Freezing transition in decaying Burgers turbulence and random matrix dualities
School of Mathematical Sciences, University of Nottingham - Nottingham NG7 2RD, UK, EU
2 CNRS-Laboratoire de Physique Théorique de l'Ecole Normale Supérieure 24 rue Lhomond, 75231 Paris Cedex, France, EU
3 Laboratoire de Physique Théorique et Modèles Statistiques, CNRS (UMR 8626), Université Paris-Sud Bât. 100, 91405 Orsay Cedex, France, EU
Accepted: 15 June 2010
We reveal a phase transition with decreasing viscosity ν at ν=νc>0 in one-dimensional decaying Burgers turbulence with a power-law–correlated random profile of Gaussian-distributed initial velocities . The low-viscosity phase exhibits non-Gaussian one-point probability density of velocities, continuously dependent on ν, 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 the 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. We also provide some insights for the finite-viscosity behaviour of velocities in the latter model.
PACS: 05.40.-a – Fluctuation phenomena, random processes, noise, and Brownian motion / 47.27.eb – Statistical theories and models
© EPLA, 2010