An algorithm is presented to compute time scales of complex processes following predetermined milestones along a reaction coordinate. A non-Markovian hopping mechanism is assumed and constructed from underlying microscopic dynamics. General analytical analysis, a pedagogical example, and numerical solutions of the non-Markovian model are presented. No assumption is made in the theoretical derivation on the type of microscopic dynamics along the reaction coordinate. However, the detailed calculations are for Brownian dynamics in which the velocities are uncorrelated in time (but spatial memory remains).

The generalized master equation or the equivalent continuous time random walk equations can be used to compute the macroscopic first passage time distribution (FPTD) of a complex stochastic system from short-term microscopic simulation data. The computation of the mean first passage time and additional low-order FPTD moments can be simplified by directly relating the FPTD moment generating function to the moments of the local FPTD matrix. This relationship can be physically interpreted in terms of steady-state relaxation, an extension of steady-state flow. Moreover, it is amenable to a statistical error analysis that can be used to significantly increase computational efficiency. The efficiency improvement can be extended to the FPTD itself by modelling it using a Gamma distribution or rational function approximation to its Laplace transform.

Abstract

A continuum level simulation of dislocation dynamics has been developed and used to study the evolution of Frank-Read sources. The model is based on isotropic elasticity theory and includes the self-stress of the dislocations, as well as the dislocation-dislocation interactions. The critical stress of a Frank-Read source is calculated as a function of the source length. The simulations are used to study the strain produced by a Frank-Read source as a function of time. It is noted that the total area swept out by the source as a function of time t scales as t3. This scaling appears to hold for a linear dependence of the dislocation velocity on stress, as well as for a stress cubed dependence. It is noted that under some conditions, a subcritically stressed Frank-Read source may produce strain for a significant length of time before complete exhaustion of its motion.