Chaotic scattering and diffusion in the Lorentz gas

Gaspard, Pierre; Baras, F.

doi:10.1103/physreve.51.5332

Cited by 95 publications

(121 citation statements)

References 47 publications

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“…The concepts used in this section have been discussed in great detail in Ref. [3,7]. In each specific system the probability P (Ω, t|x 0 ) for a given trajectory can be expressed in terms of the transition probabilities w(x|y):…”

Section: Thermodynamic Formalismmentioning

confidence: 99%

“…Several chaotic quantities can be derived from this function [3,7]. For example: the sum of all positive Lyapunov exponents is λ ≡ (+) …”

Section: Thermodynamic Formalismmentioning

confidence: 99%

“…This formalism has been successfully applied to Lorentz gases [7,8,9,10]. These are models in which independent light particles are moving among fixed scatterers.…”

Section: Introductionmentioning

confidence: 99%

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Thermodynamic formalism and localization in Lorentz gases and hopping models

et al. 1997

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The thermodynamic formalism expresses chaotic properties of dynamical systems in terms of the Ruelle pressure ψ(β). The inversetemperature like variable β allows one to scan the structure of the probability distribution in the dynamic phase space. This formalism is applied here to a Lorentz Lattice Gas, where a particle moving on a lattice of size L d collides with fixed scatterers placed at random locations. Here we give rigorous arguments that the Ruelle pressure in the limit of infinite systems has two branches joining with a slope discontinuity at β = 1. The low and high β-branches correspond to localization of trajectories on respectively the "most chaotic" (highest density) region, and the "most deterministic" (lowest density) region, i.e. ψ(β) is completely controlled by rare fluctuations in the distribution of scatterers on the lattice, and it does not carry any information on the global structure of the static disorder.As β approaches unity from either side, a localization-delocalization transition leads to a state where trajectories are extended and carry information on transport properties. At finite L the narrow region around β = 1 where the trajectories are extended scales as (ln L) −α , where α depends on the sign of 1 − β, if d > 1, and as (L ln L) −1 if d = 1. This result appears to be general for diffusive systems with static disorder, such as random walks in random environments or for 1 permanent address: CNRS, LPS, Ecole Normale Supérieure, 24 rue Lhomond, 75231 PARIS Cedex 05, France 1 the continuous Lorentz gas. Other models of random walks on disordered lattices, showing the same phenomenon, are discussed.

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Section: Thermodynamic Formalismmentioning

confidence: 99%

“…Several chaotic quantities can be derived from this function [3,7]. For example: the sum of all positive Lyapunov exponents is λ ≡ (+) …”

Section: Thermodynamic Formalismmentioning

confidence: 99%

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Thermodynamic formalism and localization in Lorentz gases and hopping models

et al. 1997

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“…(39), has yet another consequence of some considerable interest, namely, we express f as f ( r, p, t) = 1/V ( r, p, t), where V is the volume of a small phase space region about ( r, p) where the system is located at time t. Then the equation for f ,ḟ = (d − 1)αf , easily obtained from Eqs. (37,39), leads to…”

Section: B the Gaussian Thermostat Methodsmentioning

confidence: 99%

“…The escape-rate method has been applied to compute the properties of the fractal repeller for a number of model systems, including the multibaker model for diffusion in one dimensional systems [38], the two dimensional periodic Lorentz gas at sufficiently high density that the free path length of the moving particle is small compared to the dimensions of the system [36,39], to the random Lorentz gas at low densities [40,41], but still with a mean free path small compared to the size of the system, and to Lorentz lattice gases [42].…”

Section: Fokker-planck Equation Coincides With Thediffusion Equationmentioning

confidence: 99%

Deterministic chaos and the foundations of the kinetic theory of gases

Dorfman

1998

Physics Reports

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Recent work in dynamical systems theory has shown that many properties that are associated with irreversible processes in fluids can be understood in terms of the dynamical properties of reversible, Hamiltonian systems. That is, stochastic-like behavior is possible for these systems. Here we review the basic theory for this stochastic-like behavior and show how it may be used to obtain an understanding of irreversible processes in gases and fluids. Recent, closely related, work on the use of kinetic theory to calculate dynamical quantities such as Lyapunov exponents is also discussed.

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Diffusion and the Poincaré‐Birkhoff Mapping of Chaotic Systems

Gaspard¹

2002

Advances in Chemical Physics

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Chaotic scattering and diffusion in the Lorentz gas

Cited by 95 publications

References 47 publications

Thermodynamic formalism and localization in Lorentz gases and hopping models

Thermodynamic formalism and localization in Lorentz gases and hopping models

Deterministic chaos and the foundations of the kinetic theory of gases

Diffusion and the Poincaré‐Birkhoff Mapping of Chaotic Systems

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