发表机构
East China Normal University; University of California, Riverside（华东师范大学; 加州大学河滨分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对黎曼流形上的熵及其密度，在里奇曲率变号或边界非凸等情形建立系统估计，证明域与凸性偏差不大且里奇曲率负部分不太大时改进热力学第二定律仍成立，给出凹性反例并证明相关熵密度估计。
AI 中文摘要
本文针对黎曼流形上的熵及其密度建立了系统估计，重点关注里奇曲率变号或边界非凸这类更具挑战性的情形。例如，改进的热力学第二定律指出，$\boldsymbol{R}^n$中紧域内的熵随时间递增；此外，若该域为凸域，则熵是凹的。凹性等价于费舍尔信息递减的性质，该性质也适用于里奇曲率非负的黎曼流形中的凸域（参见文献\textbf{NiLei}）。鉴于熵在数学、信息论、物理学等领域的广泛应用，学界存在将该性质扩展至更广泛场景的需求，尤其是边界非凸的情形（参见文献\textbf{CFM}第3页）。在此，我们证明：若域与凸性偏差不大，且里奇曲率的负部分在显式、非微扰意义下不太大，则改进的热力学第二定律仍然成立。该证明基于一种新近的二阶对数庞加莱不等式，无需流形满足显式曲率条件。若里奇曲率的负部分过大，则给出了凹性的反例。本文还证明了熵密度的其他一些相关估计（汉密尔顿型估计）。
英文摘要
In this paper, we establish systematic estimates for the entropy and its density on Riemannian manifolds, focusing on the more challenging cases where the Ricci curvature changes sign or the boundary is nonconvex. For example, the refined second law of thermodynamics states that the entropy in a compact domain in $\mathbb R^n$ is increasing in time, furthermore, it is concave if the domain is convex. The concavity property is equivalent to the property that the Fisher information is decreasing, which also holds for convex domains in a Riemannian manifold with nonnegative Ricci curvature (cf. \cite{NiLei}). In view of the wide application of entropy in mathematics, information theory, physics, etc., there is certain desire in the community to extend the property to broader settings, especially to the case with nonconvex boundary (see e.g. \cite[p. 3]{CFM}), even in the Euclidean case. Here, we prove that the refined second law still holds if the domain is not too far from convex and the negative part of the Ricci curvature is not too large, in an explicit, nonperturbative sense, thus realizing some of the expectations. The proof is based on a new, sharp second order log Poincaré inequality that does not require explicit curvature conditions of the manifold. If the negative part of the Ricci curvature is too large, a counterexample to the concavity is given. Some other related estimates for the entropy density (Hamilton type estimates) are also proven.
CommentsThe constant in Theorem 1.1 (a) improved, a corollary added, three more references added and some wording changed. 34 pages
