The Definition and Measurement of the Topological Entropy per Unit Volume in Parabolic PDE's

TL;DR

This paper introduces a new way to define and measure the topological entropy per unit volume in parabolic PDEs, providing theoretical bounds and a practical sampling algorithm for experimental data analysis.

Contribution

It defines the topological entropy per unit volume in parabolic PDEs, proves its existence and bounds, and proposes a finite sampling algorithm for measurement.

Findings

Existence of topological entropy per unit volume in parabolic PDEs

Bound on entropy using Hausdorff dimension and expansion rate

A finite sampling algorithm for entropy measurement

Abstract

We define the topological entropy per unit volume in parabolic PDE's such as the complex Ginzburg-Landau equation, and show that it exists, and is bounded by the upper Hausdorff dimension times the maximal expansion rate. We then give a constructive implementation of a bound on the inertial range of such equations. Using this bound, we are able to propose a finite sampling algorithm which allows (in principle) to measure this entropy from experimental data.