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15 February 2025
 
  » arxiv » 1508.0236

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Effects of random environment on a self-organized critical system: Renormalization group analysis of a continuous model
N. V. Antonov ; P. I. Kakin ;
Date 2 Aug 2015
AbstractWe 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.} {f 62}: 1813 (1989)]. The advecting velocity field is Gaussian, not correlated in time, with the pair correlation function of the form $propto delta(t-t’) / k_{ot}^{d-1+xi}$, where $k_{ot}=|{f k}_{ot}|$ and ${f k}_{ot}$ 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.} {f 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.
Source arXiv, 1508.0236
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