基于动力学参考的态可转换性与涨落定理： majorization 与鞅的结合
State convertibility and fluctuation theorems from a dynamical reference: majorization meets martingales
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中文总结 AI 辅助
该研究将热力学中基于固定参考态的态可转换性准则扩展到时变参考分布，结合 majorization 与鞅理论推导参考相对熵产生的涨落定理，为验证参考演化提供了模型无关的诊断方法。
中文摘要 AI 辅助
态可转换性是判断在特定资源集合下变换是否可行的基础概念。在热力学领域，物理过程需保持通常为微正则或正则形式的参考态，这对应 majorization 和热 majorization 概念——这些准则需要构造并比较依赖态的洛伦兹曲线。本工作首先将这些概念统一并扩展到任意且可能随时间变化的参考分布 $g(t)$，引入 $g(t)$-majorization 概念；随后引入对偶图景，将态可转换性转化为一维凸序问题，这使我们能够证明：当且仅当相对布居数 $k_j(t)/g_j(t)$ 的关联实值分布通过鞅连接时，转变是容许的。在此基础上，我们推导了参考相对熵产生的精确涨落定理，其平均违反通过 $\chi^2$ 散度界验证假设与真实参考演化之间的不匹配——这是一种与模型无关的诊断方法，无需对真实参考演化进行独立表征，且能将观测到的涨落关系失效转化为参考误差的认证下界。
英文摘要
State convertibility represents a fundamental concept used to determine whether a transformation is possible given a specific set of resources. Within the field of Thermodynamics, where physical process are required to preserve a reference state typically in microcanonical or canonical form, this translates into the notions of majorization and thermo-majorization ---criteria that require constructing and comparing state-dependent Lorenz curves. In this work, we firstly unify and extend these notions to an arbitrary and possibly time-dependent reference distribution $g(t)$, introducing the concept of $g(t)$-majorization; we then introduce a dual picture whereby state convertibility is turned into a one-dimensional convex-order problem, which allows us to demonstrate that a transition is admissible if and only if the associated real-valued distributions of relative populations $ k_j(t)/g_j(t)$ are connected by a martingale. Building on it, we then derive an exact fluctuation theorem for a reference-relative entropy production whose average violation certifies, through a $χ^{2}$-divergence bound, the mismatch between an assumed and the true reference evolution---a model-independent diagnostic that requires no independent characterization of the latter and turns an observed breakdown of the fluctuation relation into a certified lower bound on the reference error.
