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Register a new userIrreversible local Markov chains with rapid convergence towards equilibrium
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We study the continuous one-dimensional hard-sphere model and present irreversible local Markov chains that mix on faster time scales than the reversible heatbath or Metropolis algorithms. The mixing time scales appear to fall into two distinct universality classes, both faster than for reversible local Markov chains. The event-chain algorithm, the infinitesimal limit of one of these Markov chains, belongs to the class presenting the fastest decay. For the lattice-gas limit of the hard-sphere model, reversible local Markov chains correspond to the symmetric simple exclusion process (SEP) with periodic boundary conditions. The two universality classes for irreversible Markov chains are realized by the totally asymmetric simple exclusion process (TASEP), and by a faster variant (lifted TASEP) that we propose here. Lifted Markov chains and the recently introduced factorized Metropolis acceptance rule extend the irreversible Markov chains discussed here to general pair interactions and to higher dimensions.
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The event-chain Monte Carlo (ECMC) method is an irreversible Markov process based on the factorized Metropolis filter and the concept of lifted Markov chains. Here, ECMC is applied to all-atom models of multi-particle interactions that include the long-ranged Coulomb potential. We discuss a line-charge model for the Coulomb potential and demonstrate its equivalence with the standard Coulomb model with tin-foil boundary conditions. Efficient factorization schemes for the potentials used in all-atom water models are presented, before we discuss the best choice for lifting schemes for factors of more than three particles. The factorization and lifting schemes are then applied to simulations of point-charge and charged-dipole Coulomb gases, as well as to small systems of liquid water. For a locally charge-neutral system in three dimensions, the algorithmic complexity is O(N log N) in the number N of particles. In ECMC, a Particle-Particle method, it is achieved without the interpolating mesh required for the efficient implementation of other modern Coulomb algorithms. An event-driven, cell-veto-based implementation samples the equilibrium Boltzmann distribution using neither time-step approximations nor spatial cutoffs on the range of the interaction potentials. We discuss prospects and challenges for ECMC in soft condensed-matter and biological physics.
We analyze the convergence of the irreversible event-chain Monte Carlo algorithm for continuous spin models in the presence of topological excitations. In the two-dimensional XY model, we show that the local nature of the Markov-chain dynamics leads to slow decay of vortex-antivortex correlations while spin waves decorrelate very quickly. Using a Frechet description of the maximum vortex-antivortex distance, we quantify the contributions of topological excitations to the equilibrium correlations, and show that they vary from a dynamical critical exponent z sim 2 at the critical temperature to z sim 0 in the limit of zero temperature. We confirm the event-chain algorithms fast relaxation (corresponding to z = 0) of spin waves in the harmonic approximation to the XY model. Mixing times (describing the approach towards equilibrium from the least favorable initial state) however remain much larger than equilibrium correlation times at low temperatures. We also describe the respective influence of topological monopole-antimonopole excitations and of spin waves on the event-chain dynamics in the three-dimensional Heisenberg model.
We analyse and interpret the effects of breaking detailed balance on the convergence to equilibrium of conservative interacting particle systems and their hydrodynamic scaling limits. For finite systems of interacting particles, we review existing results showing that irreversible processes converge faster to their steady state than reversible ones. We show how this behaviour appears in the hydrodynamic limit of such processes, as described by macroscopic fluctuation theory, and we provide a quantitative expression for the acceleration of convergence in this setting. We give a geometrical interpretation of this acceleration, in terms of currents that are emph{antisymmetric} under time-reversal and orthogonal to the free energy gradient, which act to drive the system away from states where (reversible) gradient-descent dynamics result in slow convergence to equilibrium.
We discuss a canonical structure that provides a unifying description of dynamical large deviations for irreversible finite state Markov chains (continuous time), Onsager theory, and Macroscopic Fluctuation Theory. For Markov chains, this theory involves a non-linear relation between probability currents and their conjugate forces. Within this framework, we show how the forces can be split into two components, which are orthogonal to each other, in a generalised sense. This splitting allows a decomposition of the pathwise rate function into three terms, which have physical interpretations in terms of dissipation and convergence to equilibrium. Similar decompositions hold for rate functions at level 2 and level 2.5. These results clarify how bounds on entropy production and fluctuation theorems emerge from the underlying dynamical rules. We discuss how these results for Markov chains are related to similar structures within Macroscopic Fluctuation Theory, which describes hydrodynamic limits of such microscopic models.c Fluctuation Theory, which describes hydrodynamic limits of such microscopic models.
An overarching action principle, the principle of minimal free action, exists for ergodic Markov chain dynamics. Using this principle and the Detailed Fluctuation Theorem, we construct a dynamic ensemble theory for non-equilibrium steady states (NESS) of Markov chains, which is in full analogy with equilibrium canonical ensemble theory. Concepts such as energy, free energy, Boltzmann macro-sates, entropy, and thermodynamic limit all have their dynamic counterparts. For reversible Markov chains, minimization of Boltzmann free action yields thermal equilibrium states, and hence provide a dynamic justification of the principle of minimal free energy. For irreversible Markov chains, minimization of Boltzmann free action selects the stable NESS, and determines its macroscopic properties, including entropy production. A quadratic approximation of free action leads to linear-response theory with reciprocal relations built-in. Hence, in so much as non-equilibrium phenomena can be modeled as Markov processes, minimal free action serves as a basic principle for both equilibrium and non-equilibrium statistical physics.
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