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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Universality of dynamic scaling for avalanches in disordered Ising systems

Guangping Zheng, Mo Li

Published Sep 10, 2002• 3 citations
Publisher

Abstract

Dynamic scaling for driven disordered systems is investigated in some disordered Ising models. Using Monte Carlo simulation, we find that avalanches in both random-field and random-bond Ising models follow dynamic power-law scaling in short times, and the scaling relations are universal for the systems studied. The probability distribution of the dynamic scaling exponent theta is found to have two peaks centered at theta(1) and theta(2). The short-time dynamic exponent theta(1) is invariant and universal for all avalanches while the exponent theta(2) depends on the strength of disorder. The analytical result for the early stage evolution of breakdown process in the random-field Ising model is obtained using mean-field approximation. Short-time dynamic scaling is also confirmed.

DOI
10.1103/physreve.66.036108

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Dynamic scaling for driven disordered systems is investigated in some disordered Ising models. Using Monte Carlo simulation, we find that avalanches in both random-field and random-bond Ising models follow dynamic power-law scaling in short times, and the scaling relations are universal for the systems studied. The probability distribution of the dynamic scaling exponent theta is found to have two peaks centered at theta(1) and theta(2). The short-time dynamic exponent theta(1) is invariant and universal for all avalanches while the exponent theta(2) depends on the strength of disorder. The analytical result for the early stage evolution of breakdown process in the random-field Ising model is obtained using mean-field approximation. Short-time dynamic scaling is also confirmed.
Dynamic scaling for fracture or breakdown process in disordered systems is investigated in a two-dimensional random field Ising model (RFIM). We find two evolving stages in the avalanche process in the RFIM. At the short-time regime, a power-law growth of the avalanche size Deltas is observed; and at late times, the conventional nucleation and growth process is found. At the critical point of the RFIM, the avalanche size is found to obey the dynamic scaling law Delta(s) approximately equal t((d-beta/nu)/z). From this dynamic scaling relation, the critical strength of the random field D(c) and the critical exponents, beta, nu, and z, are determined. The observed dynamics is explained by a simple nucleation theory of first-order phase transformations.
Nonequilibrium short-time dynamics of first-order phase transition in a driven-disordered system at zero temperature is investigated. In a random-bond Ising model under external field, the largest discontinuous jump of order parameter shows a power-law evolution in short times: Δ M ( t ) ∼ t θ . The scaling exponent θ is equal to ( d − β/ν)/ z , where d is the dimensionality; β, ν and z are the critical exponents of the system. θ is found to be a universal exponent for any metastable relaxation in the short-time regime. This investigation suggests that the short-time dynamics is valid for the first-order phase transition in the driven disordered system and the critical phenomenon of the disordered system can be understood in the framework of nonequilibrium dynamics.
Extensive Monte Carlo simulations are performed on a two-dimensional random field Ising model. The purpose of the present work is to study the disorder-induced changes in the properties of disordered spin systems. The time evolution of the domain growth, the order parameter, and the spin-spin correlation functions are studied in the nonequilibrium regime. The dynamical evolution of the order parameter and the domain growth shows a power law scaling with disorder-dependent exponents. It is observed that for weak random fields, the two-dimensional random field Ising model possesses long-range order. Except for weak disorder, exchange interaction never wins over pinning interaction to establish long-range order in the system.
We have investigated the critical behavior of a three-dimensional random-bond Ising model for a series of the disorder strength by a finite-time scaling combining with Monte Carlo renormalization-group method in the presence of a linearly varying temperature. The method enables us to estimate a lot of critical exponents of both static and dynamic nature independently as well as the critical temperatures. The static exponents obtained agree well with most existing results, verify both the hyperscaling and the Rushbrooke scaling laws and their combined scaling law, which in turn validate their asymptotic nature, and corroborate the universality of the relevant random fixed point with respect to the forms of disorder. The dynamic critical exponent z is estimated to 2.114(51), which is compatible with those obtained from experiments and renormalization-group analyses. The exponents at low and high disorder strengths do not satisfy all scaling laws and are argued to be crossover exponents that reflect crossover from the random fixed point to the pure and the percolation fixed point. They also indicate that the exponents that were previously suggested to be a distinct universality class for strong disorder strength in the literature may be just crossover. Our results demonstrate the effectiveness of the finite-time scaling method.