Thermal or stochastic effects are prevalent in physical, chemical, and biological systems. Particularly in small systems, noise can overpower the deterministic dynamics and lead to ``rare events'' that would never be seen in the absence of noise. One example is the thermally-driven switching of the magnetization in small memory elements.

We use Wentzell-Freidlin large deviation theory and concepts from stochastic resonance to analyze magnetic switching. A surprising and physically relevant result is that in multiple-pulse experiments, unconventional ``short-time switching pathways'' can dominate.

One advantage of the method is that it generalizes to systems with spatial variation. To discuss the complications and richness that emerge in the PDE setting, we consider the (simpler) Allen-Cahn equation. The associated action functional and its sharp-interface limit represent a new problem in the calculus of variations. We present some results, including a $\Gamma$-convergence result for the problem in one space dimension.

This talk includes joint work with Bob Kohn, Felix Otto, Yoshihiro Tonegawa, and Eric Vanden--Eijnden.