Information geometric analysis of phase transitions in complex patterns: the case of the Gray-Scott reaction-diffusion model

TL;DR

This study applies information geometry, specifically Fisher information, to analyze phase transitions in complex patterns of the Gray-Scott reaction-diffusion model, revealing boundaries between different pattern regimes without assuming a statistical model.

Contribution

It demonstrates the use of Fisher information to detect phase transitions in complex systems through non-parametric estimation, extending the application of information geometry beyond classical models.

Findings

Fisher information highlights boundaries between different pattern regimes.

High Fisher information lines correspond to phase transition boundaries.

The approach does not rely on predefined statistical models.

Abstract

The Fisher-Rao metric from Information Geometry is related to phase transition phenomena in classical statistical mechanics. Several studies propose to extend the use of Information Geometry to study more general phase transitions in complex systems. However, it is unclear whether the Fisher-Rao metric does indeed detect these more general transitions, especially in the absence of a statistical model. In this paper we study the transitions between patterns in the Gray-Scott reaction-diffusion model using Fisher information. We describe the system by a probability density function that represents the size distribution of blobs in the patterns and compute its Fisher information with respect to changing the two rate parameters of the underlying model. We estimate the distribution non-parametrically so that we do not assume any statistical model. The resulting Fisher map can be interpreted as a phase-map of the different patterns. Lines with high Fisher information can be considered as boundaries between regions of parameter space where patterns with similar characteristics appear. These lines of high Fisher information can be interpreted as phase transitions between complex patterns.