Stochastic differential equation molecular dynamics

Recently Calderon introduced a surrogate process approximation (SPA) to improve the sampling in calculation of the JE. The scheme is applied to the study of the unravelling of deca-alanine at constant temperature in a steered molecular dynamics simulation. The distribution of the work is approximated by developing a model for the dynamics using a relatively small number of real trajectories in conjunction with stochastic differential equations selected to model the process. The... 

At its most basic level, molecular dynamics is about mapping out complicated point sets using trajectories of a system of ordinary differential equations (or, in Chaps. 6-8, a stochastic-differential equation system). The sets are typically defined as the collection of probable states for a certain system. In the case of Hamiltonian dynamics, they are directly associated to a region of the energy landscape. The trajectories are the means by which we efficiently explore the energy surface. In this chapter we address the design of numerical methods to calculate trajectories. 

It is widely appreciated that chemical and biochemical reactions in the condensed phase are stochastic. It has been more than 60 years since Delbriick studied a stochastic chemical reaction system in terms of the chemical master equation. Kramers theory, which connects the rate of a chemical reaction with the molecular structures and energies of the reactants, is established as a central component of theoretical chemistry [77], Yet study of the dynamics of chemical and biochemical reaction systems, in terms of either deterministic differential equations or the stochastic CME, is not the exclusive domain of chemists. Recent developments in the simulation of reaction systems are the work of many sorts of scientists, ranging from control engineers to microbiologists, all interested in the dynamic behavior of biochemical reaction systems [199, 210],... 

The positions are obtained by numerically solving differential equations. Hence, these positions are connected in time. The positions reveal real dynamics of individual molecules. In other simulation methods, the molecular positions are not temporarily related. In other simulation methods, such as Monte Carlo simulations, the positions are generated stochastically such that a molecular configuration depends only on the previous configuration [7-10],... 

This paper has focused on two recent computer methods for discrete simulation of chemical kinetics. Beginning with the realization that truly microscopic computer experiments are not at all feasible, I have tried to motivate the development of a hierarchy of simulations in studies of a class of chemical problems which best illustrate the absolute necessity for simulation at levels above molecular dynamics. It is anticipated (optimistically ) that the parallel development of discrete event simulations at different levels of description may ultimately provide a practical interface between microscopic physics and macroscopic chemistry in complex physicochemical systems. With the addition to microscopic molecular dynamics of successively higher-level simulations intermediate between molecular dynamics at one extreme and differential equations at the other, it should be possible to examine explicitly the validity of assumptions invoked at each stage in passing from the molecular level to the stochastic description and finally to the macroscopic formulation of chemical reaction kinetics. 

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