Funnel Hopping Monte Carlo: An efficient method to overcome broken ergodicity

Abstract

Monte Carlo simulations are a powerful tool to investigate the thermodynamic properties of atomic systems. In practice however, sampling of the complete configuration space is often hindered by high energy barriers between different regions of configuration space which can make ergodic sampling completely infeasible within accessible simulation times. Although several extensions to the conventional Monte Carlo scheme have been developed, that enable the treatment of such systems, these extensions often entail substantial computational cost or rely on the harmonic approximation. In this work we propose an exact method called Funnel Hopping Monte Carlo (FHMC) that is inspired by the the ideas of smart darting but is more efficient. Gaussian mixtures are used to approximate the Boltzmann distribution around local energy minima which are then used to propose high quality Monte Carlo moves that enable the Monte Carlo simulation to directly jump between different funnels. To fit the Gaussian mixtures we developed an extended version of the expectation-maximization algorithm that is able to take advantage of the high symmetry present in many low energy configurations. We demonstrate the methods performance on the example of the 38 as well as the 75 atom Lennard-Jones clusters which are well known for their double funnel energy landscapes that prevent ergodic sampling with conventional Monte Carlo simulations. By integrating FHMC into the parallel tempering scheme we were able to reduce the number of steps required until convergence of the simulation significantly.