Nonstandard entropy production in the standard map

TL;DR

This paper studies how entropy evolves in the standard map, revealing an anomalous regime linked to mixed phase space structures, characterized using nonextensive entropy, and drawing parallels with long-range Hamiltonian systems.

Contribution

It introduces the use of generalized nonextensive entropy to characterize anomalous regimes in the standard map with mixed phase space structures.

Findings

Anomalous entropy regime appears for small |a| values.

The duration of the anomalous regime increases as a approaches zero.

Analogies are drawn with metastable states in long-range Hamiltonian systems.

Abstract

We investigate the time evolution of the entropy for a paradigmatic conservative dynamical system, the standard map, for different values of its controlling parameter $a$. When the phase space is sufficiently ``chaotic'' (i.e., for large $|a|$), we reproduce previous results. For small values of $|a|$, when the phase space becomes an intricate structure with the coexistence of chaotic and regular regions, an anomalous regime emerges. We characterize this anomalous regime with the generalized nonextensive entropy, and we observe that for values of $a$ approaching zero, it lasts for an increasingly large time. This scenario displays a striking analogy with recent observations made in isolated classical long-range $N$-body Hamiltonians, where, for a large class of initial conditions, a metastable state (whose duration diverges with $1/N\to 0$) is observed before it crosses over to the usual, Boltzmann-Gibbs regime.