Superdiffusive Heat Transport in a Class of Deterministic One-dimensional Many-Particle Lorentz Gases

Collet, Pierre; Eckmann, Jean‐Pierre; Mejía‐Monasterio, Carlos

doi:10.1007/s10955-009-9783-4

Cited by 10 publications

(23 citation statements)

References 25 publications

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“…These models have to some degree motivated the idea of energies at "sites" in the present paper. Related works in systems defined by dynamical local rules include [4,5,6,7,17]. For stochastic models, the collection of results is much larger, and we mention only some that are closer to this paper in spirit: [1,2,8,9,10,16,18,19,23,24,26,29].…”

Section: Introductionmentioning

confidence: 96%

Ergodicity and Energy Distributions for Some Boundary Driven Integrable Hamiltonian Chains

Bálint

Lin

Young

2009

Commun. Math. Phys.

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Abstract. We consider systems of moving particles in 1-dimensional space interacting through energy storage sites. The ends of the systems are coupled to heat baths, and resulting steady states are studied. When the two heat baths are equal, an explicit formula for the (unique) equilibrium distribution is given. The bulk of the paper concerns nonequilibrium steady states, i.e., when the chain is coupled to two unequal heat baths. Rigorous results including ergodicity are proved. Numerical studies are carried out for two types of bath distributions. For chains driven by exponential baths, our main finding is that the system does not approach local thermodynamic equilibrium as system size tends to infinity. For bath distributions that are sharply peaked Gaussians, in spite of the near-integrable dynamics, transport properties are found to be more normal than expected.

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Section: Introductionmentioning

confidence: 96%

Ergodicity and Energy Distributions for Some Boundary Driven Integrable Hamiltonian Chains

Bálint

Lin

Young

2009

Commun. Math. Phys.

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“…[18] and extend the framework to autonomous collisional baths. The present paper can be considered a first step towards this goal.Classical collisional baths have been also analyzed in detail as a modification of the Lorentz gas, where the scatterers are fixed disks that can rotate with some angular velocity [19,20,21,22,23]. The collisions between the gas particles and the disks are inelastic because of an exchange of energy between the incident particle and the rotation of the scatterer.…”

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confidence: 99%

“…The collisions between the gas particles and the disks are inelastic because of an exchange of energy between the incident particle and the rotation of the scatterer. This type of models has been used to study different non-equilibrium phenomena like heat transport [20,21,22] and thermalization [19,23].Collisional baths are also interesting regarding the probability distribution of the incident particles. Consider a one-dimensional ideal gas in equilibrium at temperature T .…”

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confidence: 99%

“…Unfortunately, some of the proposed models, such as the one in Ref. [22], thermalize when the velocity distribution of the incident particles is indeed Maxwellian, which, at first sight, only adds to the confusion.In this paper we aim at clarifying these issues by a careful formulation of the micro-reversibility condition and its relation to Liouville's theorem and conservation of phase-space volume in classical and semiclassical collisional reservoirs. To this end the paper is structured as follows: In the following two sections 2 and 3, we review the concept of micro-reversibility and its consequences.…”

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confidence: 99%

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Micro-reversibility and thermalization with collisional baths

Ehrich

Esposito

Barra

et al. 2020

Physica A: Statistical Mechanics and its Applications

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Micro-reversibility plays a central role in thermodynamics and statistical mechanics. It is used to prove that systems in contact with a thermal bath relax to canonical ensembles. However, a problem arises when trying to reproduce this proof for classical and quantum collisional baths, i.e. particles at equilibrium interacting with a localized system via collisions. In particular, micro-reversibility appears to be broken and some models do not thermalize when interacting with Maxwellian particles. We clarify these issues by showing that micro-reversibility needs the invariance of evolution equations under time reversal plus the conservation of phase space volume in classical and semiclassical scenarios. Consequently, all canonical variables must be considered to ensure thermalization. This includes the position of the incident particles which maps their Maxwellian distribution to the effusion distribution. Finally, we show an example of seemingly plausible collision rules that do not conserve phase-space volume, and consequently violate the second law.The first issue that we address concerns the very formulation of the micro-reversibility condition. The generic statement of micro-reversibility is that the probability to observe a transition Γ → Γ is equal to the probability of the reverse transitionΓ →Γ. Here, Γ and Γ are arbitrary microscopic states of a physical system andΓ the time-reversal state of Γ. The mathematical expression of this statement reads (we provide a more detailed description later on, in Sec. 2):However, this equality immediately poses a problem if the variable Γ is continuous. In that case, ρ(Γ |Γ) is a density in Γ , whereas ρ(Γ|Γ ) is a density inΓ. Consequently, when the two conditional probabilities are compared, one has to take into account the transformation of the volume elements dΓ and dΓ . In classical systems, Liouville's theorem warrants the conservation of volume, implying dΓ = dΓ , and resolves the problem. But micro-reversibility is also relevant for quantum and semi-classical systems with states parametrized by continuous variables Γ, such as Wigner distributions in phase space or wave packets centered around a given position and velocity [9,10]. In the first case, for instance, it has been shown that there is no equivalent Liouville-like theorem, that is, unitary evolution does not necessarily conserve the phase-space volume [11]. Therefore, it is necessary to explicitly check the micro-reversibility condition, Eq. (1), in those quantum and semi-classical scenarios.The second issue concerns the role of micro-reversibility in the relaxation of a system towards equilibrium. For a classical system in contact with a thermal bath, the micro-reversibility condition applies to micro-states Γ = (x, y) of the global system, consisting of the system itself (x) and the bath (y). The bath variables can be eliminated by multiplying Eq. (1) by the equilibrium distribution ρ eq (y) and integrating over y and y . This procedure, which we describe in detail in Sec. 2.1, is especially relevant...

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