[Paper Review] Fisher metric from relative entropy group

Ignacio S. Gomez, Ernesto P. Borges+1 more|arXiv (Cornell University)|May 28, 2018

Statistical Mechanics and Entropy30 references3 citations

TL;DR

This paper introduces the Fisher metric group, derived from the Hessian of the relative entropy group, generalizing standard Fisher information to non-extensive statistical mechanics. It proves the metric is a scalar multiple of the standard Fisher metric, with the factor determined by local properties of the entropy group. For the Tsallis class, it demonstrates softening and strengthening of scalar curvature via temperature-dependent parameters, showing easier suppression of correlations than their creation in a 2D harmonic oscillator model.

ABSTRACT

In this work we consider the Fisher metric which results from the Hessian of the relative entropy group, that we called Fisher metric group, and we obtain the corresponding ones to the Boltzmann-Gibbs, Tsallis, Kaniadakis and Abe-Borges-Roditi classes. We prove that the scalar curvature of the Fisher metric group results a multiple of the standard Fisher one, with the factor of proportionality given by the local properties of the entropy group. For the Tsallis class, the softening and strengthening of the scalar curvature is illustrated with the $2D$ correlated model, from which their associated indexes for the canonical ensemble of a pair of interacting harmonic oscillators, are obtained.

Motivation & Objective

• To generalize Fisher information geometry to non-extensive statistical mechanics using the relative entropy group.
• To derive a unified framework for Fisher metrics across universal entropy classes, including Tsallis, Kaniadakis, and Abe–Borges–Roditi.
• To establish a connection between geometric curvature and macroscopic thermodynamic parameters like temperature in correlated systems.
• To demonstrate how softening and strengthening of scalar curvature can be controlled via group parameters, particularly in the Tsallis class.

Proposed method

• Derive the Fisher metric group as the Hessian of the relative entropy group, using the Kullback-Leibler divergence as the foundational divergence measure.
• Apply the formalism to universal entropy classes, explicitly computing the Fisher metric for Boltzmann–Gibbs, Tsallis, Kaniadakis, and Abe–Borges–Roditi entropies.
• Establish that the metric tensor is a scalar multiple of the standard Fisher metric, with the proportionality factor determined by the first and second derivatives of the group function G(t) at t=0.
• Introduce softening and strengthening indices q_soft and q_str for the Tsallis class, derived from the Hessian of the relative entropy group in the context of a 2D correlated harmonic oscillator model.
• Use the canonical ensemble of two interacting harmonic oscillators to link the curvature modification parameters to the bath temperature T.
• Prove a functional relationship (2−q_soft)(2−q_str)=1, showing that softening and strengthening indices are not independent but linked by the system's temperature.

Experimental results

Research questions

• RQ1 How can the Fisher metric be generalized beyond the Boltzmann–Gibbs framework using the relative entropy group?
• RQ2 What is the mathematical structure of the Fisher metric group across different universal entropy classes?
• RQ3 How do softening and strengthening of scalar curvature relate to physical processes like correlation suppression or creation in statistical models?
• RQ4 Can the temperature of a thermal bath be used to control the curvature modification parameters in the Tsallis class?
• RQ5 What is the physical interpretation of the asymmetry between q_soft and q_str in terms of correlation dynamics?

Key findings

• The Fisher metric group is a scalar multiple of the standard Fisher metric, with the proportionality factor determined by the first and second derivatives of the group function G(t) at t=0.
• For the Tsallis class, the softening index is q_soft(T) = 2 − T/T₀ and the strengthening index is q_str(T) = 2 − T₀/T, both dependent on the bath temperature T.
• The indices satisfy the relation (2−q_soft)(2−q_str)=1, indicating they are not independent but linked through the temperature scale.
• In the 2D correlated harmonic oscillator model, q_soft remains near 1.99 for low temperatures (T/T₀ < 0.1), while q_str rapidly decreases from infinity to 1 as temperature increases.
• The results suggest that suppressing correlations in a correlated system is physically easier than inducing them in an uncorrelated one, as reflected in the slower variation of q_soft compared to q_str.
• The formalism enables a generalization of the Cramér-Rao bound, where the lower bound depends on the local properties of the entropy group via G'(0) and G''(0).