Comparing the efficiencies of stochastic isothermal molecular dynamics methods

When molecular dynamics is used to find time-dependent properties, such as correlation functions, at a prescribed 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 as quickly as possible. 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, and carry 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.