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Motivated by the presence of deep connections among dynamical equations, experimental data, physical systems, and statistical modeling, we report on a series of findings uncovered by the authors and collaborators during the last decade within the framework of the so-called Information Geometric Approach to Chaos (IGAC). The IGAC is a theoretical modeling scheme that combines methods of information geometry with inductive inference techniques to furnish probabilistic descriptions of complex systems in presence of limited information. In addition to relying on curvature and Jacobi field computations, a suitable indicator of complexity within the IGAC framework is given by the so-called information geometric entropy (IGE). The IGE is an information geometric measure of complexity of geodesic paths on curved statistical manifolds underlying the entropic dynamics of systems specified in terms of probability distributions. In this manuscript, we discuss several illustrative examples wherein our modeling scheme is employed to infer macroscopic predictions when only partial knowledge of the microscopic nature of a given system is available. Finally, we include comments on the strengths and weaknesses of the current version of our proposed theoretical scheme in our concluding remarks.

Characterizing and to some degree understanding the emergence and evolutionary development of biological systems represent one of the most compelling motivations to investigate the highly elusive concept of complexity [

Entropic inference methods [

This line of research was initially referred to as the

For a review of the MrE inference algorithm, we refer to [

In this section, we outline all available applications concerning the characterization of the complexity of geodesic paths on curved statistical manifolds within the IGAC framework. For a conventional approach to the Riemannian geometrization of classical Newtonian dynamics, we refer to [

Very preliminary concepts and applications of the IGAC appeared originally in [

Previously, the IGAC modeling framework was used to investigate the information geometric properties of a system of arbitrary nature containing

In [

In a broad sense, it is known that the primary issues addressed by the General Theory of Relativity are twofold: first, one is interested in the manner that space-time geometry evolves in response to mass-energy; second, one seeks to understand how mass-energy configurations move in such a space-time geometry. Within the IGAC approach, one focuses on how systems move in a given statistical geometry, while the evolution of the statistical geometry itself is neglected. The realization that there are two distinct and separate aspects to this scenario served as a turning point in the development of the IGAC framework that led to an intriguing result. The first formal result in this research direction was presented in [

From a more applied perspective, building upon the results found in [

We recall that it is commonly conjectured that spectral correlations of classically integrable systems are well described by Poisson statistics and that quantum spectra of classically chaotic systems are universally correlated according to Wigner-Dyson statistics. The former and the latter conjectures are known as the BGS (Bohigas-Giannoni-Schmit, [

In [

Reducing the complexity of statistical models is a very active field of research [

Inspired by the preliminary analysis presented in [

Comparing classical and quantum chaoticity (i.e., temporal complexity) and explaining the reason why the former is stronger than the latter are of great theoretical interest [

In [

We recall that, in an antiferromagnetic triangular Ising model with coupling between neighboring spins equal to

Based on these findings, it was argued in [

In this manuscript, we presented a brief survey of the main results uncovered by the authors and collaborators within the framework of the IGAC over the past decade. As pointed out in the Introductory Background, for the sake of brevity, we have omitted mathematical details in our main presentation. However, for the sake of self-consistency, we have added a number of Appendices covering the basic mathematical details necessary to critically follow the content of the manuscript. For an extended review with more mathematical details and physical interpretations on the IGAC, we refer to the recent work appearing in [

We provided here several illustrative examples of entropic dynamical models employed to infer macroscopic predictions when only limited information of the microscopic nature of a system is available. In the first example, we considered the IGAC applied to a high-dimensional Gaussian statistical model. In particular, we reported the scaling of the scalar curvature with the microscopic degrees of freedom of the system in (

We are aware of several unresolved issues within the IGAC. In what follows, we outline in a more systematic fashion a number of strengths and weaknesses of the IGAC theoretical scheme.

No arbitrariness or lack of explanation of how macrostates of a system leading to the formation of geodesic paths on the curved statistical manifold is present. Within the IGAC, the transition (that is, the updating) from an initial macrostate to a final macrostate occurs by navigating through a continuous sequence of intermediate macrostates chosen by maximizing the relative entropy between any two consecutive intermediate macrostates subject to the available information constraints.

All the dynamical information is collected into a single geometric quantity where all the available symmetries are retained: the curved statistical manifold on which the geodesic flow is induced. For instance, the sensitive dependence of trajectories on initial conditions can be analyzed from the geodesic deviation equation. Furthermore, the nonintegrability (chaoticity) of the system can be studied by investigating the existence (absence) of Killing tensors on the curved manifold.

IGAC offers a unifying theoretical setting wherein both curvature and entropic indicators of complexity are available.

IGAC represents a convenient platform for enhancing our comprehension of the role played by statistical curvature in modeling realistic processes by linking it to conventionally accepted quantities, including entropy.

From a more foundational perspective, provided that the true degrees of freedom of the system are identified, IGAC presents a serious opportunity to uncover deep insights into the foundations of modeling and inductive reasoning together with the relationship to each other.

IGE lacks a detailed comparison with other entropic complexity indicators of geometric flavor.

Despite the interpretational power of the Riemannian geometrization of dynamics, the integration of geodesic equations together with computations involving curvatures and Jacobi fields can become quite challenging, especially for higher-dimensional statistical manifolds lacking any particular degree of symmetry.

There is no fully developed quantum mechanical IGAC framework suitable for characterizing the complexity of quantum evolution.

IGAC lacks experimental evidence in support of theoretical macroscopic predictions advanced within its setting.

General results with a wide range of applicability are absent. Most macroscopic predictions are limited to specific classes of physical systems in the presence of very peculiar functional forms of relevant available information.

Despite these weaknesses, we are truly gratified that the IGAC is gradually gaining attention within the scientific community. Indeed, there seems to be an increasing number of scientists who either actively make use of, or whose work is related to, the IGAC [

In conclusion, we emphasize that it was not our intention to report in this manuscript all the available scientific investigations on complexity based upon the information geometric approach. Instead, we limited our presentation to the findings uncovered within the framework of the so-called IGAC theoretical framework. For an overview of various methods of information geometry used to quantify the complexity of physical systems in both classical and quantum settings, we refer to the recent review article in [

In this appendix, we review some basic mathematical details on the concept of curvature. Recall that an

In this manuscript, the points of

Once the Fisher-Rao information metric

We remark at this point that a geodesic on a

We point out that the negativity of the Ricci curvature

In this appendix, we present the concept of the IGE within the IGAC theoretical setting. Assume that the points

The IGE at a certain instant is essentially the logarithm of the volume of the effective parameter space explored by the system at that very instant. In order to average out the possibly highly complex fine details of the entropic dynamical description of the system on the curved statistical manifold, the temporal average has been taken into consideration. Furthermore, to eliminate the consequences of transient effects which enter the computation of the expected value of the volume of the effective parameter space, the long-term asymptotic temporal behavior is considered to conveniently describe the selected dynamical complexity indicator. In summary, the IGE is constructed to furnish an asymptotic coarse-grained inferential description of the complex dynamics of a system in the presence of only partial knowledge. For further details on the IGE, we refer to [

In this appendix, we review some basic mathematical details on the concept of Jacobi vector fields. The investigation of the instability of natural motions by way of the instability of geodesics on a suitable curved manifold is especially advantageous within the Riemannian geometrization of dynamics. In particular, the so-called Jacobi-Levi-Civita (JLC) equation for geodesic spread is a very powerful mathematical tool employed to study the stability/instability of a geodesic flow. This equation is a familiar quantity both in theoretical physics (in General Relativity, for instance) and in Riemannian geometry. The JLC equation covariantly describes how neighboring geodesics locally scatter. More specifically, the JLC equation connects curvature properties of the ambient manifold to the stability/instability of a geodesic flow. It paves the way to a wide and largely unexplored field of investigation that concerns the links among geometry, topology, and geodesic instability and therefore to chaoticity and complexity. To the best of our knowledge, the use of the JLC equation in the framework of information geometry appeared originally in [

Let us consider two neighboring geodesic paths

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Carlo Cafaro acknowledges the hospitality of the United States Air Force Research Laboratory in Rome (New York) where part of his initial contribution to this work was completed. Finally, the authors are grateful to Dr. Domenico Felice and Dr. Daniel Stevenson for helpful comments.