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Register a new userAcceleration of convergence to equilibrium in Markov chains by breaking detailed balance
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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.
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Among the Markov chains breaking detailed-balance that have been proposed in the field of Monte-Carlo sampling in order to accelerate the convergence towards the steady state with respect to the detailed-balance dynamics, the idea of Lifting consists in duplicating the configuration space into two copies $sigma=pm$ and in imposing directed flows in each copy in order to explore the configuration space more efficiently. The skew-detailed-balance Lifted-Markov-chain introduced by K. S. Turitsyn, M. Chertkov and M. Vucelja [Physica D Nonlinear Phenomena 240 , 410 (2011)] is revisited for the Curie-Weiss mean-field ferromagnetic model, where the dynamics for the magnetization is closed. The large deviations at various levels for empirical time-averaged observables are analyzed and compared with their detailed-balance counterparts, both for the discrete extensive magnetization $M$ and for the continuous intensive magnetization $m=frac{M}{N}$ for large system-size $N$.
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.
Many modern techniques employed in physics, such a computation of path integrals, rely on random walks on graphs that can be represented as Markov chains. Traditionally, estimates of running times of such sampling algorithms are computed using the number of steps in the chain needed to reach the stationary distribution. This quantity is generally defined as mixing time and is often difficult to compute. In this paper, we suggest an alternative estimate based on the Kolmogorov-Sinai entropy, by establishing a link between the maximization of KSE and the minimization of the mixing time. Since KSE are easier to compute in general than mixing time, this link provides a new faster method to approximate the minimum mixing time that could be interesting in computer sciences and statistical physics. Beyond this, our finding will also be of interest to the out-of-equilibrium community, by providing a new rational to select stationary states in out-of-equilibrium physics: it seems reasonable that in a physical system with two simultaneous equiprobable possible dynamics, the final stationary state will be closer to the stationary state corresponding to the fastest dynamics (smallest mixing time).Through the empirical link found in this letter, this state will correspond to a state of maximal Kolmogorov-Sinai entropy. If this is true, this would provide a more satisfying rule for selecting stationary states in complex systems such as climate than the maximization of the entropy production.
We consider nonlinear reaction systems satisfying mass-action kinetics with slow and fast reactions. It is known that the fast-reaction-rate limit can be described by an ODE with Lagrange multipliers and a set of nonlinear constraints that ask the fast reactions to be in equilibrium. Our aim is to study the limiting gradient structure which is available if the reaction system satisfies the detailed-balance condition. The gradient structure on the set of concentration vectors is given in terms of the relative Boltzmann entropy and a cosh-type dissipation potential. We show that a limiting or effective gradient structure can be rigorously derived via EDP convergence, i.e. convergence in the sense of the Energy-Dissipation Principle for gradient flows. In general, the effective entropy will no longer be of Boltzmann type and the reactions will no longer satisfy mass-action kinetics.
A systematic theory of product and diagonal states is developed for tensor products of $mathbb Z_2$-graded $*$-algebras, as well as $mathbb Z_2$-graded $C^*$-algebras. As a preliminary step to achieve this goal, we provide the construction of a {it fermionic $C^*$-tensor product} of $mathbb Z_2$-graded $C^*$-algebras. Twisted duals of positive linear maps between von Neumann algebras are then studied, and applied to solve a positivity problem on the infinite Fermi lattice. Lastly, these results are used to define fermionic detailed balance (which includes the definition for the usual tensor product as a particular case) in general $C^*$-systems with gradation of type $mathbb Z_2$, by viewing such a system as part of a compound system and making use of a diagonal state.
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