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Physical Review E

Avalanche distributions in the two-dimensional nonequilibrium zero-temperature random field Ising model

Djordje Spasojević, Sanja Janićević, Milan Knežević

Published Nov 17, 2011• 46 citations
Publisher

Abstract

We present in detail the scaling analysis and data collapse of avalanche distributions and joint distributions that characterize the recently evidenced critical behavior of the two-dimensional nonequilibrium zero-temperature random field Ising model. The distributions are collected in extensive simulations of systems with linear sizes up to L=131072.

DOI
10.1103/physreve.84.051119

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We present in detail the scaling analysis and data collapse of avalanche distributions and joint distributions that characterize the recently evidenced critical behavior of the two-dimensional nonequilibrium zero-temperature random field Ising model. The distributions are collected in extensive simulations of systems with linear sizes up to L=131072.
We present a numerical analysis of spanning avalanches in a two-dimensional (2D) nonequilibrium zero-temperature random field Ising model. Finite-size scaling analysis, performed for distribution of the average number of spanning avalanches per single run, spanning avalanche size distribution, average size of spanning avalanche, and contribution of spanning avalanches to magnetization jump, is augmented by analysis of spanning field (i.e., field triggering spanning avalanche), which enabled us to collapse averaged magnetization curves below critical disorder. Our study, based on extensive simulations of sufficiently large systems, reveals the dominant role of subcritical 2D-spanning avalanches in model behavior below and at the critical disorder. Other types of avalanches influence finite systems, but their contribution for large systems remains small or vanish.
We present a numerical study of the critical behavior of the nonequilibrium zero-temperature random field Ising model in two dimensions on a triangular lattice. Our findings, based on the scaling analysis and collapse of data collected in extensive simulations of systems with linear sizes up to L=65536, show that the model is in a different universality class than the same model on a quadratic lattice, which is relevant for a better understanding of model universality and the analysis of experimental data.
We explain Barkhausen noise in magnetic systems in terms of avalanches of domains near a plain old critical point in the hysteretic zero-temperature random-field Ising model. The avalanche size distribution has a universal scaling function, making nontrivial predictions of the shape of the distribution up to 50% above the critical point, where two decades of scaling are still observed. We simulate systems with up to 10003{1000}^{3}10003 domains, extract critical exponents in 2, 3, 4, and 5 dimensions, compare with our 2D and $6\ensuremath{-}\ensuremath{\epsilon}$ predictions, and compare to a variety of experiments.
We investigate the critical properties of the d=3d=3d=3 random-field Ising model with a Gaussian field distribution at zero temperature. By implementing suitable graph-theoretical algorithms, we perform a large-scale numerical simulation of the model for a vast range of values of the disorder strength hhh and system sizes $\mathcal{V}=L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L$, with $L\ensuremath{\le}156$. Using the sample-to-sample fluctuations of various quantities and proper finite-size scaling techniques we estimate with high accuracy the critical disorder strength hc{h}_{\mathrm{c}}hc​ and the correlation length exponent $\ensuremath{\nu}$. Additional simulations in the area of the estimated critical-field strength and relevant scaling analysis of the bond energy suggest bounds for the specific heat critical exponent $\ensuremath{\alpha}$ and the violation of the hyperscaling exponent $\ensuremath{\theta}$. Finally, a data collapse analysis of the order parameter and disconnected susceptibility provides accurate estimates for the critical exponent ratios $\ensuremath{\beta}/\ensuremath{\nu}$ and $\overline{\ensuremath{\gamma}}/\ensuremath{\nu}$, respectively.
The non-equilibrium random-field Ising model is well studied, yet there are outstanding questions. In two dimensions, power law scaling approaches fail and the critical disorder is difficult to pin down. Additionally, the presence of faceting on the square lattice creates avalanches that are lattice dependent at small scales. We propose two methods which we find solve these issues. First, we perform large scale simulations on a Voronoi lattice to mitigate the effects of faceting. Secondly, the invariant arguments of the universal scaling functions necessary to perform scaling collapses can be directly determined using our recent normal form theory of the Renormalization Group. This method has proven useful in cleanly capturing the complex behavior which occurs in both the lower and upper critical dimensions of systems and here captures the 2D NE-RFIM behavior well. The obtained scaling collapses span over a range of a factor of ten in the disorder and a factor of 10410^4104 in avalanche cutoff. They are consistent with a critical disorder at zero and with a lower critical dimension for the model equal to two.