Beyond Kinetic Relations

TL;DR

This paper introduces kinetic equations as a nonlocal extension of kinetic relations, providing a more comprehensive model for defect dynamics by incorporating internal time scales and system coupling.

Contribution

It proposes a new kinetic equation framework that generalizes traditional kinetic relations by including nonlocal, time-dependent effects in defect dynamics modeling.

Findings

Kinetic equations extend kinetic relations with nonlocal, time-dependent terms.

A detailed model of an overdamped defect in a lattice demonstrates the approach.

The minimal nonlocal kinetic description involves coupled ODEs for defect position and internal parameters.

Abstract

We introduce the concept of kinetic equations representing a natural extension of the more conventional notion of a kinetic relation. Algebraic kinetic relations, widely used to model dynamics of dislocations, cracks and phase boundaries, link the instantaneous value of the velocity of a defect with an instantaneous value of the driving force. The new approach generalizes kinetic relations by implying a relation between the velocity and the driving force which is nonlocal in time. To make this relations explicit one needs to integrate the system of kinetic equations. We illustrate the difference between kinetic relation and kinetic equations by working out in full detail a prototypical model of an overdamped defect in a one-dimensional discrete lattice. We show that the minimal nonlocal kinetic description containing now an internal time scale is furnished by a system of two ordinary differential equations coupling the spatial location of defect with another internal parameter that describes configuration of the core region.