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Diffusion and localization for the Chirikov typical map.

Klaus M Frahm¹, Dima L Shepelyansky

¹Université de Toulouse-UPS, F-31062 Toulouse, France.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|August 8, 2009

Summary

This summary is machine-generated.

We studied the Chirikov typical map, finding its classical chaos border and two diffusive regimes. Quantum dynamics reveal a localization length subtly dependent on classical diffusion constants across different time scales.

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Area of Science:

Quantum chaos
Statistical physics
Dynamical systems

Background:

The Chirikov standard map is a foundational model in chaos theory.
Understanding the transition from regular to chaotic behavior is crucial.
Previous studies have explored the standard map's properties extensively.

Purpose of the Study:

To analyze the classical and quantum properties of the Chirikov typical map.
To investigate the impact of random phase-shift angles on system dynamics.
To characterize the classical chaos border and quantum localization phenomena.

Main Methods:

Introduction of a finite number of random phase-shift angles to the standard map.
Analysis of classical dynamics to identify chaos borders and diffusion regimes.
Examination of quantum dynamics, focusing on Chirikov localization (dynamical localization).

Main Results:

Identified the classical chaos border k(c) approximately T^(-3/2).
Revealed two distinct regimes of diffusive behavior on short and long time scales.
Found that quantum localization length depends subtly on classical diffusion constants in these regimes.

Conclusions:

The Chirikov typical map exhibits unique classical and quantum properties due to random phase-shifts.
The interplay between classical diffusion and quantum localization is a key feature.
The study provides insights into dynamical localization in systems with mixed regular and chaotic dynamics.