A method to project the rate kinetics of high dimensional barrier crossing problems onto a tractable 1D system

Abstract

We derive a technique to project barrier crossing simulation algorithms onto a one-dimensional system. In this 1D system we assume that the dynamics are dominated by moves between immediately adjacent states. The technique extracts the key kinetic information from a barrier crossing simulation. This determines the 1D system with identical properties to the multi-dimensional simulation when projected onto a 1D order parameter. The resulting 1D system has a known analytic solution. We apply the technique to the Graham and Olmsted model of flow-induced nucleation in polymers [R. S. Graham and P. D. Olmsted, Phys. Rev. Lett., 2009, 103, 115702]. This reveals a simple pattern in the model's 1D projected kinetics, which leads to a general expression for the model's kinetics. This ultimately enables us to derive a simple, cheap and accurate analytic expression for the barrier crossing rate. Under extreme flow conditions, nucleation in this model becomes highly anisotropic, which gives a clear signature when our 1D projection technique is applied.