Numerical simulation of polynomial-speed convergence phenomenon

Abstract

We provide a hybrid method that captures the polynomial speed of convergence and polynomial speed of mixing for Markov processes. The hybrid method that we introduce is based on the coupling technique and renewal theory. We propose to replace some estimates in classical results about the ergodicity of Markov processes by numerical simulations when the corresponding analytical proof is difficult. After that, all remaining conclusions can be derived from rigorous analysis. Then we apply our results to two 1D microscopic heat conduction models. The mixing rate of these two models are expected to be polynomial but very difficult to prove. In both examples, our numerical results match the expected polynomial mixing rate well.

References in corpus (15)

• Non-Equilibrium Statistical Mechanics of Anharmonic Chains Coupled to Two Heat Baths at Different Temperatures
• Large Deviation of the Density Profile in the Steady State of the Open Symmetric Simple Exclusion Process
• Spectral gaps in Wasserstein distances and the 2D stochastic Navier--Stokes equations
• Quantitative bounds on convergence of time-inhomogeneous Markov chains
• Sufficient burn-in for Gibbs samplers for a hierarchical random effects model
• Asymptotic Behavior of Thermal Non-Equilibrium Steady States for a Driven Chain of Anharmonic Oscillators
• Nonequilibrium Energy Profiles for a Class of 1-D Models
• Heat conduction and Fourier's law by consecutive local mixing and thermalization
• Non-equilibrium steady states for chains of four rotors
• Non-equilibrium steady state and subgeometric ergodicity for a chain of three coupled rotors
• A practical criterion for positivity of transition densities
• Mixing rates of particle systems with energy exchange
• Sub-exponential mixing of random billiards driven by thermostats
• Reflections on Gibbs: From Statistical Physics to the Amistad
• Sub-exponential mixing of open systems with particle-disk interactions

Cited by in corpus (2)

• Numerical computations of geometric ergodicity for stochastic dynamics
• From billiards to thermodynamic laws: I. Stochastic energy exchange model