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Journal of Statistical Mechanics Theory and Experiment

The effect of finite driving rate on avalanche distributions

Sonja Radić, Sanja Janićević, Dragutin Jovković, Djordje Spasojević

Published Sep 1, 2021• 13 citations
Publisher

Abstract

Abstract We study distributions of spin avalanches in the three-dimensional nonequilibrium zero temperature random field Ising model driven by external magnetic field that is increased at constant rate. Our results show that avalanche distributions are affected by the driving regime and although the avalanche properties (e.g. size) remain power-law distributed above critical disorder, the scaling exponents vary with the driving rate. For slow driving, the integral distributions of nonspanning avalanches are almost the same as in the adiabatic and quasi-static driving regime and therefore described by constant values of power-law exponents. Owing to temporal overlapping and merging of avalanches, these distributions become markedly different with the increased driving rate which causes an increase of exponent values in the range of moderate driving rates and their saturation for fast driving. The integral distributions of all avalanches behave similarly but without the saturation of exponents for fast driving. Our results, based on extensive numerical simulations, could be relevant for studies of variety of systems in finite driving regime, particularly those with experimental realizations that are not easily attainable.

DOI
10.1088/1742-5468/ac1f12

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Abstract We study distributions of spin avalanches in the three-dimensional nonequilibrium zero temperature random field Ising model driven by external magnetic field that is increased at constant rate. Our results show that avalanche distributions are affected by the driving regime and although the avalanche properties (e.g. size) remain power-law distributed above critical disorder, the scaling exponents vary with the driving rate. For slow driving, the integral distributions of nonspanning avalanches are almost the same as in the adiabatic and quasi-static driving regime and therefore described by constant values of power-law exponents. Owing to temporal overlapping and merging of avalanches, these distributions become markedly different with the increased driving rate which causes an increase of exponent values in the range of moderate driving rates and their saturation for fast driving. The integral distributions of all avalanches behave similarly but without the saturation of exponents for fast driving. Our results, based on extensive numerical simulations, could be relevant for studies of variety of systems in finite driving regime, particularly those with experimental realizations that are not easily attainable.
Abstract We analyze the nonequilibrium athermal random field Ising model (RFIM) at equilateral cubic lattices of finite size L and show that the entire range of disorder consists of three distinct domains in which the model manifests different scaling behaviour. The first domain contains the values of disorder R that are below the critical disorder R c where the spanning avalanches almost surely appear when the system is driven by the external magnetic field. The spanning avalanches become unlikely for disorders above the size-dependent effective disorder R c eff ( L ) > R c , and the system response is size-independent. Between the foregoing two lies the domain of transitional disorders R c < R < R c eff ( L ) vanishing in the thermodynamic limit. In this domain, not recognized in the literature so far, all types of spanning avalanches exist, whereas for R < R c only the avalanches spanning all three dimensions are present. Like for R < R c the data collapsing is possible only for distributions having the same value of (1 − R c / R ) L 1/ ν , however with different universal scaling functions than in R < R c case; ν is the correlation length exponent. The foregoing findings follow from the extensive simulations of L ⩽ 2048 systems enabling us to propose modified values of some of the RFIM critical exponents and nonuniversal critical parameters as well as the analytic forms of universal scaling functions and different definitions of reduced magnetization and reduced magnetic field.
Abstract We study the qualitative and quantitative properties of the Barkhausen noise emerging at finite temperatures in random Ising models. The random-bond Ising Model is studied with a Wolff cluster Monte-Carlo algorithm to monitor the avalanches generated by an external driving magnetic field. Satisfactory power-law distributions are found which expand over five decades, with a temperature-dependent critical exponent which matches the existing experimental measurements. We also focus on a Ising system in which a finite fraction of defects is quenched. Also the presence of defects proves able to induce a critical response to a slowly oscillating magnetic field, though in this case the critical exponent associated with the distributions obtained with different defect fractions and temperatures seems to belong to the same universality class, with a critical exponent close to 1.
With respect to the usual thermal ferromagnetic transitions, the zero-temperature finite-disorder critical point of the random field Ising model (RFIM) has the peculiarity of involving some ‘droplet’ exponent θ that enters the generalized hyperscaling relation 2 − α = ν( d − θ). In the present paper, to better understand the meaning of this droplet exponent θ beyond its role in the thermodynamics, we discuss the statistics of low-energy excitations generated by an imposed single spin-flip with respect to the ground state, as well as the statistics of equilibrium avalanches, i.e. the magnetization jumps that occur in the sequence of ground states as a function of the external magnetic field. The droplet scaling theory predicts that the distribution dl / l 1 + θ of the linear size l of low-energy excitations transforms into the distribution ds / s 1 + θ/ d f for the size s (number of spins) of excitations of fractal dimension d f ( s ∼ l d f ). In the non-mean-field region d < d c , droplets are compact d f = d , whereas in the mean-field region d > d c , droplets have a fractal dimension d f = 2θ leading to the well-known mean-field result ds / s 3/2 . Zero-field equilibrium avalanches are expected to display the same distribution ds / s 1 + θ/ d f . We also discuss the statistics of equilibrium avalanches integrated over the external field and finite-size behaviors. These expectations are checked numerically for the Dyson hierarchical version of the RFIM, where the droplet exponent θ(σ) can be varied as a function of the effective long-range interaction J ( r ) ∼ 1/ r d + σ in d = 1.
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.
The critical behavior of the Ising model with nonconserved dynamics and an external driving field mimicking a shear profile is analyzed by studying its dynamical evolution in the short-time regime. Starting from high-temperature disordered configurations (fully disordered configurations, FDC), the critical temperature Tc{T}_{c}Tc​ is determined when the order parameter, defined as the absolute value of the transversal spin profile, exhibits a power-law behavior with an exponent that is a combination of some of the critical exponents of the transition. For each value of the shear field magnitude, labeled as $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\gamma}}$, Tc{T}_{c}Tc​ has been estimated and two stages have been found: (1) a growing stage at low values of $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\gamma}}$, where ${T}_{c}\ensuremath{\sim}{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\gamma}}}^{\ensuremath{\psi}}$ and $\ensuremath{\psi}=0.52(3)$, and (2) a saturation regime at large $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\gamma}}$. The same values of ${T}_{c}(\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\gamma}})$ were found studying the dynamical evolution from the ground-state configuration with all spins pointing in the same direction. By combining the exponents of the corresponding power laws obtained from each initial configuration, the set of critical exponents was calculated. These values, at large external field magnitude, define a critical behavior different from that of the Ising model and of other driven lattice gases.
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.