Pinning-depinning transition in dislocation dynamics

We present a series of models used to analyze the motion of dislocations in ${\mathrm{L}1\mathrm{}}_2$ intermetallic compounds, which show a yield strength anomaly (increasing strength with increasing temperature). The models are presented in terms of stochastic, finite difference equations of motion and are based on physical arguments from previous work. The solutions show that the models display a pinning-depinning transition with increasing stress. The pinned phase is shown to have a large number of possible configurations. A set of deterministic, mean-field equations are derived from the stochastic equations, and used to analyze some of the statistical properties of the models. We also examine numerical solutions to the stochastic equations, and analyze them for critical exponents and scaling. Finite-size effects are likely to be very important in the practical application of these models.