Effective dynamics along given reaction coordinates, and reaction rate theory

Abstract

In molecular dynamics and related fields one considers dynamical descriptions of complex systems in full (atomic) detail. In order to reduce the overwhelming complexity of realistic systems (high dimension, large timescale spread, limited computational resources) the projection of the full dynamics onto some reaction coordinates is examined in order to extract statistical information like free energies or reaction rates. In this context, the effective dynamics that is induced by the full dynamics on the reaction coordinate space has attracted considerable attention in the literature. In this article, we contribute to this discussion: we first show that if we start with an ergodic diffusion process whose invariant measure is unique then these properties are inherited by the effective dynamics. Then, we give equations for the effective dynamics, discuss whether the dominant timescales and reaction rates inferred from the effective dynamics are accurate approximations of such quantities for the full dynamics, and compare our findings to results from approaches like Mori-Zwanzig, averaging, or homogenization. Finally, by discussing the algorithmic realization of the effective dynamics, we demonstrate that recent algorithmic techniques like the "equation-free" approach and the "heterogeneous multiscale method" can be seen as special cases of our approach.

Reliable Approximation of Long Relaxation Timescales in Molecular Dynamics

“…The results presented subsequently can be extended to more general reversible diffusion processes with a state-dependent noise intensity matrix, cf. [23]. However, for the sake of simplicity of presentation we restrict our considerations to the specific case (1) typically studied in molecular dynamics.…”

“…Following this work, we characterize the approximation quality for the (low-lying) eigenvalues of the infinitesimal generator. This permits us to study the connection between the effective dynamics considered in [23] and Galerkin discretization schemes for the transfer operator. Secondly, following the work [7,8], we study the variational approach for the infinitesimal generator.…”

“…Various advanced methods such as umbrella sampling [11,12], metadynamics [13,14], blue moon sampling [15], the adaptive biasing force method [16], or temperature-accelerated molecular dynamics (TAMD) [17], as well as trajectory-based techniques like milestoning [18], transition interface sampling [19], or forward flux sampling [20] may serve as some examples. These methods result in free energy barriers, transition rates, or first mean passage times for the rare events of interest; they are complemented by several approaches to the effective dynamics of the reaction coordinate space [21][22][23] that allow for significantly faster simulation of these rare events [24][25][26] including details of the underlying molecular mechanisms. Surprisingly, our analytic tools, originally developed for discrete coarse graining approaches, can also be utilized for evaluating the approximation quality of reaction coordinate sampling approaches to the effective dynamics.…”

Variational Characterization of Free Energy: Theory and Algorithms

“…Letting now P denote the path space measure that is generated by the Brownian motion (B s ) s≥0 in the parameter dependent SDE (31), we can express Jarzynski's equality (30) by…”

“…An idea, inspired by the adaptive biasing force (ABF) algorithm [26][27][28] is to estimate γ on the fly and then iteratively refine the estimate in the course of the simulation using a suitable parametric representation [29,30]. If good collective variables or reaction coordinates are known, it is further possible to choose a representation that depends only on these variables and still obtain low variance estimators [31,32].…”