Freezing Transition in Decaying Burgers Turbulence and Random Matrix Dualities

TL;DR

This paper uncovers a phase transition in decaying Burgers turbulence with power-law correlated initial velocities, revealing non-Gaussian velocity distributions and a spontaneous replica symmetry breaking, supported by analytical and numerical evidence.

Contribution

It introduces a novel phase transition in Burgers turbulence linked to a freezing transition in a related statistical mechanics model, extending duality relations from Random Matrix Theory.

Findings

Identifies a critical viscosity  where the phase transition occurs.

Derives analytical expressions for low-order cumulants of velocity.

Numerically confirms non-Gaussian velocity distributions and RSB phenomena.

Abstract

We reveal a phase transition with decreasing viscosity $\nu$ at \nu=\nu_c>0 in one-dimensional decaying Burgers turbulence with a power-law correlated random profile of Gaussian-distributed initial velocities <v(x,0)v(x',0)>\sim|x-x'|^{-2}. The low-viscosity phase exhibits non-Gaussian one-point probability density of velocities, continuously dependent on \nu, reflecting a spontaneous one step replica symmetry breaking (RSB) in the associated statistical mechanics problem. We obtain the low orders cumulants analytically. Our results, which are checked numerically, are based on combining insights in the mechanism of the freezing transition in random logarithmic potentials with an extension of duality relations discovered recently in Random Matrix Theory. They are essentially non mean-field in nature as also demonstrated by the shock size distribution computed numerically and different from the short range correlated Kida model, itself well described by a mean field one step RSB ansatz. We also provide some insights for the finite viscosity behaviour of velocities in the latter model.