Influence of turbulent mixing on critical behavior of directed percolation process: Effect of compressibility

Abstract

Universal behavior is a typical emergent feature of critical systems. A paramount model of the nonequilibrium critical behavior is the directed bond percolation process that exhibits an active-to-absorbing state phase transition in the vicinity of a percolation threshold. Fluctuations of the ambient environment might affect or destroy the universality properties completely. In this work, we assume that the random environment can be described by means of compressible velocity fluctuations. Using field-theoretic models and renormalization group methods, we investigate large-scale and long-time behavior. Altogether, 11 universality classes are found, out of which 4 are stable in the infrared limit and thus macroscopically accessible. In contrast to the model without velocity fluctuations, a possible candidate for a realistic three-dimensional case, a regime with relevant short-range noise, is identified. Depending on the dimensionality of space and the structure of the turbulent flow, we calculate critical exponents of the directed percolation process. In the limit of the purely transversal random force, critical exponents comply with the incompressible results obtained by previous authors. We have found intriguing nonuniversal behavior related to the mutual effect of compressibility and advection.

Randomly stirred perfect gas

“…In (3) the irreducible part is explicitly written as a separate term and contains the space dimension three in the coefficient of the Kronecker symbol. In d dimensions the coefficient is thus 2/d instead of 2/3 used -quite surprisingly -in earlier work [10,11,12,13,14]. This is reflected in the connection between the dynamic viscosities η and ζ on one hand and kinematic viscosities µ and ν on the other.…”

“…which leads to the model used in the earlier RG analyses of compressible fluid [11,12,13,14]. In the next section it will be shown that this is not the model suitable for description of perfect gas.…”

“…In ( 13) the internal energy is assumed to vanish at the reference value the density ρ = ρ 0 , while in (14) the internal energy vanishes with the density. As a consistency check calculate the time derivative of the function u 2…”

“…Attempts to carry out a renormalization group (RG) analysis of stochastic hydrodynamics of compressible fluid have been based on the idea to close the system of the Navier-Stokes equation and the continuity equation with assumption of "isothermal" fluid (i.e. a fluid, in which the density is a linear function of the pressure: (p − p) = c 2 (ρ − ρ), where c is the speed of sound,) [10,11,12,13,14].…”