Emergence of asperity-like energy concentration in a stochastic Langevin framework

Modeling earthquake rupture dynamics often requires stochastic approaches to address the impracticality of obtaining analytical solutions for asymmetric many-body systems. Following Langevin's approach, we propose a stochastic dynamic model for the earthquake rupture process, where complexity in degrees of freedom is reduced by introducing a random force to account for uncertainties in fault plane heterogeneity and structural collisions. In this coarse-grained framework, the random term captures unresolved heterogeneity and interactions at a macroscopic system scale; it does not assert that rupture at the scale of specific fault patches is inherently random. Treating the tectonic process as a Coulomb friction process allows this Langevin equation to be viewed as a stochastic variant of Newton’s second law, attributing physical significance to the sample paths.

However, applying a zero-dimensional (0-D) stochastic framework to complex faulting raises a critical conceptual challenge: can a model lacking explicit spatial dimensions reproduce the highly heterogeneous energy distribution observed in nature? Intuition suggests that the exponential slip distribution derived from a 0-D process may not exhibit tail behavior sufficient to satisfy the standard asperity criterion, where a small fraction of the fault area releases most of the seismic energy. To validate the physical basis of the model, we first examine the spectral properties of the synthetic velocity fluctuations. Results demonstrate that the model output is not arbitrary white noise; rather, the velocity spectra exhibit a Lorentzian form characterized by a single corner frequency. This spectral structure indicates that system memory is governed by a characteristic timescale determined by the load ratio, reflecting a competition between frictional dissipation (which erases memory) and external driving (which sustains motion).

Furthermore, we evaluate the steady-state slip distribution derived from the corresponding Fokker–Planck equation against empirical scaling relations for asperities. Adopting the criterion which defines an asperity as regions where slip exceeds 1.5 times the average, and using squared slip as an upper-bound proxy for energy release under elastic loading, we calculate the theoretical energy concentration. The model predicts that the top ∼22% of the statistical "area" contributes ∼81% of the total energy. This theoretical prediction lies within the 20–30% range observed empirically for asperity area fractions. These findings suggest that the concentration of energy in asperities can emerge from stochastic frictional dynamics, arising from the exponential tail of the slip distribution without explicit modeling of spatial heterogeneity.