| In this thesis, we study the dynamics of a manifold being pulled through random media. We consider the effect of the bulk degrees of freedom of the medium on the motion of the manifold. In the first chapter, we review what is known about this system when bulk elasticity is included but not elastic waves. Then the manifold undergoes a continuous transition from a pinned to a moving phase as the applied force F is increased through its critical value.; In the second chapter, we study this transition when the effects of elastic waves are included. Elastic waves create temporary overshoots in the stresses on the manifold above their static value. We study these stress overshoots within a mean field theory and demonstrate that, even in the presence of overshoots, the depinning transition stays continuous. Just like in the transition without elastic waves, there is no hysteresis, at least in mean field theory. However, for larger stress overshoots, the transition becomes discontinuous with hysteresis after passing through a novel tricritical point. These results are supported by the study of avalanches---precursor events to depinning.; In the third chapter, we numerically study the effects of small stress overshoots beyond mean field theory. If the transition is approached from the moving phase, it is continuous. However, as F is increased back up again, unlike the case without elastic waves, we find that the manifold's velocity stays zero until a much larger force, where it jumps into the moving phase discontinuously. This unusual hysteresis is understood through several simpler stress overshoot models, and we find that M is an irrelevant parameter despite the hysteresis. We also study the avalanches and find evidence for the discontinuity in the transition as F is increased to its critical value.; In the fourth chapter, we present a heuristic derivation of the first passage time exponent for the integral of a random walk. This result is applicable to the distribution of avalanche sizes studied in Chapter 2. Based on this derivation we construct an estimation scheme for the first passage time exponent for the nth integral of a random walk. |
