二阶平均下扩散过程一般热力学泛函的鞅结构
Martingale structure for general thermodynamic functionals of diffusion processes under second-order averaging
摘要：
在研究多尺度随机过程的非平衡态热力学过程中，发现了一些新的隐藏的热力学结构。在这里，我们揭示了鞅结构的一般热力学泛函的非齐次奇摄动扩散过程下的二阶平均，其中一般的热力学泛函被定义为对数Radon-Nykodim导数之间的法律的原始过程和可比的过程（前向情况下）或其时间反演（后向情况下）。在正演情形下，我们证明了热力学泛函的正则部分和反常部分是正交鞅。在后向情形下，虽然正则部分可能不是鞅，但我们证明了反常部分仍然是鞅。借助于鞅结构，我们证明了一般热力学泛函的正则部分和反常部分所满足的积分涨落定理。进一步的扩展和应用随机热力学也进行了讨论，包括鞅结构和波动定理的正常和异常部分的熵产生和管家热的存在或不存在的奇变量。
英文摘要：
Novel hidden thermodynamic structures have recently been uncovered during the investigation of nonequilibrium thermodynamics for multiscale stochastic processes. Here we reveal the martingale structure for a general thermodynamic functional of inhomogeneous singularly perturbed diffusion processes under second-order averaging, where a general thermodynamic functional is defined as the logarithmic Radon-Nykodim derivative between the laws of the original process and a comparable process (forward case) or its time reversal (backward case). In the forward case, we prove that the regular and anomalous parts of a thermodynamic functional are orthogonal martingales. In the backward case, while the regular part may not be a martingale, we prove that the anomalous part is still a martingale. With the aid of the martingale structure, we prove the integral fluctuation theorem satisfied by the regular and anomalous parts of a general thermodynamic functional. Further extensions and applications to stochastic thermodynamics are also discussed, including the martingale structure and fluctuation theorems for the regular and anomalous parts of entropy production and housekeeping heat in the absence or presence of odd variables.
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