Spatiotemporal chaos: the microscopic perspective

TL;DR

This paper provides numerical evidence linking the sum of Lyapunov exponents in the complex Ginzburg-Landau equation to the space average of the squared macroscopic field, revealing a simple underlying relationship.

Contribution

It introduces an explicit formula for the time-dependent Lyapunov exponents and connects microscopic chaos measures to macroscopic field properties.

Findings

Sum of Lyapunov exponents equals a simple function of the space-averaged squared field

Explicit formula for time-dependent Lyapunov exponents derived

Numerical evidence supports the theoretical relationship

Abstract

Extended nonequilibrium systems can be studied in the framework of field theory or from dynamical systems perspective. Here we report numerical evidence that the sum of a well-defined number of instantaneous Lyapunov exponents for the complex Ginzburg-Landau equation is given by a simple function of the space average of the square of the macroscopic field. This relationship follows from an explicit formula for the time-dependent values of almost all the exponents.