Local-mode approximations in the Frenkel-Kontorova or sine-Gordon chain

Local-mode-approximation methods previously developed for the study of dislocation or kink dynamics in the modified Frenkel-Kontorova model or sine-Gordon chain in the absence of viscosity are here extended to the case in which viscous forces are present. It is found that the effect of viscosity is to increase the accuracy of the local-mode approximation at $T=0$ K. Computer simulation results are presented for the average dislocation velocity as a function of applied stress for $T>0$ K as well as for $T=0$ K. The local-mode approximation leads to a pendulum analogy, involving inelastic collisions between successive pendulums, which shows a close relation between the Brownian motion of kinks in the sine-Gordon chain and the motion of a single Brownian particle in a periodic potential.