Gaps between avalanches in 1D 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-neighbour ferromagnetic RFIM. We map the sequence of avalanches in this system to a non-homogeneous 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 $-\infty$ to $+\infty$. We show that $P(ΔH)$ tends to a constant $\mathcal{C}(R)$ as $ΔH \to 0^+$, which displays a non-trivial behaviour 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 behaviour $P(ΔH) = 0$ up to a system size dependent offset value $ΔH_{\text{off}}$, and $P(ΔH) \sim (ΔH - ΔH_{\text{off}})^θ$ as $ΔH \to H_{\text{off}}^+$. We perform numerical simulations on this model and determine $θ\approx 0.95(5)$. We also discuss mechanisms which would lead to a non-zero exponent $θ$ for general spin models with quenched random fields.