Dynamic Critical approach to Self-Organized Criticality

A dynamic scaling Ansatz for the approach to the Self-Organized Critical (SOC) regime is proposed and tested by means of extensive simulations applied to the Bak-Sneppen model (BS), which exhibits robust SOC behavior. Considering the short-time scaling behavior of the density of sites ($\rho(t)$) below the critical value, it is shown that i) starting the dynamics with configurations such that $\rho(t=0) \to 0$ one observes an {\it initial increase} of the density with exponent $\theta = 0.12(2)$; ii) using initial configurations with $\rho(t=0) \to 1$, the density decays with exponent $\delta = 0.47(2)$. It is also shown that he temporal autocorrelation decays with exponent $C_a = 0.35(2)$. Using these, dynamically determined, critical exponents and suitable scaling relationships, all known exponents of the BS model can be obtained, e.g. the dynamical exponent $z = 2.10(5)$, the mass dimension exponent $D = 2.42(5)$, and the exponent of all returns of the activity $\tau_{ALL} = 0.39(2)$, in excellent agreement with values already accepted and obtained within the SOC regime.