Title: A variational formulation of stochastic thermodynamics. Part I: Finite-dimensional systems Show Chinese title

Title: 随机热力学的变分公式化。 第一部分：有限维系统

Abstract: In this paper, we develop a variational foundation for stochastic thermodynamics of finite-dimensional, continuous-time systems. Requiring the second law (non-negative average total entropy production) systematically yields a consistent thermodynamic structure, from which novel generalized fluctuation-dissipation relations emerge naturally, ensuring local detailed balance. This principle extends key results of stochastic thermodynamics including an individual trajectory level description of both configurational and thermal variables and fluctuation theorems in an extended thermodynamic phase space. It applies to both closed and open systems, while accommodating state-dependent parameters, nonlinear couplings between configurational and thermal degrees of freedom, and cross-correlated noise consistent with Onsager symmetry. This is achieved by establishing a unified geometric framework in which stochastic thermodynamics emerges from a generalized Lagrange-d'Alembert principle, building on the variational structure introduced by Gay-Balmaz and Yoshimura [Phil. Trans. R. Soc. A 381, 2256 (2023)]. Irreversible and stochastic forces are incorporated through nonlinear nonholonomic constraints, with entropy treated as an independent dynamical variable. This work provides a novel approach for thermodynamically consistent modeling of stochastic systems, and paves the way to applications in continuum systems such as active and complex fluids. Show Chinese abstract

Abstract: 在本文中，我们为有限维连续时间系统的随机热力学建立了变分基础。 系统地要求第二定律（平均总熵产生非负）产生了一致的热力学结构，从中自然地出现了新的广义涨落-耗散关系，确保局部详细平衡。 这一原理扩展了随机热力学的关键结果，包括配置变量和热变量在个体轨迹层次上的描述，以及扩展热力学相空间中的涨落定理。 它适用于封闭系统和开放系统，同时适应状态依赖参数、配置自由度和热自由度之间的非线性耦合，以及符合Onsager对称性的交叉相关噪声。 这是通过建立一个统一的几何框架实现的，在该框架中，随机热力学来源于广义的拉格朗日-达朗贝尔原理，基于Gay-Balmaz和Yoshimura [Phil. Trans. R. Soc. A 381, 2256 (2023)] 引入的变分结构。 不可逆和随机力通过非线性非完整约束引入，熵被当作独立的动力学变量处理。 这项工作为随机系统的热力学一致建模提供了一种新方法，并为在连续系统（如活性和复杂流体）中的应用铺平了道路。