Diffusive properties of persistent walks on cubic lattices with application to periodic Lorentz gases

Gilbert, Thomas; Nguyën, Huu Chuong; Sanders, David P.

doi:10.1088/1751-8113/44/6/065001

Cited by 13 publications

(31 citation statements)

References 28 publications

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“…Extensive numerical investigations of the master equation (15) with the stochastic kernel (23) were presented in Ref. [9], supporting with high precision the validity of Eq.…”

Section: Three-dimensional Billiard For Heat Transportmentioning

confidence: 58%

“…which is an exact result for the stochastic system evolved by the master equation (15) and does not make explicit use of the form (16), except for some symmetries 9 . The corresponding result (13) for the billiard dynamics is obtained by plugging back the proper timescale of binary collisions and letting ρ m → ρ c so that the separation of timescales (12) is effective.…”

Section: Heat Transport In Two-dimensional Confining Billiardsmentioning

confidence: 99%

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A two-stage approach to relaxation in billiard systems of locally confined hard spheres

Gaspard

Gilbert

2012

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We consider the three-dimensional dynamics of systems of many interacting hard spheres, each individually confined to a dispersive environment, and show that the macroscopic limit of such systems is characterized by a coefficient of heat conduction whose value reduces to a dimensional formula in the limit of vanishingly small rate of interaction. It is argued that this limit arises from an effective loss of memory. Similarities with the diffusion of a tagged particle in binary mixtures are emphasized. The derivation of the macroscopic transport equations of hydrodynamics and the computation of the associated coefficients for systems described at the microscopic level by Hamilton's equations of classical mechanics is a central problem of non-equilibrium statistical physics. The periodic Lorentz gas provides an example where this program can be achieved and Fick's law of diffusion established 1-3 . Furthermore, by tweaking the system's geometry so that tracer particles hop from cell to cell at nearly vanishing rates, memory effects disappear and the dynamics of tracers is well approximated by a continuous time random walk 4 . In this regime, the diffusion coefficient takes on a simple limiting value, given by a dimensional formula 5 , by which we mean that its expression reduces to the square of the length scale of the cell separation multiplied by the hopping rate. Similarly, it was found that heat transport in systems of confined hard disks with rare interactions reduces to a stochastic process of energy exchanges which obeys Fourier's law of heat conduction, with the coefficient of heat conductivity also given by a dimensional formula 6-8 , where the timescale is that of energy exchanges among particles in neighboring cells. Here we extend these findings to mechanical systems of confined hard spheres, whose stochastic limit was already studied elsewhere 9 , and review in some details the analogy with the problem of mass transport in the periodic Lorentz gas.

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“…Extensive numerical investigations of the master equation (15) with the stochastic kernel (23) were presented in Ref. [9], supporting with high precision the validity of Eq.…”

Section: Three-dimensional Billiard For Heat Transportmentioning

confidence: 58%

Section: Heat Transport In Two-dimensional Confining Billiardsmentioning

confidence: 99%

A two-stage approach to relaxation in billiard systems of locally confined hard spheres

Gaspard

Gilbert

2012

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…It consists of approximating the Taylor-Green-Kubo formula by including memory in a self-consistent, persistent way. Recently this method has been worked out for chaotic diffusion in Hamiltonian particle billiards [28,32,33]. Here we apply this scheme to the different case of a one-dimensional map, and we obtain a series of approximations analytically and then numerically.…”

Section: Introductionmentioning

confidence: 99%

Capturing correlations in chaotic diffusion by approximation methods

Knight

Klages

2011

Phys. Rev. E

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We investigate three different methods for systematically approximating the diffusion coefficient of a deterministic random walk on the line which contains dynamical correlations that change irregularly under parameter variation. Capturing these correlations by incorporating higher order terms, all schemes converge to the analytically exact result. Two of these methods are based on expanding the Taylor-Green-Kubo formula for diffusion, whilst the third method approximates Markov partitions and transition matrices by using a slight variation of the escape rate theory of chaotic diffusion. We check the practicability of the different methods by working them out analytically and numerically for a simple one-dimensional map, study their convergence and critically discuss their usefulness in identifying a possible fractal instability of parameter-dependent diffusion, in case of dynamics where exact results for the diffusion coefficient are not available.

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“…More sophisticated models extend this to take into account some correlations, for example Markov chains with finite memory [46][47][48]. There is also a study of three dimensional Lorentz gases using this approach [49] including both finite and infinite horizon regimes (Sec. 4 below).…”

Section: Open Problemmentioning

confidence: 99%

“…Here, (21) that the normal diffusion coefficient exists (at least in terms of mean square displacement). As with finite horizon this is not accessible in closed form, but may be approximated using correlated random walks [49]. As discussed in Ref.…”

Section: Local Limit Theoremmentioning

confidence: 99%

Diffusion in the Lorentz Gas

Dettmann¹

2014

Commun. Theor. Phys.

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General rightsThis document is made available in accordance with publisher policies. Please cite only the published version using the reference above. The Lorentz gas, a point particle making mirror-like reflections from an extended collection of scatterers, has been a useful model of deterministic diffusion and related statistical properties for over a century. This survey summarises recent results, including periodic and aperiodic models, finite and infinite horizon, external fields, smooth or polygonal obstacles, and in the Boltzmann-Grad limit. New results are given for several moving particles and for obstacles with flat points. Finally, a variety of applications are presented.

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Diffusive properties of persistent walks on cubic lattices with application to periodic Lorentz gases

Cited by 13 publications

References 28 publications

A two-stage approach to relaxation in billiard systems of locally confined hard spheres

A two-stage approach to relaxation in billiard systems of locally confined hard spheres

Capturing correlations in chaotic diffusion by approximation methods

Diffusion in the Lorentz Gas

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