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Records in the classical and quantum standard map. (English) Zbl 1352.37119

Summary: Record statistics is the study of how new highs or lows are created and sustained in any dynamical process. The study of the highest or lowest records constitute the study of extreme values. This paper represents an exploration of record statistics for certain aspects of the classical and quantum standard map. For instance the momentum square or energy records is shown to behave like that of records in random walks when the classical standard map is in a regime of hard chaos. However different power laws is observed for the mixed phase space regimes. The presence of accelerator modes are well-known to create anomalous diffusion and we notice here that the record statistics is very sensitive to their presence. We also discuss records in random vectors and use it to analyze the quantum standard map via records in their eigenfunction intensities, reviewing some recent results along the way.

MSC:

37E05 Dynamical systems involving maps of the interval
37E30 Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces
81S05 Commutation relations and statistics as related to quantum mechanics (general)
82D55 Statistical mechanics of superconductors
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