Note on the Kaplan-Yorke dimension and linear transport coefficients

TL;DR

This paper explores the relationships between the Kaplan-Yorke dimension, Lyapunov exponents, and transport coefficients in nonequilibrium dissipative systems, providing new formulas and confirming them with simulations.

Contribution

It introduces a new expression linking the Kaplan-Yorke dimension with transport coefficients and discusses phase space dimension extensivity in dissipative systems.

Findings

Derived a new formula for transport coefficients using the Kaplan-Yorke dimension.

Established a relation between the Kaplan-Yorke dimension and transport coefficients.

Confirmed theoretical results with computer simulations of an atomic fluid.

Abstract

A number of relations between the Kaplan-Yorke dimension, phase space contraction, transport coefficients and the maximal Lyapunov exponents are given for dissipative thermostatted systems, subject to a small external field in a nonequilibrium stationary state. A condition for the extensivity of phase space dimension reduction is given. A new expression for the transport coefficients in terms of the Kaplan-Yorke dimension is derived. Alternatively, the Kaplan-Yorke dimension for a dissipative macroscopic system can be expressed in terms of the transport coefficients of the system. The agreement with computer simulations for an atomic fluid at small shear rates is very good.