Accelerated importance sampling of energy landscapes with discrete symmetries and barriers

When the energy landscape of a thermodynamic system consists of multiple wells separated by large barriers, the convergence rate of the standard Metropolis Markov chain Monte Carlo (MCMC) method can suffer from slow convergence. Here we show that in many cases, energy barriers are a consequence of discrete (approximate) symmetries in the Hamiltonian and develop a group-theoretic based sampling algorithm which uses knowledge of these symmetries to accelerate convergence of MCMC. We show that this group theoretic sampling approach is a generalization of the well-known and highly successful clustering algorithms. Importantly, this method does not rely on the symmetries being exact, but merely approximate, so that the approach is more broadly applicable and can consider the effects of symmetry breaking. We present results from Hamiltonians with discrete reflection, rotational, and translational approximate symmetries for which the proposed method convergences much more rapidly than either standard or umbrella sampling. We conclude by outlining how the proposed method is well-suited for studying polymers subject to electromagnetic fields with potential phase transitions, which has far reaching applications in energy harvesting, soft bio-inspired robotics and biomedical devices.