Gaps between avalanches in one-dimensional random-field Ising models

Abstract

We analyze the statistics of gaps (ΔH) between successive avalanches in one-dimensional random-field Ising models (RFIMs) in an external field H at zero temperature. In the first part of the paper we study the nearest-neighbor ferromagnetic RFIM. We map the sequence of avalanches in this system to a nonhomogeneous Poisson process with an H-dependent rate ρ(H). We use this to analytically compute the distribution of gaps P(ΔH) between avalanches as the field is increased monotonically from -∞ to +∞. We show that P(ΔH) tends to a constant C(R) as ΔH→0^{+}, which displays a nontrivial behavior with the strength of disorder R. We verify our predictions with numerical simulations. In the second part of the paper, motivated by avalanche gap distributions in driven disordered amorphous solids, we study a long-range antiferromagnetic RFIM. This model displays a gapped behavior P(ΔH)=0 up to a system size dependent offset value ΔH_{off}, and P(ΔH)∼(ΔH-ΔH_{off})^{θ} as ΔH→H_{off}^{+}. We perform numerical simulations on this model and determine θ≈0.95(5). We also discuss mechanisms which would lead to a nonzero exponent θ for general spin models with quenched random fields.

Distribution of interevent avalanche times in disordered and frustrated spin systems

“…It likewise van- ishes for the SKM and VBM. This value for the exponent is the same as the one-dimensional RFIM [17]. It has been argued that this is a consequence of the mapping between the RFIM and a depinning process when the dimensionality is less than 5 [39].…”

“…We also calculate the pseudogap exponent θ, given by P (∆B) ∼ (∆B) θ as ∆B → 0. This follows the procedure described in recent literature [17] on the one-dimensional RFIM.…”

“…7, showing C(R), exhibits a peak for R ≈ 3.7. Note that for the analytical calculation of the one-dimensional case, C(R) peaks at R ≈ 1 [17]. The flatness of P (∆B) as ∆B → 0 implies that the pseudogap exponent θ = 0 for all R in the RFIM.…”

“…The use of simulations allows us to generalize to higher dimensions a recent analytical study by Nampoothiri et al [17] on the interevent time distribution of the one-dimensional random-field Ising model. The central result of that work was the computation of the distribution of times P (∆B) of the magnetic field change ∆B between spin avalanches.…”

“…(If the magnetic field is increased at constant rate, ∆B is proportional to time.) It was found that P (∆B) ∼ (∆B) θ as ∆B → 0 [17], with θ = 0 for the short-range ferromagnetic random-field Ising model, whereas θ = 0.95 for the long-range antiferromagnetic case. Other studies of the distribution of gaps between events have been conducted in On a large scale, the curves appears to represent a smooth evolution of the magnetization M (B).…”