Ergodicity and Energy Distributions for Some Boundary Driven Integrable Hamiltonian Chains

Bálint, Péter; Lin, Kevin K.; Young, Lai Sang

doi:10.1007/s00220-009-0918-x

Cited by 11 publications

(22 citation statements)

References 29 publications

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“…That does not, however, explain why they should be mixtures, or close to mixtures. Similar results have been observed for models with different interactions, see for instance [7][8][9]. To gain some intuition into how mixtures come about as marginal distributions in nonequilibrium chains, we recall below a stochastic model which bears a certain resemblance to Model II, and which also has mixtures as its single-site marginals.…”

Section: Nonequilibrium Steady Statessupporting

confidence: 59%

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Nonequilibrium Steady States of Some Simple 1-D Mechanical Chains

Ryals

Young

2012

J Stat Phys

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We study nonequilibrium steady states of some 1-D mechanical models with N moving particles on a line segment connected to unequal heat baths. For a system in which particles move freely, exchanging energy as they collide with one another, we prove that the mean energy along the chain is constant and equal to 1 2 √ T L T R where T L and T R are the temperatures of the two baths. We then consider systems in which particles are trapped, i.e., each confined to its designated interval in the phase space, but these intervals overlap to permit interaction of neighbors. For these systems, we show numerically that the system has well defined local temperatures and obeys Fourier's Law (with energy-dependent conductivity) provided we vary the masses randomly to enable the repartitioning of energy. Dynamical systems issues that arise in this study are discussed though their resolution is beyond reach.

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Section: Nonequilibrium Steady Statessupporting

confidence: 59%

“…Theorem 2 [9] The model above has a unique steady state with respect to which (i) the mean energy profile is linear, i.e.…”

Section: Nonequilibrium Steady Statesmentioning

confidence: 99%

Nonequilibrium Steady States of Some Simple 1-D Mechanical Chains

Ryals

Young

2012

J Stat Phys

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“…Rigorous derivations of heat conduction laws for mechanical particle models coupled to heat reservoirs remain a mathematical challenge. A variety of models have been introduced in the past [2,9,11,13,18,19,20]; nearly all of the proposed derivations of the Fourier Law are partial solutions based on unproven assumptions [3,4,5,6,13,21]. Developing proofs of these assumptions would require deep understanding of the properties of systems in non-equilibrium, i.e., coupled to several unequal heat reservoirs.…”

Section: Introductionmentioning

confidence: 99%

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

Yarmola

2014

J Stat Phys

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We consider a class of mechanical particle systems with deterministic particle-disk interactions coupled to Gibbs heat reservoirs at possibly different temperatures. We show that there exists a unique (non-equilibrium) steady state. This steady state is mixing, but not exponentially mixing, and all initial distributions converge to it. In addition, for a class of initial distributions, the rates of converge to the steady state are sub-exponential.

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“…The inequality (1) ensures that the values of V at the random images of (r, v ⊥ ) are, on average, smaller than V (r, v ⊥ ) when V (r, v ⊥ ) is large, and, on average, smaller than a constant when V (r, v ⊥ ) is small. This implies the dynamics enters the 'center' of the phase space, represented by some level set of V , regularly with tight control on the length of excursions from the 'center' [6,10].…”

Section: 1mentioning

confidence: 99%

Ergodic Properties of Random Billiards Driven by Thermostats

Khanin¹,

Yarmola²

2013

Commun. Math. Phys.

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Abstract:We consider a class of mechanical particle systems interacting with thermostats. Particles move freely between collisions with disk-shaped thermostats arranged periodically on the torus. Upon collision, an energy exchange occurs, in which a particle exchanges its tangential component of the velocity for a randomly drawn one from the Gaussian distribution with the variance proportional to the temperature of the thermostat. In the case when all temperatures are equal one can write an explicit formula for the stationary distribution. We consider the general case and show that there exists a unique absolutely continuous stationary distribution. Moreover under rather mild conditions on the initial distribution the corresponding Markov dynamics converges to the equilibrium with exponential rate. One of the main technical difficulties is related to a possible overheating of moving particle. However as we show in the paper non-compactness of the particle velocity can be effectively controlled.

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Ergodicity and Energy Distributions for Some Boundary Driven Integrable Hamiltonian Chains

Cited by 11 publications

References 29 publications

Nonequilibrium Steady States of Some Simple 1-D Mechanical Chains

Nonequilibrium Steady States of Some Simple 1-D Mechanical Chains

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

Ergodic Properties of Random Billiards Driven by Thermostats

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