发表机构
The Ohio State University; Amazon.com Inc.（俄亥俄州立大学; 亚马逊公司）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出最小作用量框架，以吉布斯分布为参考，用信息几何解释分布演化，通过信息投影降低动力学代价，并用朗伯W函数实现闭式解，统一了最小作用量、信息几何与非平衡热力学。
AI 中文摘要
我们发展了一个最小作用量框架，用于描述在受约束的增量变化下，概率分布如何从平衡态演化到指定的非平衡态。以吉布斯分布作为平衡参考，该框架赋予概率单纯形的几何结构以直接的物理意义：与平衡态的距离对应非平衡自由能，而连续分布之间的变化则携带信息动力学代价。相对熵的毕达哥拉斯结构随后提供了本工作的核心洞见。它表明，通过顺序信息投影选择的中间分布可以降低大转变的动力学代价，并建立了沿路径的动力学支出与到达目标分布所累积的自由能之间的能量守恒类关系。受此几何结构启发，我们通过连续信息投影构造了一条贪婪的最小作用量路径，利用朗伯W函数获得了每个投影的闭式表征，并建立了有限步性能保证。我们进一步表明，状态依赖的代价可以通过重塑底层吉布斯参考自然地纳入，从而为路径惩罚提供了作为有效能量景观修改的热力学解释。综合来看，这些结果通过最小作用量、信息几何和非平衡热力学为分布演化提供了一个统一视角。
英文摘要
We develop a least-action framework for describing how a probability distribution can evolve from an equilibrium state to a prescribed nonequilibrium state under constrained incremental changes. Taking a Gibbs distribution as the equilibrium reference, the framework gives a direct physical meaning to the geometry of the probability simplex: distance from equilibrium corresponds to nonequilibrium free energy, while changes between successive distributions carry an informational kinetic cost. The Pythagorean structure of relative entropy then provides the central insight of the work. It shows that intermediate distributions chosen via sequential information projections can reduce the kinetic cost of large transitions and establishes an energy-conservation-like relation between the kinetic expenditure along a path and the free energy accumulated in reaching the target distribution. Motivated by this geometry, we construct a greedy least-action path through successive information projections, obtain a closed-form characterization of each projection through the Lambert W function, and establish a finite-step performance guarantee. We further show that state-dependent costs can be incorporated naturally by reshaping the underlying Gibbs reference, providing a thermodynamic interpretation of path penalties as modifications of the effective energy landscape. Together, these results provide a unified view of distributional evolution through least action, information geometry, and nonequilibrium thermodynamics.
