Phase space structure and dynamics for the Hamiltonian isokinetic thermostat

TL;DR

This paper analyzes the phase space structure and dynamics of a Hamiltonian isokinetic thermostat, linking it to reaction rate theory, and evaluates its ergodic behavior through model systems at various temperatures.

Contribution

It establishes connections between the thermostat's phase space dynamics and unimolecular reaction rate theory, providing new insights into ergodicity and lifetime distributions.

Findings

Approximate ergodicity condition is satisfied at all studied temperatures.

High temperatures exhibit non-exponential lifetime distributions.

Low temperatures show more nearly exponential lifetime distributions.

Abstract

We investigate the phase space structure and dynamics of a Hamiltonian isokinetic thermostat, for which ergodic thermostat trajectories at fixed (zero) energy generate a canonical distribution in configuration space. Model potentials studied consist of a single bistable mode plus transverse harmonic modes. Interpreting the bistable mode as a reaction (isomerization) coordinate, we establish connections with the theory of unimolecular reaction rates, in particular the formulation of isomerization rates in terms of gap times. The distribution of gap times (or associated lifetimes) for a microcanonical ensemble initiated on the dividing surface is of great dynamical significance; an exponential lifetime distribution is usually taken to be an indicator of `statistical' behavior. Moreover, comparison of the magnitude of the phase space volume swept out by reactive trajectories as they pass through the reactant region with the total phase space volume (classical density of states) for the reactant region provides a necessary condition for ergodic dynamics. We compute gap times, associated lifetime distributions, mean gap times, reactive fluxes, reactive volumes and total reactant phase space volumes for model systems with 3 degrees of freedom, at three different temperatures. At all three temperatures, the necessary condition for ergodicity is approximately satisfied. At high temperatures a non-exponential lifetime distribution is found, while at low temperatures the lifetime is more nearly exponential. The degree of exponentiality of the lifetime distribution is quantified by computing the information entropy deficit with respect to pure exponential decay. The efficacy of the Hamiltonian isokinetic thermostat is examined by computing coordinate distributions averaged over single long trajectories initiated on the dividing surface.