A Stochastic Model for Earthquake Rupture

Analytical solutions for the dynamics of asymmetric many-body systems are impractical to obtain, and numerical solutions usually exhibit chaotic behavior if interactions between bodies are considered. To address these challenges, stochastic approaches have been widely employed in modelling many-body systems. Following Langevin’s approach, we propose a stochastic dynamic model for the earthquake rupture process, in which complexity in degrees of freedom is reduced by introducing a random force that accounts for uncertainties in faulting 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. Under this approach, treating the tectonic process as a Coulomb friction process allows the proposed Langevin equation to be viewed as a stochastic variant of Newton’s second law, thereby attributing physical significance to the equation through the realization of stochastic processes as sample paths. This study analyzes synthetic events generated numerically with the Langevin equation to determine the energy–duration relationship and solves the corresponding Fokker–Planck equation to obtain the theoretical rupture slip distribution. The results for the energy–duration relationship show a consistent scaling law within our additive-noise, Coulomb-damping framework, with the scaling exponent varying under different slip velocity thresholds used to define synthetic events. Regarding the slip distribution, solutions obtained assuming relatively large external driving forces align closely with the truncated exponential model, characterizing rupture models of large earthquake events worldwide. The proposed Langevin equation provides a physical basis that offers insights into the scaling parameter of such laws and suggests a connection between the scaling parameter and the dissipative and environmental noise effects of the faulting system.