Gutenberg-Richter-like relations in physical systems

We analyze regional earthquake energy statistics from the Southern California and Japan seismic catalogs and find scale-invariant energy distributions characterized by an exponent $\tau \simeq 1.67$. To quantify how closely scale-invariant dynamics with different exponent values resemble real earthquakes, we generate synthetic energy distributions over a wide range of $\tau$ under conditions of constant activity. Earthquake-like behavior, in a broad sense, is obtained for $1.5 \leqslant \tau < 2.0$. When energy variations are further restricted to be within a factor of ten relative to real earthquakes, the admissible range narrows to $1.58 \leqslant \tau \leqslant 1.76$. We identify the physical mechanisms governing the dynamics in the different regimes: fault dynamics characterized by a balance between slow energy accumulation and release through scale-free events in the earthquake-like regime; externally supplied energy relative to a slowly driven fault for $\tau < 1.5$; and dominance of small events in the energy budget for $\tau > 2$