Critical behaviour of the randomly stirred dynamical Potts model: Novel universality class and effects of compressibility

Abstract

Critical behaviour of a nearly critical system, subjected to vivid turbulent mixing, is studied by means of the field theoretic renormalization group. Namely, relaxational stochastic dynamics of a non-conserved order parameter of the Ashkin-Teller-Potts model, coupled to a random velocity field with prescribed statistics, is considered. The mixing is modelled by Kraichnan's rapid-change ensemble: time-decorrelated Gaussian velocity field with the power-like spectrum $\propto k^{-d-\xi}$. It is shown that, depending on the symmetry group of the underlying Potts model, the degree of compressibility and the relation between the exponent $\xi$ and the space dimension $d$, the system exhibits various types of infrared (long-time, large-scale) scaling behaviour, associated with four different infrared attractors of the renormalization group equations. In addition to known asymptotic regimes (equilibrium dynamics of the Potts model and the passively advected scalar field), existence of a new, strongly non-equilibrium type of critical behaviour is established. That "full-scale" regime corresponds to the novel type of critical behaviour (universality class), where the self-interaction of the order parameter and the turbulent mixing are equally important. The corresponding critical dimensions depend on $d$, $\xi$, the symmetry group and the degree of compressibility. The dimensions and the regions of stability for all the regimes are calculated in the leading order of the double expansion in two parameters $\xi$ and $\varepsilon=6-d$. Special attention is paid to the effects of compressibility of the fluid, because they lead to nontrivial qualitative crossover phenomena.