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.

最近在研究多尺度随机过程的非平衡热力学过程中发现了新的隐藏热力学结构。在这里，我们揭示了二阶平均下非均匀奇异扰动扩散过程的一般热力学泛函的鞅结构，其中一般热力学泛函被定义为原始过程定律和可比过程定律之间的对数 Radon-Nykodim 导数（向前case) 或其时间反转 (backward case)。在前向情况下，我们证明热力学泛函的规则部分和异常部分是正交的鞅。在后向情况下，虽然规则部分可能不是鞅，但我们证明异常部分仍然是鞅。借助鞅结构，我们证明了一般热力学泛函的正则和反常部分满足的积分涨落定理。还讨论了随机热力学的进一步扩展和应用，包括在不存在或存在奇数变量的情况下，熵产生和内务热的规则和异常部分的鞅结构和波动定理。