Lyapunov Exponents, Transport and the Extensivity of Dimensional Loss

TL;DR

This paper establishes a relation connecting Lyapunov exponents, entropy production, and transport in systems under thermal gradients, revealing how microscopic chaos relates to macroscopic behavior and demonstrating the extensivity of dimensional loss.

Contribution

It introduces an explicit relation linking microscopic Lyapunov exponents with macroscopic transport and entropy production, including finite size corrections and numerical verification.

Findings

The product of maximum Lyapunov exponents and dimensional loss is unique and macroscopic.

Dimensional loss is extensive in systems with bulk behavior.

Finite size corrections to the relations are computed and validated numerically.

Abstract

An explicit relation between the dimensional loss ($\Delta D$), entropy production and transport is established under thermal gradients, relating the microscopic and macroscopic behaviors of the system. The extensivity of $\Delta D$ in systems with bulk behavior follows from the relation. The maximum Lyapunov exponents in thermal equilibrium and $\Delta D$ in non-equilibrium depend on the choice of heat-baths, while their product is unique and macroscopic. Finite size corrections are also computed and all results are verified numerically.