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

Chris G. Antonopoulos; Tassos Bountis; Lambros Drossos

doi:10.1016/j.apnum.2015.07.003

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 language	English
Pages (from-to)	110-119
Number of pages	10
Journal	Applied Numerical Mathematics
Volume	104
Early online date	29 Jul 2015
DOIs	https://doi.org/10.1016/j.apnum.2015.07.003
Publication status	Published - Jun 2016

Bibliographical 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.

Keywords

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

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Cite this

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title = "Coupled symplectic maps as models for subdiffusive processes in disordered Hamiltonian lattices",

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.",

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

author = "Antonopoulos, {Chris G.} and Tassos Bountis and Lambros Drossos",

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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N1 - 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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N2 - 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.

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

KW - multi-dimensional maps

KW - McMillan map

KW - Klein-Gordon disordered Hamiltonian

KW - chaotic and diffusive motion

KW - q-Gaussians

KW - Tsallis entropy

U2 - 10.1016/j.apnum.2015.07.003

DO - 10.1016/j.apnum.2015.07.003

M3 - Article

SN - 1873-5460

VL - 104

SP - 110

EP - 119

JO - Applied Numerical Mathematics

JF - Applied Numerical Mathematics

ER -

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

Abstract

Bibliographical note

Keywords

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