Scaling behavior in interacting systems: joint effect of anisotropy and compressibility

TL;DR

This study investigates how anisotropy and compressibility in turbulent flows influence the scaling behavior of two classical non-linear systems, revealing that anisotropy enhances non-linear effects and stabilizes certain regimes.

Contribution

It provides a theoretical analysis of the joint effects of anisotropy and compressibility on the scaling regimes of two models using the renormalization group approach.

Findings

Anisotropy enhances non-linearities in the models.

More stable non-trivial regimes are found with anisotropy.

Four candidate behaviors for observable regimes are identified.

Abstract

Motivated by the ubiquity of turbulent flows in realistic conditions, effects of turbulent advection on two models of classical non-linear systems are investigated. In particular, we analyze model A (according to the Hohenberg-Halperin classification [1]) of a non-conserved order parameter and a model of the direct bond percolation process. Having two paradigmatic representatives of distinct stochastic dynamics, our aim is to elucidate to what extent velocity fluctuations affect their scaling behavior. The main emphasis is put on an interplay between anisotropy and compressibility of the velocity flow on their respective scaling regimes. Velocity fluctuations are generated by means of the Kraichnan rapid-change model, in which the anisotropy is due to a distinguished spatial direction n and a correlator of the velocity field obeys the Gaussian distribution law with prescribed statistical properties. As the main theoretical tool, the field-theoretic perturbative renormalization group is adopted. Actual calculations are performed in the leading (one-loop) approximation. Having obtained infra-red stable asymptotic regimes, we have found four possible candidates for macroscopically observable behavior for each model. In contrast to the isotropic case, anisotropy brings about enhancement of non-linearities and non-trivial regimes are proved to be more stable.