Upscaling a model for the thermally-driven motion of screw dislocations

TL;DR

This paper develops a stochastic model for thermally-driven screw dislocation motion in a cylindrical domain, deriving asymptotic behaviors and explicit rate functionals that connect microscopic lattice effects to macroscopic dislocation dynamics.

Contribution

It introduces a novel Markov jump process model with explicit energy barrier asymptotics and identifies key dimensionless parameters governing dislocation behavior.

Findings

Derivation of energy barrier asymptotics for lattice-based models

Identification of two key dimensionless parameters

Explicit description of the most probable dislocation paths

Abstract

We formulate and study a stochastic model for the thermally-driven motion of interacting straight screw dislocations in a cylindrical domain with a convex polygonal cross-section. Motion is modelled as a Markov jump process, where waiting times for transitions from state to state are assumed to be exponentially distributed with rates expressed in terms of the potential energy barrier between the states. Assuming the energy of the system is described by a discrete lattice model, a precise asymptotic description of the energy barriers between states is obtained. Through scaling of the various physical constants, two dimensionless parameters are identified which govern the behaviour of the resulting stochastic evolution. In an asymptotic regime where these parameters remain fixed, the process is found to satisfy a Large Deviations Principle. A sufficiently explicit description of the corresponding rate functional is obtained such that the most probable path of the dislocation configuration may be described as the solution of Discrete Dislocation Dynamics with an explicit anisotropic mobility which depends on the underlying lattice structure.