UC Berkeley

Using tools of statistical mechanics, it is routine to average over the distribution of microscopic configurations to obtain equilibrium free energies. These free energies teach us about the most likely molecular arrangements and the probability of observing deviations from the norm. Frequently, it is necessary to interrogate the probability not just of static arrangements, but of dynamical events, in which case analogous statistical mechanical tools may be applied to study the distribution of molecular trajectories. Numerical study of these trajectory spaces requires algorithms which efficiently sample the possible trajectories. We study in detail one such Monte Carlo algorithm, transition path sampling, and use a non- equilibrium statistical mechanical perspective to illuminate why the algorithm cannot easily be adapted to study some problems involving long-timescale dynamics. Algorithmically generating highly-correlated trajectories, a necessity for transition path sampling, grows exponentially more challenging for longer trajectories unless the dynamics is strongly-guided by the “noise history,” the sequence of random numbers representing the noise terms in the stochastic dynamics. Langevin dynamics of Weeks-Chandler-Andersen (WCA) particles in two dimensions lacks this strong noise guidance, so it is challenging to use transition path sampling to study rare dynamical events in long trajectories of WCA particles. The spin flip dynamics of a two-dimensional Ising model, on the other hand, can be guided by the noise history to achieve efficient path sampling. For systems that can be efficiently sampled with path sampling, we show that it is possible to simultaneously sample both the paths and the (potentially vast) space of non-equilibrium protocols to efficiently learn how rate constants vary with protocols and to identify low-dissipation protocols.

When high-dimensional molecular dynamics can be coarse-grained and represented by a simplified dynamics on a low-dimensional state space, the trajectory space may also be analytically studied using methods of large deviation theory. We review these methods and introduce a simple class of dynamical models whose dynamical fluctuations we compute exactly. The simplest such model is an asymmetric random walker on a one-dimensional ring with a single heterogeneous link connecting two sites of the ring. We derive conditions for the existence of a dynamic phase transition, which separates two dynamical phases—one localized and the other delocalized. The presence of distinct classes trajectories results in profoundly non-Gaussian fluctuations in dynamical quantities. We discuss the implications of such large dynamical fluctuations in the context of simple stochastic models for biological growth.