Strong overlap of deterministic and stochastic dynamics in a super-diffusive regime

TL;DR

This paper demonstrates that a deterministic map called the slicer map exhibits statistical behaviors similar to a stochastic Lévy-Lorentz gas, especially in superdiffusive regimes, by analyzing higher-order position autocorrelation functions.

Contribution

It analytically derives the generalized position autocorrelation functions of the slicer map and compares their scaling with those of the Lévy-Lorentz gas, revealing strong dynamical similarities.

Findings

Slicer map and Lévy-Lorentz gas share key superdiffusive features.

Higher-order autocorrelation functions exhibit similar scaling behaviors.

Deterministic slicer map can predict multi-time correlations in superdiffusive systems.

Abstract

We consider deterministic dynamics, known as the slicer map (SM), which exhibits normal and anomalous diffusion by varying a single parameter. The statistics of the position moments and the low-order position autocorrelation function (PACF) of the SM closely overlap with those of a stochastic process called the L\'evy-Lorentz gas (LLg), particularly in the normal and strongly superdiffusive anomalous regimes. However, matching low-order statistics alone cannot fully characterize the microscopic dynamics or distinguish underlying process classes. To demonstrate how these dynamics strongly overlap, we focus on the scaling of higher-order PACF, which provides a more detailed characterization. In this paper, we analytically derive the generalized PACF of the SM and explore its scaling forms under different temporal relationships. Specifically, we derive several scalings of the 3-point PACF by analyzing intriguing relations between three times. We compare these scalings with the power-law tails of the numerically estimated 3-point PACF of the LLg. This comparison provides a detailed description of the correlation scalings of the SM, demonstrating that the SM shares key features with the LLg. Our findings establish the SM as a deterministic analog of the LLg, enabling efficient prediction of multi-time position correlations in superdiffusive systems.