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

Abstract

We analyze the statistics of gaps (Δ H ) between successive avalanches in one-dimensional random-field Using 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 non homogeneous Poisson process with an H -dependent rate ρ (H ) . We use this to analytically compute the distribution of gaps P (Δ H ) between avalanches as the field is increased monotonically from -∞ to +∞ . We show that P (Δ H ) tends to a constant C (R ) as Δ H →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 gaped behavior P (Δ H )=0 up to a system size dependent offset value Δ Hoff , and P (Δ H ) ˜(ΔH -Δ Hoff) θ as Δ H →Hoff+ . 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.