概率流的路径依赖熵拉格朗日：平衡 - 熵路由与可组合信息势
Path-Dependent Entropic Lagrangian for Probability Flows: Balance--Entropy Routing and Composable Information Potentials
AI总结：
研究概率分布相关问题，开发路径依赖熵拉格朗日演算，通过受限生成器等扩展概率路径演化，得出热态关系等，KL/香农扇区恢复相关定律，可组合信息和结构势控制多种特性，数值示例验证多项内容。
AI中文摘要：
概率分布在信息论、统计推断和现代概率学习中至关重要。最大熵在规定约束下选择概率状态，但未说明该状态如何达成、概率如何传输，以及沿路径如何考虑耗散和外部信息交换。我们开发了一种路径依赖熵拉格朗日演算，通过受限生成器、上限历史项和显式平衡 - 熵端口路由，将静态状态选择扩展到概率路径演化。构建得出热态关系、保守概率平衡和标准迁移封闭下的非负产生。其KL/香农扇区将最大熵和贝叶斯定律恢复为静态无通量状态，而随时间变化的信息势将内部耗散与供应的信息功率分离。可组合信息和结构势控制尾部、稀疏性、鲁棒性、正则化和非局部多模态，且不改变核算架构。两个数值示例验证了质量守恒、能量分解和总自由能账目。
英文摘要：
Probability distributions are central to information theory, statistical inference, and modern probabilistic learning. Maximum entropy selects a probability state under prescribed constraints, but it does not specify how that state is reached, how probability is transported, or how dissipation and external information exchange are accounted for along the path. We develop a path-dependent entropic Lagrangian calculus that extends static state selection to probability-path evolution through restricted generators, upper-limit history terms, and explicit balance--entropy port routing. The construction yields the thermal state relation, conservative probability balance, and nonnegative production under standard mobility closure. Its KL/Shannon sector recovers maximum-entropy and Bayesian laws as stationary no-flux states, while time-dependent information potentials separate internal dissipation from supplied information power. Composable information and structural potentials control tails, sparsity, robustness, regularization, and nonlocal multimodality without changing the accounting architecture. Two numerical examples verify mass conservation, energy decomposition, and the total free-energy ledger.
