Mixing rates of particle systems with energy exchange

  title={Mixing rates of particle systems with energy exchange},
  author={A. Grigo and K. Khanin and D. Sz{\'a}sz},
A fundamental problem of non-equilibrium statistical mechanics is the derivation of macroscopic transport equations in the hydrodynamic limit. The rigorous study of such limits requires detailed information about rates of convergence to equilibrium for finite sized systems. In this paper, we consider the finite lattice {1, 2, ..., N}, with an energy xi ∈ (0, ∞) associated with each site. The energies evolve according to a Markov jump process with nearest neighbour interaction such that the… Expand
Thermal conductivity for stochastic energy exchange models
We consider a class of stochastic models for energy transport and study relations between the thermal conductivity and some static observables, such as the static conductivity, which is defined asExpand
This paper studies a billiards-like microscopic heat conduction model, which describes the dynamics of gas molecules in a long tube with thermalized boundary. We numerically investigate the law ofExpand
On the Limiting Markov Process of Energy Exchanges in a Rarely Interacting Ball-Piston Gas
We analyse the process of energy exchanges generated by the elastic collisions between a point-particle, confined to a two-dimensional cell with convex boundaries, and a ‘piston’, i.e. aExpand
Dynamical contribution to the heat conductivity in stochastic energy exchanges of locally confined gases
We present a systematic computation of the heat conductivity of the Markov jump process modeling the energy exchanges in an array of locally confined hard spheres at the conduction threshold. BasedExpand
Equilibration of energy in slow–fast systems
It is shown that violation of ergodicity in the fast dynamics can drive the whole system to equilibrium, and a set of mechanical toy models—the springy billiards—are introduced and stochastic processes corresponding to their adiabatic behavior are described. Expand
Polynomial convergence to equilibrium for a system of interacting particles
We consider a stochastic particle system in which a finite number of particles interact with one another via a common energy tank. Interaction rate for each particle is proportional to the squareExpand
Spectral gap for stochastic energy exchange model with nonuniformly positive rate function
We give a lower bound on the spectral gap for a class of stochastic energy exchange models. In 2011, Grigo et al. introduced the model and showed that, for a class of stochastic energy exchangeExpand
Nonequilibrium steady states for a class of particle systems
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 chainsExpand
On the polynomial convergence rate to nonequilibrium steady-states
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 nonequilibriumExpand
From deterministic dynamics to thermodynamic laws II: Fourier's law and mesoscopic limit equation.
  • Yao Li
  • Physics, Mathematics
  • 2019
This paper consider the mesoscopic limit of a stochastic energy exchange model that is numerically derived from deterministic dynamics. The law of large numbers and the central limit theorems areExpand


On the derivation of Fourier’s law in stochastic energy exchange systems
We present a detailed derivation of Fourier's law in a class of stochastic energy exchange systems that naturally characterize two-dimensional mechanical systems of locally confined particles inExpand
On the Hydrodynamic Limit of a Scalar Ginzburg-Landau Lattice Model: The Resolvent Approach
A d-dimensional lattice system of continuous spins is considered, the evolution law is given by an infinite system of locally coupled stochastic differential equations. In view of the construction ofExpand
Heat conduction and Fourier's law in a class of many particle dispersing billiards
We consider the motion of many confined billiard balls in interaction and discuss their transport and chaotic properties. In spite of the absence of mass transport, due to confinement, energyExpand
Heat transport in stochastic energy exchange models of locally confined hard spheres
We study heat transport in a class of stochastic energy exchange systems that characterize the interactions of networks of locally trapped hard spheres under the assumption that neighbouringExpand
Scaling Limits of Interacting Particle Systems
1. An Introductory Example: Independent Random Walks.- 2. Some Interacting Particle Systems.- 3. Weak Formulations of Local Equilibrium.- 4. Hydrodynamic Equation of Symmetric Simple ExclusionExpand
Nonequilibrium Energy Profiles for a Class of 1-D Models
As a paradigm for heat conduction in 1 dimension, we propose a class of models represented by chains of identical cells, each one of which contains an energy storage device called a ``tank''. EnergyExpand
Kinetically Constrained Lattice Gases
Kinetically constrained lattice gases (KCLG) are interacting particle systems which show some of the key features of the liquid/glass transition and, more generally, of glassy dynamics. TheirExpand
Heat flow in an exactly solvable model
A chain of one-dimensional oscillators is considered. They are mechanically uncoupled and interact via a stochastic process which redistributes the energy between nearest neighbors. The total energyExpand
Determination of the spectral gap for Kac's master equation and related stochastic evolution
We present a method for bounding, and in some cases computing, the spectral gap for systems of many particles evolving under the influence of a random collision mechanism. In particular, the methodExpand
Determination of the Spectral Gap in the Kac Model for Physical Momentum and Energy-Conserving Collisions
The Kac model describes the local evolution of a gas of N particles with three-dimensional velocities by a random walk in which the steps correspond to binary collisions that conserve momentum as well as energy, and the Kac conjecture concerns the spectral gap in the one-step transition operator Q. Expand