无穷维哈密顿流形上的熵几何与归一化均值
Entropy Geometry and Normalized Means on Infinite-Dimensional Hamiltonian Manifolds
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中文总结 AI 辅助
该研究提出无穷维哈密顿系统平衡统计力学的几何-分析框架，用归一化均值推广概率测度，构造熵与自由能泛函，结合实例验证其性质，为相关领域提供理论支撑。
中文摘要 AI 辅助
我们提出了一种用于无穷维哈密顿系统平衡统计力学的几何-分析框架。在无法获得合适σ-可加不变测度的情形下，我们使用归一化均值，其可推广概率测度与归一化积分。该构造在弱辛弗雷歇流形上生成熵与自由能泛函，并在明确的可容许性与分离假设下，给出指数族平衡态的存在性与唯一性。这些态在保持参考均值与平衡权重的哈密顿流下是平稳的，且当参考均值为泊松不变时满足经典泊松-KMS恒等式。在局部指数正则性假设下，对数配分函数是光滑且凸的，其黑塞矩阵由协方差形式给出；它在热力学零方向模下严格凸，并通过勒让德-芬切尔对偶在广延变量域上诱导出凹熵。我们以电流群Map(M,G)与微分同胚群Diff(M)上的H^s测地线方程为例说明该框架，包括流体动力学与场论实例。
英文摘要
We propose a geometric--analytic framework for equilibrium statistical mechanics on infinite-dimensional Hamiltonian systems. In situations where no suitable $σ$-additive invariant measure is available, we use \emph{normalized means}, which generalize probability measures and normalized integrals. This construction yields entropy and free-energy functionals on weak symplectic Fréchet manifolds and gives existence and uniqueness of exponential-family equilibrium states under explicit admissibility and separation assumptions. These states are stationary under Hamiltonian flows preserving both the reference mean and the equilibrium weight, and satisfy a classical Poisson--KMS identity when the reference mean is Poisson invariant. Under a local exponential regularity assumption, the logarithmic partition functional is smooth and convex, with Hessian given by the covariance form. It is strictly convex modulo thermodynamically null directions and, through Legendre--Fenchel duality, induces a concave entropy on the domain of extensive variables. We illustrate the framework with $H^s$-geodesic equations on current groups $\operatorname{Map}(M,G)$ and diffeomorphism groups $\operatorname{Diff}(M)$, including hydrodynamic and field-theoretic examples.
