Central limit theorem for the functional of jump Markov process

Central limit theorem for the functional of jump Markov process Nguyễn Văn Hữu, Vương Quân Hoàng và Trần Minh NgọcBáo cáo: Hội nghị toàn quốc lần thứ III “Xác suất - Thống kê: Nghiên cứu, ứng dụng và giảng dạy” (tr. 34)Ba Vì, Hà Tây, ngày 12-14 tháng 05 năm 2005Viện Toán họcTrường Đại học Khoa học tự nhiên / Đại học Quốc gia Hà Nội

Representing stochastic damage evolution in disordered media as a jump Markov process using the fiber bundle model

The damage evolution in quasi-brittle materials is inherently stochastic due to the presence of strong disorder in the form of heterogeneities, voids, and microcracks. The final macroscopic failure is foreshadowed by accumulation of a significant amount of distributed damage that results in precursory events observed as avalanches in experiments and simulations. Simulations on spring lattice models of disordered media have been widely used to understand the collective nature of the quasi-brittle material failure process. In this study, we use the jump Markov process to model stochastic damage evolution, which is informed by the avalanche size distributions for a given material. The jump Markov process is defined based on the probability distributions of the jump sizes, the wait-time between consecutive jumps, and the failure strength. The fiber bundle model is used as an example to obtain the required inputs and test the viability of the proposed approach. The stochasticity and size-dependence of the damage evolution process is inherently captured through the inputs provided for the jump Markov process. The avalanche and strength distributions are used to describe the effect of microscopic information present in the form of disorder, on the macroscopic damage evolution behavior.

Jump Markov models and transition state theory: the quasi-stationary distribution approach

We are interested in the connection between a metastable continuous state space Markov process (satisfyinge.g.the Langevin or overdamped Langevin equation) and a jump Markov process in a discrete state space. More precisely, we use the notion of quasi-stationary distribution within a metastable state for the continuous state space Markov process to parametrize the exit event from the state. This approach is useful to analyze and justify methods which use the jump Markov process underlying a metastable dynamics as a support to efficiently sample the state-to-state dynamics (accelerated dynamics techniques). Moreover, it is possible by this approach to quantify the error on the exit event when the parametrization of the jump Markov model is based on the Eyring–Kramers formula. This therefore provides a mathematical framework to justify the use of transition state theory and the Eyring–Kramers formula to build kinetic Monte Carlo or Markov state models.