Coupled symplectic maps as models for subdiffusive processes in disordered Hamiltonian lattices

Chris G. Antonopoulos, Tassos Bountis, Lambros Drossos

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Abstract

We investigate dynamically and statistically diffusive motion in a chain of linearly coupled 2-dimensional symplectic McMillan maps and find evidence of subdiffusion in weakly and strongly chaotic regimes when all maps of the chain possess a saddle point at the origin and the central map is initially excited. In the case of weak coupling, there is either absence of diffusion or subdiffusion with q>1-Gaussian probability distributions, characterizing weak chaos. However, for large enough coupling and already moderate number of maps, the system exhibits strongly chaotic (q≈1) subdiffusive behavior, reminiscent of the subdiffusive energy spreading observed in a disordered Klein-Gordon Hamiltonian. Our results provide evidence that coupled symplectic maps can exhibit physical properties similar to those of disordered Hamiltonian systems, even though the local dynamics in the two cases is significantly different.
Original languageEnglish
Pages (from-to)110-119
Number of pages10
JournalApplied Numerical Mathematics
Volume104
Early online date29 Jul 2015
DOIs
Publication statusPublished - Jun 2016

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Hamiltonians
Subdiffusion
Disordered Systems
Model
Weak Coupling
Saddlepoint
Physical property
Chaos theory
Probability distributions
Hamiltonian Systems
Gaussian distribution
Chaos
Probability Distribution
Physical properties
Linearly
Motion
Energy

Keywords

  • complex statistics
  • multi-dimensional maps
  • McMillan map
  • Klein-Gordon disordered Hamiltonian
  • chaotic and diffusive motion
  • q-Gaussians
  • Tsallis entropy

Cite this

Coupled symplectic maps as models for subdiffusive processes in disordered Hamiltonian lattices. / Antonopoulos, Chris G.; Bountis, Tassos; Drossos, Lambros.

In: Applied Numerical Mathematics, Vol. 104, 06.2016, p. 110-119.

Research output: Contribution to journalArticle

Antonopoulos, Chris G. ; Bountis, Tassos ; Drossos, Lambros. / Coupled symplectic maps as models for subdiffusive processes in disordered Hamiltonian lattices. In: Applied Numerical Mathematics. 2016 ; Vol. 104. pp. 110-119.
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abstract = "We investigate dynamically and statistically diffusive motion in a chain of linearly coupled 2-dimensional symplectic McMillan maps and find evidence of subdiffusion in weakly and strongly chaotic regimes when all maps of the chain possess a saddle point at the origin and the central map is initially excited. In the case of weak coupling, there is either absence of diffusion or subdiffusion with q>1-Gaussian probability distributions, characterizing weak chaos. However, for large enough coupling and already moderate number of maps, the system exhibits strongly chaotic (q≈1) subdiffusive behavior, reminiscent of the subdiffusive energy spreading observed in a disordered Klein-Gordon Hamiltonian. Our results provide evidence that coupled symplectic maps can exhibit physical properties similar to those of disordered Hamiltonian systems, even though the local dynamics in the two cases is significantly different.",
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note = "Acknowledgements One of us (T. B.) acknowledges many interesting discussions on coupled maps with Professor C. Tsallis. We are also grateful to the anonymous referees for their constructive feedback that helped us improve the manuscript and to the HPCS Laboratory of the TEI of Western Greece for providing the computer facilities where all our simulations were performed. C. G. A. was partially supported by the “EPSRC EP/I032606/1” grant of the University of Aberdeen. This research has been co-financed by the European Union (European Social Fund - ESF) and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF) - Research Funding Program: THALES - Investing in knowledge society through the European Social Fund.",
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AB - We investigate dynamically and statistically diffusive motion in a chain of linearly coupled 2-dimensional symplectic McMillan maps and find evidence of subdiffusion in weakly and strongly chaotic regimes when all maps of the chain possess a saddle point at the origin and the central map is initially excited. In the case of weak coupling, there is either absence of diffusion or subdiffusion with q>1-Gaussian probability distributions, characterizing weak chaos. However, for large enough coupling and already moderate number of maps, the system exhibits strongly chaotic (q≈1) subdiffusive behavior, reminiscent of the subdiffusive energy spreading observed in a disordered Klein-Gordon Hamiltonian. Our results provide evidence that coupled symplectic maps can exhibit physical properties similar to those of disordered Hamiltonian systems, even though the local dynamics in the two cases is significantly different.

KW - complex statistics

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