Gaps between avalanches in 1D Random Field Ising Models

Abstract

We analyze the statistics of gaps ($\Delta H$) 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 $\rho (H)$. We use this to analytically compute the distribution of gaps $P(\Delta H)$ between avalanches as the field is increased monotonically from $-\infty$ to $+\infty$. We show that $P(\Delta H)$ tends to a constant $\mathcal{C}(R)$ as $\Delta H\to 0^{+}$, 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 $P(\Delta H)=0$ up to a system size dependent offset value $\Delta H_{\text{off}}$, and $P(\Delta H)\sim (\Delta H-\Delta H_{\text{off}}{)}^{\theta }$ as $\Delta H\to H_{\text{off}}^{+}$. We perform numerical simulations on this model and determine $\theta \approx 0.95(5)$. We also discuss mechanisms which would lead to a non-zero exponent $\theta$ for general spin models with quenched random fields.

arXiv comment16 pages, 16 figures

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• Distribution of interevent avalanche times in disordered and frustrated spin systems
• Hysteresis and return point memory in the random field Blume Capel model
• Discontinuity in the distribution of field increments between avalanches in non-abelian random field Blume-Emery-Griffiths model with no passing violation