Driven Depinning of Strongly Disordered Media and Anisotropic Mean-Field Limits

Extended systems driven through strong disorder are modeled generically using coarse-grained degrees of freedom that interact elastically in the directions parallel to the drive and slip along at least one of the directions transverse to the motion. In the limit of infinite-range elastic and viscous coupling this model has a tricritical point separating a region where the depinning is continuous, in the universality class of elastic depinning, from a region where depinning is hysteretic. Many of the collective transport models discussed in the literature are special cases of the generic model.

Figure 1

A two-dimensional realization of the anisotropic driven medium. Degrees of freedom are labeled by discrete indices $\ell$ (parallel to driving force $F$) and $i$ (transverse to $F$). Each degree of freedom interacts with its neighbors via elastic couplings in the longitudinal direction and via viscous or similar slip couplings in the transverse direction.

Figure 2

Velocity curves and phase diagrams for the mean-field slip model. Part (a) shows $v$ vs $F$ curves for the purely viscous case and $\rho (h)=e^{-h}$, for two values of viscous coupling $\eta$. Part (b) shows the phase diagram from (a), with curves indicating jumps in the adiabatic response (i.e., $dv/dF\to \infty$) when the force $F$ is increased from $F=0$ [$F^{\uparrow }(\eta )$] and decreased from a sliding state [$F^{\downarrow }(\eta )$]. The critical point $({\eta }_c,F_c)\approx (9.61,0.278)$. Part (c) shows example $v$ vs $F$ curves when there is elasticity, $K=1$, for piecewise linear pinning forces and $\rho (h)=e^{-h}$. The curves are again continuous for $\eta <{\eta }_c$, but have a depinning transition at $F_T$, with critical behavior identical to the $\eta =0$ case. When $\eta >{\eta }_c$, the depinning is abrupt and the $v$-$F$ curves are hysteretic. Part (d) summarizes the $K=1$, $\rho (h)=e^{-h}$ results in a phase diagram, indicating depinning from an initial pinned state at $F^{\uparrow }=F_T$ and repinning from a sliding state at a lower force $F^{\downarrow }(\eta )$. The point $({\eta }_c,F_T)\approx (3.74,0.298)$ is a tricritical point.