Note on a Metrically Transitive System

We show that simple diffusive systems, such as the Lorentz gas and multibaker maps are perfectly compatible with the laws of irreversible thermodynamics, despite the fact that the moving particles, or their equivalents, in these models do not interact with each other, and that the dynamics takes place in low-dimensional phase spaces. The interaction of moving particles with scatterers provides the dynamical mechanism responsible for an approach to equilibrium, under appropriate conditions. This analysis provides a refutation of the criticisms expressed recently by Cohen and Rondoni [Physica A 306 (2002) 117-128].

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“…The baker map was introduced in the thirties in the work by Seidel[13] who seems to have been inspired by Birkhoff.…”

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confidence: 99%

Diffusive Lorentz gases and multibaker maps are compatible with irreversible thermodynamics

Gaspard

Nicolis

Dorfman

2003

Physica A: Statistical Mechanics and its Applications

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“…We refer to it as the 2D-baker map. It was introduced by Seidel [18], where the square is divided into q equal vertical strips. Seidel used q = 10.…”

Section: Baker Mapsmentioning

confidence: 99%

Piecewise linear maps with heterogeneous chaos

2021

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Chaotic dynamics can be quite heterogeneous in the sense that in some regions the dynamics are unstable in more directions than in other regions. When trajectories wander between these regions, the dynamics is complicated. We say a chaotic invariant set is heterogeneous when arbitrarily close to each point of the set there are different periodic points with different numbers of unstable dimensions. We call such dynamics heterogeneous chaos (or hetero-chaos). While we believe it is common for physical systems to be hetero-chaotic, few explicit examples have been proved to be hetero-chaotic. Here we present two explicit dynamical systems that are particularly simple and tractable with computer. It will give more intuition as to how complex even simple systems can be. Our maps have one dense set of periodic points whose orbits are 1D unstable and another dense set of periodic points whose orbits are 2D unstable. Moreover, they are ergodic relative to the Lebesgue measure.

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“…A natural approach to understanding complicated systems is to analyse a collection of simple examples that retain some essential features of the complexity of the original systems. The baker map [6] is one of the simplest models of chaotic dynamical systems. It divides the unit square into p ⩾ 2 equal vertical strips, and maps each strip to a horizontal one by squeezing it vertically by the factor p and stretching it horizontally by the same factor.…”

Section: Introductionmentioning

confidence: 99%

The twisted baker map

2023

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As a model to provide a hands-on, elementary understanding of ‘vortex dynamics’, we introduce a piecewise linear non-invertible map called a twisted baker map. We show that the set of hyperbolic repelling periodic points with complex conjugate eigenvalues and that without complex conjugate eigenvalues are simultaneously dense in the phase space. We also show that these two sets equidistribute with respect to the normalised Lebesgue measure, in spite of a non-uniformity in their Lyapunov exponents.

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Note on a Metrically Transitive System

Cited by 9 publications

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Diffusive Lorentz gases and multibaker maps are compatible with irreversible thermodynamics

Diffusive Lorentz gases and multibaker maps are compatible with irreversible thermodynamics

Piecewise linear maps with heterogeneous chaos

The twisted baker map

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