On the Concavity of Tsallis Entropy along the Heat Flow

2026-01-19

AI Summary

This work overcomes the long-standing limitation of studying the concavity of Tsallis entropy along the heat flow exclusively in one dimension, establishing its concavity for the first time in arbitrary dimensions. By analyzing the second-order time derivative of Tsallis entropy under heat flow evolution and employing refined integration-by-parts techniques together with high-dimensional differential estimates, the authors develop a novel method to control the second derivative and derive a new class of functional inequalities. This advancement not only extends the theory of entropy concavity to higher-dimensional settings but also significantly broadens the applicability of nonlinear entropy functionals in geometric and analytic contexts.

Technology Category

• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Qualitative Reasoning

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

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.

Problem

• Tsallis entropy
• heat flow
• concavity
• functional inequality
• higher dimensions

Innovation

• Tsallis entropy
• heat flow
• concavity
• functional inequality
• higher dimensions