Information Dimensions of Simple Four-Dimensional Flows

• Autorzy: Hoover William Graham , Hoover Carol Griswold , Travis Karl Patrick

Identyfikatory

DOI

10.12921/cmst.2023.0000017

• Języki publikacji: EN

Abstrakty

EN

Baker Maps have long served as pedagogical tools for understanding chaos and fractal phase-space distributions. Recent work [1], following earlier efforts from 1997 [2], shows that the Kaplan-Yorke formula for information dimension disagrees with direct computation for some simple compressible Baker Maps. Here we extend this map work to simple continuous flows. We compare pointwise information dimensions to the Kaplan-Yorke dimension for a simple four-dimensional flow [3] controlling both ⟨p 4 ⟩ and ⟨p 2 ⟩: { q˙ = p ; ˙p = −q − ξp3 − ζp ; ˙ξ = p 4 − 3p 2 ; ˙ζ = p 2 − T }. Precisely similar sets of Gaussian points could be generated with Metropolis’ Monte-Carlo simulations of harmonic oscillators in Gibbs’ canonical ensemble with f(q) = e −q 2/2 / p (2π). Remarkably, we show that the dependence of the pointwise information dimension for the Gaussian distribution is linear in the inverse of the logarithm of the mesh spacing, ∝ 1/ ln(1/δ). The Hoover-Holian Gaussian oscillator problem [3] can be generalized [2–4] to some nonequilibrium steady-state problems by introducing a temperature-gradient parameter ϵ. In that case the temperature T varies from 1 − ϵ to 1 + ϵ : T = 1 + ϵ tanh(q) so that both conservative (ϵ = 0) and dissipative (ϵ > 0) flows result.

Słowa kluczowe

EN

fractals; Baker Map; information dimension; Kaplan-Yorke dimension

• Wydawca: Institute of Bioorganic Chemistry Scientific Publishers OWN, Polish Academy of Sciences
• Czasopismo: Computational Methods in Science and Technology
• Rocznik: 2023
• Tom: Vol. 29, No. 1-4
• Strony: 21--26
• Opis fizyczny: Bibliogr. 6 poz., rys.

Twórcy

autor

Hoover William Graham

• Ruby Valley Research Institute 601 Highway Contract 60 Ruby Valley, Nevada 89833, USA

autor

Hoover Carol Griswold

• Ruby Valley Research Institute 601 Highway Contract 60 Ruby Valley, Nevada 89833, USA

autor

Travis Karl Patrick

• University of Sheffield Department of Materials Science and Engineering Sheffield S1 3JD, United Kingdom

Bibliografia

• [1] W.G. Hoover, C.G. Hoover, A Quarter Century of Baker Map Exploration, Computational Methods in Science and Technology 29, 5–16 (2023).
• [2] H.A. Posch, W.G. Hoover, Time-Reversible Dissipative Attractors In Three And Four Phase-Space Dimensions, Physical Review E 55, 6803–6810 (1997).
• [3] W.G. Hoover, B.L. Holian, Kinetic Moments Method for the Canonical Ensemble Distribution, Physics Letters A 211, 253–257 (1996).
• [4] W.G. Hoover, Compressible Baker Maps and Their Inverses. A Memoir for Francis Hayin Ree [1936–2020], Computational Methods in Science and Technology 26, 5–13 (2020).
• [5] T. Tél, M. Gruiz, Chaotic Dynamics; An Introduction Based on Classical Mechanics, Cambridge University Press (2006).
• [6] J.C. Sprott, W.G. Hoover, C.G. Hoover, Heat Conduction, and the Lack Thereof, in Time-Reversible Dynamical Systems: Generalized Nosé-Hoover Oscillators with a Temperature Gradient, Physical Review E 89, 042914 (2014).

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2024).