对应的非线性热力学形式体系

中文摘要 AI 辅助

我们在紧致度量空间上为对应（correspondences）发展了一种非线性热力学形式体系。连续能量在连续状态对的经验测度上进行评估。平稳转移对的测度论压力是其核熵与两个连续状态联合分布能量之和。对于满足遍历丰足条件的前向扩张对应，我们证明了变分原理以及非线性平衡对的存在性。我们进一步证明，渐近最大化吉布斯系综的两坐标投影的极限是平衡对的两坐标分布的平均。当能量依赖于可容许对空间上的有限多个势函数时，Legendre正则性假设允许我们通过自洽方程和全局最大化条件在线性平衡对中识别非线性平衡对。在这些假设下，一维中的实解析性和线性平衡对的唯一性意味着非线性平衡集的有限性。有限状态示例展示了连续和一级相变、亚稳定性以及具有相同单坐标边际但不同两坐标分布的相。基于Taylor展开的准则确定了相关的临界指数。

英文摘要

We develop a nonlinear thermodynamic formalism for correspondences on compact metric spaces. A continuous energy is evaluated on empirical measures of successive pairs of states. The measure-theoretic pressure of a stationary transition pair is the sum of its kernel entropy and the energy of the joint distribution of two successive states. For forward expansive correspondences satisfying an ergodic-abundance condition, we prove a variational principle and the existence of nonlinear equilibrium pairs. We further prove that limits of the two-coordinate projections of asymptotically maximizing Gibbs ensembles are averages of the two-coordinate distributions of equilibrium pairs. When the energy depends on finitely many potentials on the space of admissible pairs, Legendre regularity assumptions allow us to identify nonlinear equilibrium pairs among linear equilibrium pairs through a self-consistency equation and a global maximization condition. Under these assumptions, real analyticity in one dimension and uniqueness of linear equilibrium pairs imply finiteness of the nonlinear equilibrium set. Finite-state examples exhibit continuous and first-order transitions, metastability, and phases that have identical one-coordinate marginals but distinct two-coordinate distributions. A criterion based on Taylor expansions determines the associated critical exponents

发表机构

• College of Science, Northwest A&F University（西北农林科技大学理学院）
• School of Mathematics, Northwest University（西北大学数学学院）

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