Information Geometry in Gaussian Random Fields: Searching for an Arrow of Time in Complex Systems

Alexandre Levada

doi:10.3390/ecea-1-c003

Abstract

Random fields are characterized by intricate non-linear relationships between their elements over time. However, what is a reasonable intrinsic definition for time in such complex systems? Here, we discuss the problem of characterizing the notion of time in isotropic pairwise Gaussian random fields. In particular, we are interested in studying the behavior of these fields when temperature deviates from infinity. Our investigations are focused in the relation between entropy and Fisher information, by the definition of the Fisher curve. The results suggest the emergence of an arrow of time as a consequence of asymmetrical geometric deformations in the random field model's metric tensor. In terms of information geometry, the process of taking a random field from a lower entropy state A to a higher entropy state B and then bringing it back to A, induces a natural intrinsic one-way direction of evolution. In practice, there are different trajectories in the information space, suggesting that the deformations induced by the metric tensor into the parametric space (manifold) are not reversible for positive and negative displacements in the inverse temperature parameter direction. In other words, there is only one possible orientation to move through different entropic states along a Fisher curve.

Highlights

Since the origins of the human race, the concept of time has always intrigued mankind
We addressed the problem of measuring the emergence of an arrow of time in Gaussian random field models
Investigations about the relation between two important information-theoretic measures, entropy and Fisher information, led us to the definition of the Fisher curve of a random field, a parametric trajectory embbeded in an information space, which characterizes the system behavior in terms of variations in the inverse temperature parameter

Summary

Introduction

Since the origins of the human race, the concept of time has always intrigued mankind. Along centuries of evolution many philosophers and researchers have studied this fascinating but seemingly obscure topic [1,2,3]. Why does time seem to flow in one single direction? Is the passage of time merely an illusion? In an attempt to study the effect of the passage of time in complex systems, this paper proposes to investigate a reasonable way to characterize an intrinsic notion of time in random fields composed by Gaussian variables. Our study focuses on an information-theoretic perspective, motivated by the connection between Fisher information and the geometric structure of stochastic models, provided by information geometry [4,5]. The proposed framework is mostly based on a data-driven approach, that is, we make use of intensive computational simulations to achieve our conclusions

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Results

Conclusion