在自然科學領域，隨機模型(stochastic model)被廣泛應用於描述系統的定量與定性關係。由於地震破裂的複雜性與尺度，決定性的力學方法存在著許多限制。有別於決定性的模型，本研究提出一個本質上包含隨機因子的統計力學形式之地震破裂過程之模型，探討地震滑移背後可能的物理機制。 Thingbaijam 及 Mai (2016) 將世界各地的地震破裂模型之樣式公式化，展示地震各地之滑移分布遵守指數截略分布(Truncated Exponential distribution, TEX)，惟迄今仍缺乏物理定律說明此現象。受到該研究的啟發，我們提出一個基於朗之萬方程式(Langevin equations)之隨機過程模型，其所產生之破裂滑移分布在統計性質上與世界各地的地震事件觀測一致，符合Thingbaijam 及 Mai (2016) 研究結果。 本研究主張在一個地震中，穩態破裂過程可由朗之萬方程式描述，並從此觀點重新詮釋Thingbaijam and Mai (2016)的研究結果，如下所列： TEX擬合參數 u_c 與平均滑移量u_avg相依。 於破裂模型邊緣裁切能增進滑移分布對TEX函數的擬合結果。 地震破裂滑移的TEX分布。
研究結果展示特定朗之萬方程式於有限穩態過程的解析解與地震滑移之指數截略分布(TEX)相同，TEX函數中的擬合參數 u_c 可與朗之萬方程式中的擴散係數與阻尼係數(diffusion and damping coefficient)之比值相關聯。結果暗示著朗之萬方程式可能是主導地震破裂過程的關鍵，而方程式背後存在多種可能的物理意涵，提供我們進一步發掘隨機破裂過程背後物理意義之線索。 ;Due to the capability of simplifying chaotic-like dynamics, stochastic modeling approaches have been widely applied in many domains of natural science. In this study, we propose a stochastic dynamics model for earthquake rupture that intrinsically includes fluctuations in the environment as well as uncertainties in the heterogeneity of the faulting plane in the random variable, and reveal that the responsible Langevin equation (LE) may have a fundamental role in interpreting the physics of earthquake rupture process. Specifically, both analytical and numerical solutions of the governing equation meet the empirically observed Truncated Exponential (TEX) feature of rupture slip distribution. We claim that the stationary part of the earthquake rupture process is governed by the Langevin equation, and accordingly explain the phenomena/facts listed below: (1) The scaling relationship between the parameter uc of TEX and the average co-seismic slips uavg. (2) The overall improvement of the goodness-of-fit of TEX resulting from trimming the original rupture model at edges. (3) The commonly observed truncated exponential distribution for earthquake rupture slips. The result of this study further implies that the fitting parameter uc in TEX is directly related to the ratio of the diffusion and friction coefficient of the Langevin equation, and gives us clues for investigating the physically-based laws underpinning the stochastic rupture process.
