Concavity of Tsallis Entropy and Tsallis Entropy Power along Heat Flow

Abstract

We study the evolution of Tsallis entropy along the heat flow and establish concavity results in arbitrary dimensions. Extending earlier one-dimensional results, we prove that Tsallis entropy is concave along the heat flow for $q\in(0,3]$ in dimension one and for $q\in[1,3]$ in higher dimensions. The upper endpoint $q=3$ is sharp in every dimension. The proof is based on a nonlinear transformation of the heat equation, a sharp dimension-free functional inequality with constant $C_u=3$, and a rigorous justification of the integration-by-parts identities used in the argument. The sharp inequality is proved by an explicit integration-by-parts sum-of-squares identity, rather than by a computer-assisted semidefinite-programming search. As consequences, we recover a generalized de Bruijn identity, prove monotonicity of the associated $q$-Fisher information along the heat flow, and establish concavity results for Tsallis entropy power, including the Shannon entropy-power case and Costa's EPI as an endpoint. We also obtain an asymptotic entropy-power concavity statement for general initial data and a sharp auxiliary functional inequality which may be of independent analytic interest.