Concavity of Tsallis Entropy and Tsallis Entropy Power along Heat Flow

TL;DR

This paper proves the concavity of Tsallis entropy along heat flow in multiple dimensions, extending previous one-dimensional results, and introduces new analytic methods and inequalities in non-additive entropy frameworks.

Contribution

It establishes the concavity of Tsallis entropy in higher dimensions for a range of q, using a fully analytic approach that avoids computer-assisted methods.

Findings

Tsallis entropy is concave in time for certain q values in multiple dimensions.

The paper recovers a generalized de Bruijn identity and shows monotonicity of q-Fisher information.

It derives new concavity properties for Tsallis entropy power and a novel functional inequality.

Abstract

We study the evolution of Tsallis entropy along the heat flow and establish its concavity in arbitrary dimensions. Extending prior results that were restricted to the one-dimensional setting, we prove that the Tsallis entropy is concave in time for a nontrivial range of the entropic index $q$ in both the one-dimensional and higher-dimensional settings. The analysis is based on a nonlinear transformation, together with a novel estimate for the second-order time derivative of the entropy and a rigorous justification of the integration-by-parts identities required in the argument. Our approach is fully analytic and avoids the use of computer-assisted methods that have limited previous works in higher dimensions. As consequences, we recover a generalized de Bruijn identity, establish the monotonicity of the associated $q$-Fisher information along the heat flow, and derive concavity properties for the Tsallis entropy power, including asymptotic results under general initial conditions. In addition, our method yields a new functional inequality that may be of independent interest. These results contribute to the broader program of extending classical information-theoretic inequalities beyond the Shannon framework to non-additive entropy settings.