Abstract
We study effects of random fluid motion on a system in a self-organized critical state. The latter is described by the continuous stochastic model, proposed by Hwa and Kardar [{\it Phys. Rev. Lett.} {\bf 62}: 1813 (1989)]. The advecting velocity field is Gaussian, not correlated in time, with the pair correlation function of the form $\propto \delta(t-t') / k_{\bot}^{d-1+\xi}$, where $k_{\bot}=|{\bf k}_{\bot}|$ and ${\bf k}_{\bot}$ is the component of the wave vector, perpendicular to a certain preferred direction -- the $d$-dimensional generalization of the ensemble introduced by Avellaneda and Majda [{\it Commun. Math. Phys.} {\bf 131}: 381 (1990)]. Using the field theoretic renormalization group we show that, depending on the relation between the exponent $\xi$ and the spatial dimension $d$, the system reveals different types of large-scale, long-time scaling behaviour, associated with the three possible fixed points of the renormalization group equations. They correspond to ordinary diffusion, to passively advected scalar field (the nonlinearity of the Hwa--Kardar model is irrelevant) and to the "pure" Hwa--Kardar model (the advection is irrelevant). For the special choice $\xi=2(4-d)/3$ both the nonlinearity and the advection are important. The corresponding critical exponents are found exactly for all these cases.
Highlights
Numerous physical systems reveal self-similar behaviour over extended ranges of the spatial or temporal scales with highly universal exponents
An essentially different example is given by the phenomenon of selforganized criticality (SOC), typical of open nonequilibrium systems with dissipative transport; see [3] and references therein
We study effects of the turbulent advection on a driven diffusive system in a selforganized critical state by means of the field theoretic renormalization group (RG)
Summary
Numerous physical systems reveal self-similar (scaling) behaviour over extended ranges of the spatial or temporal scales with highly universal exponents. Is the kernel of the inverse linear operation D−v 1 for the correlation function Dv in (4) This allows one to apply the field theoretic renormalization theory and the renormalization group to our stochastic problem. G and w are renormalized analogues of the bare parameters in (9), which play the role of the coupling constants (dimensionless expansion parameters) in the renormalized perturbation theory, and a, b are numerical coefficients. Their precise values are unimportant (they can be absorbed by the redefinition of g and w), but it is important for the following that they are positive: a, b > 0. Where the ellypsis stands for the higher-order corrections in g and w
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