Diffusion Limit for a Slow-Fast Standard Map

Consider the map ( x , z ) ↦ ( x + ϵ - α sin ( 2 π x ) + ϵ - ( 1 + α ) z , z + ϵ sin ( 2 π x ) ) , which is conjugate to the Chirikov standard map with a large parameter. The parameter value α = 1 is related to “scattering by resonance” phenomena. For suitable α , we obtain a central limit theorem f...

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Bibliographic Details
Published in:Communications in mathematical physics 2020-02, Vol.374 (1), p.187-210
Main Authors: Blumenthal, Alex, De Simoi, Jacopo, Zhang, Ke
Format: Article
Language:English
Subjects:
Classical and Quantum Gravitation
Complex Systems
Diffusion rate
Mathematical and Computational Physics
Mathematical Physics
Parameters
Physics
Physics and Astronomy
Quantum Physics
Relativity Theory
Resonance scattering
Theorems
Theoretical
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Summary:Consider the map ( x , z ) ↦ ( x + ϵ - α sin ( 2 π x ) + ϵ - ( 1 + α ) z , z + ϵ sin ( 2 π x ) ) , which is conjugate to the Chirikov standard map with a large parameter. The parameter value α = 1 is related to “scattering by resonance” phenomena. For suitable α , we obtain a central limit theorem for the slow variable z for a (Lebesgue) random initial condition. The result is proved by conjugating to the Chirikov standard map and utilizing the formalism of standard pairs. Our techniques also yield for the Chirikov standard map a related limit theorem and a “finite-time” decay of correlations result.
ISSN:0010-3616
1432-0916
DOI:10.1007/s00220-019-03600-7