随机热力学这一新兴领域将经典的熵、热和功的概念推广到非平衡系统。一个值得注意的发现是，热力学第二定律通常只有在针对随机轨迹系综取适当平均后才成立。由此得到的平均熵产生率量化了系统偏离热力学平衡的程度。本文研究半透界面的存在如何增加单个扩散粒子的平均熵产生。从粒子概率密度的Gibbs-Shannon熵出发，我们证明半透界面或膜$\calS$使平均熵产生率增加的量等于通过界面的通量与界面两侧概率密度之比的对数的乘积沿$\calS$的积分。因此，熵产生率在热力学平衡时为零，但在弛豫到平衡的过程中可以非零，或者当存在非零稳态非平衡态（NESS）时也可以非零。我们以圆上带随机重置的扩散为例说明后者，并证明界面平均熵产生率是重置率和渗透率的非单调函数。最后，我们利用所谓的snapping out Brownian motion给出界面熵产生率的概率解释。这也使我们能够构造熵产生的随机版本。
英文摘要：
The emerging field of stochastic thermodynamics extends classical ideas of entropy, heat and work to non-equilibrium systems. One notable finding is that the second law of thermodynamics typically only holds after taking appropriate averages with respect to an ensemble of stochastic trajectories. The resulting average rate of entropy production then quantifies the degree of departure from thermodynamic equilibrium. In this paper we investigate how the presence of a semipermeable interface increases the average entropy production of a single diffusing particle. Starting from the Gibbs-Shannon entropy for the particle probability density, we show that a semipermeable interface or membrane $\calS$ increases the average rate of entropy production by an amount that is equal to the product of the flux through the interface and the logarithm of the ratio of the probability density on either side of the interface, integrated along $\calS$. The entropy production rate thus vanishes at thermodynamic equilibrium, but can be nonzero during the relaxation to equilibrium, or if there exists a nonzero stationary equilibrium state (NESS). We illustrate the latter using the example of diffusion with stochastic resetting on a circle, and show that the average rate of interfacial entropy production is a nonmonotonic function of the resetting rate and the permeability. Finally, we give a probabilistic interpretation of the interfacial entropy production rate using so-called snapping out Brownian motion. This also allows us to construct a stochastic version of entropy production.
Entanglement Requires Fluctuations at Conformal Interfaces
Institute for Advanced Study, Kyushu University（九州大学高等研究院）；Department of Physics, Kyushu University（九州大学物理系）；RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS)（理化学研究所跨学科理论与数学科学中心(iTHEMS)）
