Algorithms for Measuring Extremely Rare Events in Statistical Physics

An event that occurs infrequently is called a "rare event." Some rare events can be of significant interest. Rare events occur in numerous contexts: disease extinction, diffusion into a labyrinth, first-order phase transitions, protein folding, queue overflow, etc. Because rare events occur infrequently, they are nearly impossible to accurately measure using direct simulation methods. We have developed several algorithms for measuring these rare events. Our algorithms fall into two categories: algorithms for measuring rare diffusion events and algorithms for measuring long transition times. We use the algorithms in the first category to measure the complete distribution of diffusing random walkers landing on the surface of a fractal; this is called the harmonic measure. We obtain the harmonic measure for two- and three-dimensional random fractals, focusing on critical percolation and diffusion limited aggregation (DLA) clusters. Using these methods we obtain extremely small probabilities, as small as 10 -4600 and find excellent agreement with theory where possible. The subject of rare long transition times has a long history and has received considerable theoretical and computational interest. Dozens of algorithms have been developed for measuring these events, the vast majority of which can only be applied to systems with detailed balance, e.g., systems with energy landscapes. We developed several general algorithms that can be applied to systems with and without detailed balance. We use these methods to measure the time it takes a disease to go extinct within a population, the poisoning time of the Ziff-Gulari-Barshad (ZGB) model of heterogeneous catalysis, and the transition time of a bi-stable non-equilibrium model. In addition to transition times, these methods give insight into the transition pathway, which we use to determine the most likely transition path in the ZGB model.