Non-Ergodicity of Nosé-Hoover Chain Dynamics

Ergodicity of the dynamics is a prerequisite for obtaining ensemble averaged results from a single time trajectory, making it a key requirement for calculating statistical mechanical properties through molecular dynamics simulations. Nosé-Hoover chain (NHC) thermostat is a popular way of constraining temperature in molecular dynamics simulations. Previously, it was thought that NHC imparts ergodicity to the dynamics, thereby removing the deficiencies of several other thermostats. However, very recently, doubt has been casted on the ergodic nature of dynamics due to a two-chain NHC thermostat. In this work, we build upon this doubt and show, with the help of a single harmonic oscillator, that even after long time duration, the conditional univariate distribution functions of position and velocity do not reach a Gaussian distribution in several Poincare sections. The deviations from the Gaussian distribution are computed using the symmetric form of Kullback-Leibler divergence, Hellinger distance and by analysing the moments. The significant deviation of the conditional univariate distribution functions from a Gaussian distribution suggests that the dynamics due to NHC thermostat is non-ergodic although it constrains the temperature of the single harmonic oscillator to a desired value in a long time averaged sense.