#Artificial academy 2 lag full#
Mathematically, the line of maximum probability flux is a curve in the phase space, and a calculation of the full committor is practically infeasible. (2) This observation provides a critical insight in the formulation of the string method, (3−5) which seeks to determine the dominant “reaction tube” that contains most of the probability current between A and B.
This analysis leads to the observation that the principal lines of a reactive probability current between the states A and B are largely determined by the equilibrium probability times the local gradiant of the committor.
(1) This “committor” probability, which can be determined on the basis of the backward dynamical propagation, then becomes a critical ingredient in efforts to formulate a theoretical framework seeking to treat such problems. Further analysis shows that the steady-state probability of the states under such nonequilibrium conditions can be expressed as the product of the equilibrium probability of the states times the probability that a trajectory initiated at the same position will be reactive and first reach the state B before ever reaching the state A. In the context of a multistate Markov model, the steady-state flux from A to B can be expressed as the net sum of productive transitions across a dividing surface between the two end states. Many of the key concepts can be formulated by considering a prototypical system comprising two dominant metastable states A and B. The present formulation provides a powerful theoretical framework to characterize the optimal pathway between two metastable states of a system.Ī central problem in computational biophysics is the characterization of the long-time kinetic behavior of molecular systems. The time-correlation functions calculated by swarms-of-trajectories along the string pathway constitutes a natural extension of these developments. The effective propagator with finite lag time is amenable to an eigenvalue-eigenvector spectral analysis, as elaborated previously in the context of position-based Markov models. It is argued that the conditions for Markovity within a subspace of collective variables may not be satisfied with an arbitrary short time-step and that proper kinetic behaviors appear only when considering the effective propagator for longer lag times. The theoretical formulation is exploited to clarify the foundation of the string method with swarms-of-trajectories, which relies on the mean drift of short trajectories to determine the optimal transition pathway. A quadratic expression for the steady-state flux between the two metastable states can serve as a robust variational principle to determine an optimal approximate committor expressed in terms of a set of collective variables. Dimensionality reduction to a subspace of collective variables yields familiar expressions for the propagator, committor, and steady-state flux. The kinetics of a dynamical system comprising two metastable states is formulated in terms of a finite-time propagator in phase space (position and velocity) adapted to the underdamped Langevin equation.