Records in the classical and quantum standard map

TL;DR

This paper investigates record statistics in classical and quantum standard maps, revealing how chaos, mixed regimes, and accelerator modes influence extreme value behaviors and energy records, with implications for understanding quantum eigenfunctions.

Contribution

It introduces a detailed analysis of record statistics in classical and quantum standard maps, highlighting the effects of chaos, mixed phases, and accelerator modes on extreme value behaviors.

Findings

Energy records behave like random walk records in chaotic regimes.

Different power laws are observed in mixed phase space regimes.

Record statistics are sensitive to accelerator modes and anomalous diffusion.

Abstract

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 {\it quantum} standard map via records in their eigenfunction intensities, reviewing some recent results along the way.