A Stochastic Model for Earthquake Rupture

Abstract 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.By treating the tectonic process as a Coulomb friction process, the proposed Langevin equation can 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 suggest the existence of a universal scaling law, 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 establishes a physical foundation that elucidates the physics governing the scaling parameter of such laws and reveals the connection between the scaling parameter and the dissipative and environmental noise effects of the faulting system.