Effects of Random Environment on a Self-Organized Critical System: Renormalization Group Analysis of a Continuous Model

Department of Theoretical Physics, St. Petersburg State University, Uljanovskaja 1, Petrodvorez, 198504, St. Petersburg, Russia

^a e-mail: n.antonov@spbu.ru
^b e-mail: p.kakin@spbu.ru

Published online: 9 February 2016

Abstract

We study effects of the 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 [Phys. Rev. Lett. 62: 1813 (1989)]. The advecting velocity field is Gaussian, not correlated in time, with the pair correlation function of the form ∝ δ(t − t′)/k_⊥^d-1+ξ , where k_⊥ = |k_⊥| and k_⊥ 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 [Commun. Math. Phys. 131: 381 (1990)]. Using the field theoretic renormalization group we show that, depending on the relation between the exponent ξ 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 case ξ = 2(4 − d)/3 both the nonlinearity and the advection are important. The corresponding critical exponents are found exactly for all these cases.