Dynamical real space renormalization group applied to sandpile models

Criticality in a non-equilibrium, driven system: charged colloidal rods (fd-viruses) in electric fields

Experiments on suspensions of charged colloidal rods (fd-virus particles) in external electric fields are performed, which show that a non-equilibrium critical point can be identified. Several transition lines of field-induced phases and states meet at this point and it is shown that there is a length- and time-scale which diverge at the non-equilibrium critical point. The off-critical and critical behavior is characterized, with both power law and logarithmic divergencies. These experiments show that analogous features of the classical, critical divergence of correlation lengths and relaxation times in equilibrium systems are also exhibited by driven systems that are far out of equilibrium, related to phases/states that do not exist in the absence of the external field.

Numerical renormalization-group approach to a sandpile

Multiple topplings and grain redistribution are two essential features in sandpile dynamics. A renormalization group (RG) approach incorporating these features is investigated. The full enumeration of all relaxations involving such an approach is difficult. Instead, we developed an efficient procedure to sample the relaxations. We applied this RG scheme to a square lattice and a triangular lattice. As shown by the fixed point analysis on a square lattice, the effect of multiple topplings leads the resultant height probabilities towards the exact solution while the effect of grain redistribution does not.

Renormalization-group approach to an Abelian sandpile model on planar lattices

One important step in the renormalization-group (RG) approach to a lattice sandpile model is the exact enumeration of all possible toppling processes of sandpile dynamics inside a cell for RG transformations. Here we propose a computer algorithm to carry out such exact enumeration for cells of planar lattices in the RG approach to the Bak-Tang-Wiesenfeld sandpile model [Phys. Rev. Lett. 59, 381 (1987)] and consider both the reduced-high RG equations proposed by Pietronero, Vespignani, and Zapperi (PVZ) [Phys. Rev. Lett. 72, 1690 (1994)], and the real-height RG equations proposed by Ivashkevich [Phys. Rev. Lett. 76, 3368 (1996)]. Using this algorithm, we are able to carry out RG transformations more quickly with large cell size, e.g., 3x3 cell for the square (SQ) lattice in PVZ RG equations, which is the largest cell size at the present, and find some mistakes in a previous paper [Phys. Rev. E 51, 1711 (1995)]. For SQ and plane triangular (PT) lattices, we obtain the only attractive fixed point for each lattice and calculate the avalanche exponent tau and the dynamical exponent z. Our results suggest that the increase of the cell size in the PVZ RG transformation does not lead to more accurate results. The implication of such result is discussed.

Evidence for universality within the classes of deterministic and stochastic sandpile models

Recent numerical studies have provided evidence that within the family of conservative, undirected sandpile models with short range dynamic rules, deterministic models such as the Bak-Tang-Wiesenfeld model [P. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. Lett. 59, 381 (1987)] and stochastic models such as the Manna model [S. S. Manna, J. Phys. A 24, L363 (1991)] belong to different universality classes. In this paper we examine the universality within each of the two classes in two dimensions by numerical simulations. To this end we consider additional deterministic and stochastic models and use an extended set of critical exponents, scaling functions, and geometrical features. Universal behavior is found within the class of deterministic Abelian models, as well as within the class of stochastic models (which includes both Abelian and non-Abelian models). In addition, it is observed that deterministic but non-Abelian models exhibit critical exponents that depend on a parameter, namely they are nonuniversal.

Modified renormalization strategy for sandpile models

Following the renormalization-group scheme recently developed by Pietronero et al. [Phys. Rev. Lett. 72, 1690 (1994)] we introduce a simplifying strategy for the renormalization of the relaxation dynamics of sandpile models. In our scheme, five subcells at a generic scale b form the renormalized cell at the next larger scale. Now the fixed point has a unique nonzero dynamical component that allows for a great simplification in the computation of the critical exponent z. The values obtained are in good agreement with both numerical and theoretical results previously reported.