Anomalous transport and observable average in the standard map

TL;DR

This paper investigates the relationship between finite time observable averages and transport properties in the standard map, revealing a law linking the scaling exponent of distribution maxima to the transport exponent.

Contribution

It introduces a novel connection between the scaling behavior of observable averages and transport exponents in low-dimensional Hamiltonian systems using the standard map.

Findings

The maximum of distribution scales with time according to a specific exponent.

A law relating the slope of bc(q) at q=0 to the distribution scaling exponent lpha is proposed.

The study provides insights into transport phenomena in mixed phase space systems.

Abstract

The distribution of finite time observable averages and transport in low dimensional Hamiltonian systems is studied. Finite time observable average distributions are computed, from which an exponent $\alpha$ characteristic of how the maximum of the distributions scales with time is extracted. To link this exponent to transport properties, the characteristic exponent $\mu(q)$ of the time evolution of the different moments of order $q$ related to transport are computed. As a testbed for our study the standard map is used. The stochasticity parameter $K$ is chosen so that either phase space is mixed with a chaotic sea and islands of stability or with only a chaotic sea. Our observations lead to a proposition of a law relating the slope in $q=0$ of the function $\mu(q)$ with the exponent $\alpha$.