Abstract
The field theoretic renormalization group is applied to the strongly nonlinear stochastic advection-diffusion equation. The turbulent advection is modelled by the Kazantsev–Kraichnan “rapid-change” ensemble. As a requirement of the renormalizability, the model necessarily involves infinite number of coupling constants (“charges”). The one-loop counterterm is calculated explicitly. The corresponding renormalization group equation demonstrates existence of a pair of two-dimensional surfaces of fixed points in the infinite-dimensional parameter space. If the surfaces contain infrared attractive regions, the problem allows for the large-scale, long-time scaling behaviour. For the first surface (advection is irrelevant), the critical dimensions of the scalar field Δθ, the response field Δθ′ and the frequency Δω are nonuniversal (through the dependence on the effective couplings) but satisfy certain exact identities. For the second surface (advection is relevant), the dimensions are universal and they are found exactly.
Highlights
Critical behaviour and phase transitions in systems far from thermodynamic equilibrium have attracted constant attention over decades; see, e.g., [1,2,3,4,5,6,7,8,9,10,11,12,13] and the literature cited therein
We study strongly nonlinear diffusion of a scalar field θ = θ(x) = θ(t, x) in a turbulent environment described by the velocity field v = {vi(x)}, i = 1, . . . , d, where d is the arbitrary dimension of space
This problem is not related to any kind of phase transition, but it is expected that various correlation and response functions exhibit scaling behaviour in the infrared (IR) range
Summary
Critical behaviour and phase transitions in systems far from thermodynamic equilibrium have attracted constant attention over decades; see, e.g., [1,2,3,4,5,6,7,8,9,10,11,12,13] and the literature cited therein. We reformulate the stochastic nonlinear advection-diffusion equation as a certain field theoretic model with infinitely many couplings. We show that this full-scale model can be considered multiplicatively renormalizable in a wider, than usual, sense of the term. This allows one to derive the renormalization group equations, to investigate possible asymptotic regimes of the problem and to obtain some exact relations and expressions for the critical dimensions in scaling laws. If those surfaces contain infrared attractive regions, the problem allows for the large-scale, long-time scaling behaviour.
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