| This dissertation presents new results concerning two models with deterministic dynamics but involving a stochastic initial condition or environment. The first concerns a well-known family of PDE called scalar conservation laws, where, given a random initial condition, we provide a statistical description of the solution as a stochastic process in the spatial variable at later time. This confirms a special case related to a conjecture by Menon and Srinivasan [G. Menon and R. Srinivasan. "Kinetic Theory and Lax Equations for Shock Clustering and Burgers Turbulence". In: Journal of Statistical Physics 140.6 (2010)]. The second involves a model from condensed matter physics for a one-dimensional elastic structure driven through a periodic environment with quenched phase disorder. In collaboration with M. Mungan [D.C. Kaspar and M. Mungan. Subthreshold behavior and avalanches in an exactly solvable charge density wave system". In: EPL (Europhysics Letters) 103.4 (2013) and D.C. Kaspar and M. Mungan. "Exact results for a toy model exhibiting dynamic criticality". (2014). Submitted.] the author obtained some basic results for this model and a more detailed description for an approximation to it which is a nonstandard sandpile system. We report some of these results here, with some additional introductory material targeted at a mathematical, rather than physical, audience. In each case some questions for future inquiry are identified, and it is argued that rigorous analysis of toy models such as those considered here is productive in the continued development of the field of statistical mechanics. |
