Comparing the Efficiencies of Stochastic Isothermal Molecular Dynamics Methods

Abstract

Molecular dynamics typically incorporates a stochastic-dynamical device, a "thermostat," in order to drive the system to the Gibbs (canonical) distribution at a prescribed temperature. When molecular dynamics is used to compute time-dependent properties, such as autocorrelation functions or diffusion constants, at given temperature, there is a conflict between the need for the thermostat to perturb the time evolution of the system as little as possible and the need to establish equilibrium rapidly. In this article we define a quantity called the "efficiency" of a thermostat which relates the perturbation introduced by the thermostat to the rate of convergence of average kinetic energy to its equilibrium value. We show how to estimate this quantity analytically, carrying out the analysis for several thermostats, including the Nosé-Hoover-Langevin thermostat due to Samoletov et al (J.Stat. Phys. 128, 1321-1336 , 2007) and a generalization of the "stochastic velocity rescaling" method suggested by Bussi et al (J. Chem. Phys 126, 014101, 2007). We find efficiency improvements (proportional to the number of degrees of freedom) for the new schemes compared to Langevin Dynamics.Numerical experiments are presented which precisely confirm our theoretical estimates.

The Langevin Limit of the Nosé-Hoover-Langevin Thermostat

“…We also point out that the reduced Langevin equation (21) is equivalent to the momentum directed Langevin thermostat described in [12] as a generalization of the stochastic velocity rescaling thermostat of Bussi et al [2,3].…”

“…It is remarkable that the limiting Langevin process that we derive here by means of a stochastic singular perturbation analysis [19] has been heuristically proposed before e.g. by Leimkuhler et al [12] and Bussi et al [2,3] to construct a thermostat with a mild effect on the dynamics, only along the direction of motion.…”

Stochastic-dynamical thermostats for constraints and stiff restraints

“…In this vein, we here demonstrate in Sect. 4 that results of [19] on the smaller autocorrelation error of the Nosé-Hoover-Langevin method compared to Langevin dynamics carry over to the constrained setting.…”

“…staying as close as possible to microcanonical dynamics. This topic has been studied in detail in a recent article [19]. If the convergence to equilibrium of two methods is similar, then the problem is to compare the accuracy of autocorrelation functions produced by the methods, as a measure of the efficiency of the thermostat.…”

“…In [19] it was shown that the NHL dynamics allows a more accurate computation of velocity auto-correlation functions (VAF) in the asymptotic limit of small correlation times.…”