Non-Reversible Markov-Chain Monte Carlo: Theoretical Foundations and Applications in Molecular Simulation

Abstract

The overarching objective of this doctoral thesis is the development of a rigorous paradigm for molecular simulation based on non-reversible Markov-chain Monte Carlo (MCMC) algorithms. The numerical exploration of the thermodynamic equilibrium of complex molecular systems, such as proteins in water, is of enormous importance to numerous fields including physics, chemistry, biology, and engineering. Based on the atomic hypothesis that all matter consists of atoms, molecular-mechanics models describe complex molecular systems by classical atomic sites that interact through empirical potential-energy functions. This enables efficient computer simulations of large-scale molecular systems. As a benchmark for different computational methods, dipolar water models are of particular interest because they appear in many molecular simulations as an explicit aqueous solution and because they typically contain the performance-limiting long-range interactions. <br /> This doctoral thesis discusses theoretical foundations of non-reversible Markov chains in the context of the event-chain Monte Carlo (ECMC) algorithm for an ultimate application to molecular simulation. Non-reversible direction sweeps in an analytically tractable simplified model of a dipole profoundly modify the Markov-chain trajectory and introduce persistent rotations in both directions. The comparison of mixing times indicates that introducing direction sweeps into ECMC can yield faster rotation dynamics of the dipole. For a collection of dipoles, the rotation dynamics are characterized through the integrated autocorrelation time of the polarization. A large-scale numerical benchmark considers thousands of parameter sets for different ECMC variants and reveals remarkable speed differences among them. Escape times from almost locally stable hard-disk configurations are proposed as a model for the analysis of local MCMC algorithms. A scaling theory for the escape times of various ECMC variants separates them into two entirely different classes. The significant speedup of some ECMC variants is rooted in their lack of an intrinsic scale and their event-driven nature. <br /> Motivated by the previous systematic evaluations of the manifold of ECMC variants, this doctoral thesis generalizes the Newtonian ECMC variant to molecular systems. This generalization preserves the fundamental properties of ECMC which enable rigorous molecular simulations in the canonical ensemble. The Boltzmann distribution is strictly sampled by realizing a non-equilibrium system with steady-state probability flows. Long-range interactions are treated without approximations. This doctoral thesis is accompanied by an implementation of generalized Newtonian ECMC in the <span style="font-variant: small-caps">JeLLyFysh</span> application. Simulations of <em>N</em> long-range-interacting water molecules confirm the expected <em>O</em> (<em>N</em> log <em>N</em>) computational complexity. This matches the complexity of state-of-the-art molecular simulation with the widely-used molecular-dynamics (MD) method. However, MD treats long-range interactions inaccurately. <span style="font-variant: small-caps">JeLLyFysh</span> reaches a break-even point with respect to a long-developed standard MD code below machine precision. This proves the competitive efficiency of ECMC or, more generally, of non-reversible MCMC algorithms in a rigorous paradigm for molecular simulation that is free of approximations and unphysical artifacts. It thus promises to become a gold standard for critical applications.