Yang Ying-Jen, Qian Hong
Department of Applied Mathematics, University of Washington, Seattle, Washington 98195-3925, USA.
Phys Rev E. 2020 Feb;101(2-1):022129. doi: 10.1103/PhysRevE.101.022129.
Stochastic entropy production, which quantifies the difference between the probabilities of trajectories of a stochastic dynamics and its time reversals, has a central role in nonequilibrium thermodynamics. In the theory of probability, the change in the statistical properties of observables due to reversals can be represented by a change in the probability measure. We consider operators on the space of probability measures that induce changes in the statistical properties of a process, and we formulate entropy production in terms of these change-of-probability-measure (CPM) operators. This mathematical underpinning of the origin of entropy production allows us to achieve an organization of various forms of fluctuation relations: All entropy production has a nonnegative mean value, admit the integral fluctuation theorem, and satisfy a rather general fluctuation relation. Other results such as the transient fluctuation theorem and detailed fluctuation theorems then are derived from the general fluctuation relation with more constraints on the operator of entropy production. We use a discrete-time, discrete-state-space Markov process to draw the contradistinction among three reversals of a process: time reversal, protocol reversal, and the dual process. The properties of their corresponding CPM operators are examined, and the domains of validity of various fluctuation relations for entropy production in physics and chemistry are revealed. We also show that our CPM operator formalism can help us rather easily extend other fluctuation relations for excess work and heat, discuss the martingale properties of entropy production, and derive the stochastic integral formulas for entropy production in constant-noise diffusion process with Girsanov theorem. Our formalism provides a general and concise way to study the properties of entropy-related quantities in stochastic thermodynamics and information theory.
随机熵产生量化了随机动力学轨迹与其时间反演轨迹概率之间的差异，在非平衡态热力学中起着核心作用。在概率论中，由于反演导致的可观测量统计性质的变化可以用概率测度的变化来表示。我们考虑概率测度空间上引起过程统计性质变化的算子，并根据这些概率测度变化（CPM）算子来表述熵产生。熵产生起源的这种数学基础使我们能够对各种形式的涨落关系进行整理：所有熵产生都具有非负平均值，服从积分涨落定理，并满足一个相当普遍的涨落关系。然后，诸如瞬态涨落定理和详细涨落定理等其他结果是从对熵产生算子有更多约束的一般涨落关系推导出来的。我们使用离散时间、离散状态空间的马尔可夫过程来区分过程的三种反演：时间反演、协议反演和对偶过程。研究了它们相应的CPM算子的性质，揭示了物理和化学中熵产生的各种涨落关系的有效范围。我们还表明，我们的CPM算子形式主义可以帮助我们相当容易地扩展其他关于超额功和热的涨落关系，讨论熵产生的鞅性质，并利用吉尔萨诺夫定理推导恒定噪声扩散过程中熵产生的随机积分公式。我们的形式主义为研究随机热力学和信息论中与熵相关量的性质提供了一种通用且简洁的方法。
