Records and Occupation Time Statistics for Area-Preserving Maps.

Area of Science:

• Physics
• Dynamical Systems
• Statistical Mechanics

Background:

• Characterizing deterministic systems exhibiting stochastic-like features is a key challenge in dynamics.
• Transport properties in deterministic systems on non-compact phase spaces are widely studied, including normal and anomalous transport.

Purpose of the Study:

• To investigate transport properties, records statistics, and occupation time statistics in two area-preserving maps: the Chirikov-Taylor standard map and the Casati-Prosen triangle map.
• To compare the behavior of these maps with known results for random walks and explore potential generalized laws.

Main Methods:

• Numerical investigation of area-preserving maps (Chirikov-Taylor standard map and Casati-Prosen triangle map).
• Analysis of transport properties, records statistics, and occupation time statistics.

Main Results:

• For the standard map with a chaotic sea, diffusive transport, records statistics, and occupation time fractions align with simple symmetric random walks.
• The triangle map exhibits anomalous transport and similar anomalies in its records statistics.
• Occupation time statistics and persistence probabilities in the triangle map are consistent with a generalized arcsine law and transient dynamics.

Conclusions:

• Deterministic systems can exhibit complex behaviors, including diffusive and anomalous transport, mirroring stochastic processes.
• The study confirms and extends findings for the standard map, while revealing anomalous behaviors in the triangle map consistent with generalized statistical laws.