Distribution of Avalanche Sizes in the Hysteretic Response of Random Field Ising Model on a Bethe Lattice at Zero Temperature

TL;DR

This paper derives exact equations for avalanche size distributions in the zero-temperature random-field Ising model on a Bethe lattice, revealing exponential decay and critical behavior near discontinuities.

Contribution

It provides explicit solutions for avalanche distributions on Bethe lattices with specific disorder distributions, highlighting critical phenomena and phase transition characteristics.

Findings

Avalanche size distribution decays exponentially for large s.

Discontinuities in magnetization occur for z > 3 with small disorder.

Near discontinuities, Prob(s) scales as s^{-3/2}.

Abstract

We consider the zero-temperature single-spin-flip dynamics of the random-field Ising model on a Bethe lattice in the presence of an external field h. We derive the exact self-consistent equations to determine the distribution Prob(s) of avalanche sizes s, as the external field increases from large negative to positive values. We solve these equations explicitly for a rectangular distribution of the random fields for a linear chain and the Bethe lattice of coordination number z=3, and show that in these cases, Prob(s) decreases exponentially with s for large s for all h on the hysteresis loop. We found that for z >3 and for small disorder, the magnetization shows a first order discontinuity for several continuous and unimodel distributions of random fields. The avalanche distribution Prob(s) varies as s^{-3/2} for large s near the discontinuity.