gibbs measure with short-range interactions.
• To establish a thermodynamic limit for turbulence statistics by modeling amplitude and phase increments as spin variables.
• To demonstrate that the false stability of the K41 fixed point in independent multiplier models arises from the unphysical independence assumption.
• To derive and test new scaling laws and derivative relations predicted by the existence of a thermodynamic limit in the Gibbsian framework.

Proposed method

• Introduce spin variables: σₙ = ln(wₙ) for amplitude ratios and Uₙ = exp(iΔₙ) for dynamical phases, where wₙ = ρₙ/ρₙ₋₁ and Δₙ = −θₙ + θₙ₋₁ + θₙ₋₂.
• Propose that the joint distribution of spin variables follows a translation-invariant Gibbs measure with a short-range potential Φ, derived from the dynamics of the shell model.
• Use exact constraints from the shell model dynamics to show that independent multipliers lead to a dynamically unstable K41 fixed point, invalidating the i.i.d. assumption.
• Apply spectral theory of Markov chains to analyze energy and helicity fluxes, identifying leading and subleading eigenvalues μ₍₃₎ and μ₍₃₎′ of the transfer operator T₍₃₎.
• Construct a solvable Markov chain model with non-negative transition kernels and non-zero helicity flux, showing that negative subleading eigenvalues are possible without violating realizability.
• Perform direct numerical simulations of the SABRA shell model to test predictions of the Gibbs hypothesis, including scaling exponents and derivative relations.

Experimental results

Research questions

• RQ1 Can the steady-state statistics of turbulence multipliers be described by a Gibbs measure with short-range interactions, rather than assuming statistical independence?
• RQ2 What are the consequences of assuming statistical independence for the stability of the K41 fixed point in shell models?
• RQ3 How do energy and helicity fluxes behave in a Markov chain model with correlated multipliers, and can they remain constant across scales?
• RQ4 What role do long-range correlations in amplitude and phase variables play in generating intermittency and non-Gaussian statistics?
• RQ5 Can a thermodynamic limit exist in a non-equilibrium system like turbulence, and what scaling laws emerge from such a limit?

Key findings

• The assumption of independent multipliers leads to a dynamically unstable K41 fixed point, which is stabilized only by introducing finite-range correlations, proving that independence is unphysical.
• Numerical simulations of the shell model support the Gibbs hypothesis, showing that the system's statistics are consistent with a unique, high-temperature Gibbs measure with rapidly decaying interactions.
• The system is in a paramagnetic phase with no long-range spin order, indicating that the Gibbs measure is in a unique-phase regime despite non-equilibrium dynamics.
• Novel scaling laws and derivative relations are predicted by the existence of a thermodynamic limit, which are validated by simulation data.
• The helicity flux can be non-zero in a Markov chain model with correlated multipliers, even when the subleading eigenvalue μ₍₃₎′ is negative, contradicting realizability constraints in independent models.
• A solvable Markov chain model with non-negative transition kernels and non-vanishing helicity flux is constructed, demonstrating that negative μ₍₃₎′ is realizable and consistent with turbulence statistics.