Emergence of distinguishability of patterns of collisions of particles in a non-equilibrium chaotic system

Vidgop, Alexander Jonathan; Fouxon, Itzhak

doi:10.1016/j.physa.2014.06.017

“…Distributions of particle positions converge quickly to Gaussianity as the number of particles is increased. Similar results with non-trivial patterns are found in stochastic models in higher dimension [8,9].…”

Section: Introductionsupporting

confidence: 87%

Compression and information entropy of binary strings from the collision history of three hard balls

Matej¹,

Ackland²

2023

Preprint

0

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We investigate how to measure and define the entropy of a simple chaotic system, three hard spheres on a ring. A novel approach is presented, which does not assume the ergodic hypothesis. It consists of transforming the particles' collision history into a sequence of binary digits. We then investigate three approaches which should demonstrate the non-randomness of these collision-generated strings compared with random number generator created strings: Shannon entropy, diehard randomness tests and compression percentage. We show that the Shannon information entropy is unable to distinguish random from deterministic strings. The Diehard test performs better, but for certain mass-ratios the collision-generated strings are misidentified as random with high confidence. The zlib and bz2 compression algorithms are efficient at detecting non-randomness and low information content, with compression efficiencies that tend to 100% in the limit of infinite strings. Thus "compression algorithm entropy" is non-extensive for this the chaotic system, in marked contrast to the extensive entropy determined from phase-space integrals by assuming ergodicity.

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“…Distributions of particle positions converge quickly to Gaussianity as the number of particles is increased. Similar results with non-trivial patterns are found in stochastic models in higher dimension [10,11]. This system has been shown to be equivalent to a billiard problem in a triangular stadium [12].…”

Section: Introductionsupporting

confidence: 78%

Compression and information entropy of binary strings from the collision history of three hard balls

Matej

¹

,

Ackland

²

2023

J. Phys. Commun.

1

0

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We investigate how to measure and define the entropy of a simple chaotic system, three hard spheres on a ring. A novel approach is presented, which does not assume the ergodic hypothesis. It consists of transforming the particles’ collision history into a sequence of binary digits. We then investigate three approaches which should demonstrate the non-randomness of these collision-generated strings compared with random number generator created strings: Shannon entropy, diehard randomness tests and compression percentage. We investigate how to measure and define the entropy of a simple chaotic system, three hard spheres on a ring. A novel approach is presented, which does not assume the ergodic hypothesis. It consists of transforming the particles’ collision history into a sequence of binary digits. We then investigate three approaches which should demonstrate the non-randomness of these collision-generated strings compared with random number generator created strings: Shannon entropy, diehard randomness tests and compression percentage. We show that the Shannon information entropy is unable to distinguish random from deterministic strings. The Diehard test performs better, but for certain mass-ratios the collision-generated strings are misidentified as random with high confidence. The zlib and bz2 compression algorithms are efficient at detecting non-randomness and low information content, with compression efficiencies that tend to 100% in the limit of infinite strings. Thus ”compression algorithm entropy” is non-extensive for this chaotic system, in marked contrast to the extensive entropy determined from phase-space integrals by assuming ergodicity.

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“…Recent implications include hard-spheres gas particles colliding with the same neighbors for an extended period of time. 18 Other examples where the arcsine law is obeyed include the time of maximal displacement in 1D Brownian motion, 19,20 lead changes within competitive team sports games, 21 and the probability distribution of longitudinal displacements of tracer particles in split flow. 22 In summary, we have reported on an elementary derivation of first and last return times which also serves as an introduction to a variety of important and broadly applicable concepts such as recurrence, first passage, heavy-tailed distributions, and the arcsine law.…”

Section: Fig 3 Heavy-tailed Distributionsmentioning

confidence: 99%

An elementary derivation of first and last return times of 1D random walks

Kostinski

¹

,

Amir

²

2016

American Journal of Physics

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Random walks, and in particular, their first passage times, are ubiquitous in nature. Using direct enumeration of paths, we find the first return time distribution of a 1D random walker, which is a heavy-tailed distribution with infinite mean. Using the same method we find the last return time distribution, which follows the arcsine law. Both results have a broad range of applications in physics and other disciplines.

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Emergence of distinguishability of patterns of collisions of particles in a non-equilibrium chaotic system

Cited by 3 publications

References 13 publications

Compression and information entropy of binary strings from the collision history of three hard balls

Compression and information entropy of binary strings from the collision history of three hard balls

Compression and information entropy of binary strings from the collision history of three hard balls

An elementary derivation of first and last return times of 1D random walks

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