Tuning coupling: Discrete changes in runaway avalanche sizes in disordered media

Hysteretic systems may exhibit a runaway avalanche in which a large fraction of the constituents of the system collectively change state. It would be very valuable to understand the role that interaction strength between constituents plays in the size of such catastrophic runaway avalanches. We use a simple model, the random field Ising model, to study how the size of the runaway avalanche changes as the coupling between spins, J, is tuned. In particular, we calculate P(S), the distribution of size changes S in the runaway avalanche size as J comes close to a critical value J_c, and find that the distribution scales as P(S)∼S^-τD(Sσ(J/J _c-1)), with τ and σ critical exponents and D(x) a universal scaling function. In mean field theory we find τ=3/2, σ=1/2, and D(x)=exp[-⁽3x⁾1^/σ/2]. On the basis of these results and previous studies, we also predict that for three dimensions τ=1.6 and σ=0.24.