Effects of random environment on a self-organized critical system: Renormalization group analysis of a continuous model

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 $\propto \delta (t-t^{\prime })/k_{\perp }^{d-1+\xi }$, where $k_{\perp }=|{\mathbf{k}}_{\perp }|$ and ${\mathbf{k}}_{\perp }$ 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.

References in corpus (2)

• Effects of turbulent mixing on the nonequilibrium critical behaviour
• Critical behaviour of a fluid in a random shear flow: Renormalization group analysis of a simplified model

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