Analysing First Passage Time Distributions of Large Ill-Conditioned Energy Landscapes
In this thesis we develop theory and associated computational tools to investigate the kinetics of competing pathways on multifunnel energy landscapes. Multifunnel landscapes are associated with molecular switches and multifunctional materials, and are expected to exhibit multiple relaxation time scales. Energy landscapes, consisting of local minima connected by transition states, can be represented by a kinetic transition network. Each local minimum is represented by a node in the network, and two nodes are joined by an edge if a transition state directly connects their associated minima. Each edge has a forwards and backwards branching probability, and a waiting time is associated with each node. This provides a continuous time discrete state Markov chain that represents the network. Our focus is on investigating, understanding, and improving algorithms to compute first passage time (FPT) distributions. First, we introduce new insights into FPT distributions, including showing how the distribution depends on initial conditions, and how features can be assigned to specific kinetic traps. When the state space gets too large, finding the full FPT distribution becomes computationally expensive. We introduce partial Graph Transformation, a network reduction tool that conserves the mean first passage time, and approximately preserves the full first passage time distribution. When there are significant differences in the fastest and slowest transition timescales in the network, the system becomes ill-conditioned. In practical terms, the separation of timescales increases as the system temperature is reduced, which leads to loss of precision in linear algebra computations. We introduce a method to reconstruct the FPT distribution in the ill-conditioned regime, by combining accurate treatment of mean first passage time computations with the reliable short time parts of the FPT distribution from linear algebra approaches. We test our theoretical and computational developments on two model landscapes, and a Lennard-Jones cluster.
Energy Landscapes, First Passage Times, Graph Transformation, Ill-Conditioned Regime, Kinetic Transition Networks, Markov Chains, Network Reduction
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