Rare event statistics: competition between pathways for overdamped stochastic dynamical systems

<p dir="ltr">The study of rare events, the events having near-zero probability of occurring, finds various applications in of many biological, chemical, and physical systems. Despite their apparent irrelevance, when rare events occur, they can have substantial and even catastrophic repercussions. In this thesis, we study rare events in overdamped stochastic dynamical systems, a class of mathematical models dealing with stochastic transitions between equilibrium states in a minimal model with two possible competing pathways, starting from a local potential energy minimum and eventually finding the global minimum defined by a potential energy landscape. These stochastic transitions are subsequently extended to consider the case of a pair of interacting particles. The transition depends on a trade-off between state space distance and energy barrier height. One pathway has a higher barrier but a shorter distance, while the alternative has a lower barrier but requires a longer distance. The preferred pathway depends on available time: shorter times favor the route with a higher energy barrier.</p><p dir="ltr">We compute the distribution of barriers crossing points and the time it takes to reach the potential minimum. The dynamics of interacting particles are studied using two spherical symmetric pair potentials; Gaussian and Lennard-Jones focusing on position distribution and arrival time statistics after one of the particles reaches the global minimum. We examine the effects of repulsive and attractive interactions. For a single particle, we find that varying the temperature can switch the most likely pathway. However, despite the fact that the system has complex configuration space statistics, arrival times show a simple almost exponential distribution. For interacting particles, the arrival time again follows an exponential distribution with a much narrower range than in the simple case, attributed to particle ordering statistics. Comparison of particle position distributions for the Lennard-Jones case using Jensen-Shannon divergence and histograms shows both similarities and differences, influenced by internal potential parameters. We also explore how the parameter of the exponential distribution depends on these parameters when modeling particle arrival times.</p>