AI 中文总结
本研究以旋转随机朗道模型为对象，利用Martin-Siggia-Rose路径积分证明其涌现$\boldsymbol{\rm Z}_2$对称性可导出关联涨落与响应的Ward恒等式，该恒等式与高温涨落-耗散关系一致，明确了涨落-响应关系的粗粒化随机动力学起源。
AI 中文摘要
本工作研究涨落-响应关系仅由粗粒化随机动力学涌现的程度，以及哪些方面依赖于系统的额外信息。为解决该问题，我们研究旋转随机朗道模型——一个可精确求解的系统，描述恒定磁场中过阻尼带电布朗粒子与旋转环境耗散耦合，其稳态支持循环概率流。利用Martin-Siggia-Rose路径积分，我们证明存在一个涌现的$\boldsymbol{\rm Z}_2$对称性变换，实现时间反演动力学并使作用量改变一个边界项。与Crooks涨落定理对比，该边界项对应稳态构型间跃迁相关的熵。将理论耦合到外源后，同一对称性导出关联涨落与响应的Ward恒等式，这些恒等式完全源于粗粒化随机理论，不固定噪声强度。最后，施加爱因斯坦关系后，我们证明它们与旋转Kubo-Martin-Schwinger条件对微观吉布斯系综隐含的高温涨落-耗散关系一致。
英文摘要
In this work, we investigate the extent to which fluctuation--response relations emerge from coarse-grained stochastic dynamics alone, and which aspects instead depend on additional information about the system. To address this question, we study the rotating stochastic Landau model, an exactly solvable system describing an overdamped charged Brownian particle in a constant magnetic field, coupled dissipatively to a rotating environment, whose steady state supports circulating probability currents. Using the Martin--Siggia--Rose path integral, we show that there is an emergent $\mathbb{Z}_2$ symmetry transformation that implements the time-reversed dynamics and changes the action by a boundary term. Comparison with the Crooks fluctuation theorem identifies this term with the entropy associated with transitions between steady-state configurations. After coupling the theory to external sources, the same symmetry yields Ward identities relating fluctuations and response. These identities follow entirely from the coarse-grained stochastic theory and do not fix the noise strength. Finally, upon imposing the Einstein relation, we show that they coincide with the high-temperature fluctuation--dissipation relations implied by the rotating Kubo--Martin--Schwinger condition for a microscopic Gibbs ensemble.
