On the Concavity of Tsallis Entropy along the Heat Flow

TL;DR

This work proves the concavity of the Tsallis entropy $S_q$ along the heat flow in higher dimensions by transforming the evolving density via $u_t = \phi_t^{1/(1+\delta)}$ and analyzing the second time derivative of $\int u_t^2 dx$. A rigorous integration-by-parts framework is established to justify core calculations, and a novel estimate relating $\int \Delta u_t \frac{|\nabla u_t|^2}{u_t} dx$ to $\int |\nabla u_t|^4/u_t^2 dx$ is developed. The main results give dimension-dependent ranges for the entropic index $q$ that guarantee concavity: for $d=1$, $q \in [1,3]$, and for $d>1$, $q \in [1,\frac{2(\sqrt{5}+1)}{\sqrt{5}}]$, alongside a new functional inequality $\int \frac{\|\nabla u\|^4}{u^2} dx \le (2\sqrt{5}+6) \int |\Delta u|^2 dx$ on the torus. These findings advance the understanding of entropy evolution in multi-dimensional settings and open avenues for extending the approach to overdamped Langevin dynamics and related variational problems.

Abstract

We demonstrate the concavity of the Tsallis entropy along the heat flow for general dimensions, expanding upon the findings of Wu et al 2025 and Hung 2022, which were previously limited to the one-dimensional case. The core of the proof is a novel estimate of the terms in the second-order time derivative, and a rigorous validation of integration by parts. The resulting bound establishes a new functional inequality, which may be of interest for other areas of mathematical analysis.

Theorems & Definitions (14)

• Definition 1
• Theorem 2
• Lemma 3
• Lemma 4
• Lemma 5
• Lemma 6
• Proposition 7
• Lemma 8
• Remark 9
• Remark 10