Sub-exponential Mixing of Open Systems with Particle–Disk Interactions

Yarmola, Tatiana

doi:10.1007/s10955-014-1014-y

Cited by 8 publications

(9 citation statements)

References 27 publications

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“…We prove that the Markov process generated by the generalized KMP model has a mixing rate ∼ t −2 and a convergence rate ∼ t −1 to the NESS. The closest related results we know are the slower-than-exponential convergence to the NESS in [38,37,8,9] and the polynomial convergence to the equilibrium in [25]. In addition to the upper bound of convergence, we also showed that the speed of convergence to NESS has a lower bound t −1−γ for any γ > 0.…”

Section: Introductionsupporting

confidence: 55%

On the polynomial convergence rate to nonequilibrium steady states

Li¹

2018

Ann. Appl. Probab.

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We consider a stochastic energy exchange model that models the 1D microscopic heat conduction in the nonequilibrium setting. In this paper, we prove the existence and uniqueness of the nonequilibrium steady state (NESS) and, furthermore, the polynomial speed of convergence to the NESS. Our result shows that the asymptotic properties of this model and its deterministic dynamical system origin are consistent. The proof uses a new technique called the induced chain method. We partition the state space and work on both the Markov chain induced by an "active set" and the tail of return time to this "active set".2010 Mathematics Subject Classification. Primary 60J25, 82C05; Secondary 37N05, 60G07, 82C35.

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Section: Introductionsupporting

confidence: 55%

On the polynomial convergence rate to nonequilibrium steady states

Li¹

2018

Ann. Appl. Probab.

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“…We refer readers to Section 5 and Section 6 for more engaged discussion about microscopic heat conduction models and their connections to deterministic dynamical systems. Besides these two models, other microscopic heat conduction models that have subexponential mixing rate include the particle model in [50,49], the rotor model in [9,10], and the anharmonic chain in [21]. Other examples of sub-exponential rate of convergence have also been observed in various models like MCMC algorithms and random walks [26,47,36].…”

Section: Introductionmentioning

confidence: 99%

Numerical Simulation of Polynomial-Speed Convergence Phenomenon

2017

J Stat Phys

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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 seek numerical justification for the ergodicity of 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.

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“…Under certain assumptions, these models can be described by stochastic differential equations, and results that include the existence and uniqueness of nonequilibirum steady states have been shown [3], as have exponential convergence and other statistical properties [16,17]; see also [1]. A second group consists of results for Hamiltonian models similar to those in [4], with additional assumptions or special features to make the problem more tractable (as noted earlier, Hamiltonian models are much harder), they include: [2,20], which prove ergodicity of the invariant measure assuming existence; [9], which proves existence and uniqueness for a model in which all energy exchanges are exclusively with 'thermostats' (or heat baths); and, [21], which treats a model with special geometry. A third group of results we know of consists of [10] and variants of this model [11,15].…”

Section: Introductionmentioning

confidence: 99%

Nonequilibrium steady states for a class of particle systems

Young²

2014

Nonlinearity

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This paper contains rigorous results on nonequilibrium steady states for a class of particle systems coupled to unequal heat baths. These stochastic models are derived from the mechanical chains studied by Eckmann and Young by randomizing certain quantities while retaining other features of the model. Our results include the existence and uniqueness of nonequilibrium steady states, their relation to Lebesgue measure, tail bounds on total energy and number of particles in the system, and exponential convergence to steady states from suitable initial conditions.

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Sub-exponential Mixing of Open Systems with Particle–Disk Interactions

Cited by 8 publications

References 27 publications

On the polynomial convergence rate to nonequilibrium steady states

On the polynomial convergence rate to nonequilibrium steady states

Numerical Simulation of Polynomial-Speed Convergence Phenomenon

Nonequilibrium steady states for a class of particle systems

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