Baldassarri Andrea, Puglisi Andrea
Institute for Complex Systems CNR, University of Rome "La Sapienza", P.le Aldo Moro 2, 00185 Rome, Italy.
Department of Physics, University of Rome "La Sapienza", P.le Aldo Moro 2, 00185 Rome, Italy.
Entropy (Basel). 2025 Jul 2;27(7):718. doi: 10.3390/e27070718.
We investigate the bridge problem for stochastic processes, that is, we analyze the statistical properties of trajectories constrained to begin and terminate at a fixed position within a time interval τ. Our primary focus is the time-reversal symmetry of these trajectories: under which conditions do the statistical properties remain invariant under the transformation t→τ-t? To address this question, we compare the stochastic differential equation describing the bridge, derived equivalently via Doob's transform or stochastic optimal control, with the corresponding equation for the time-reversed bridge. We aim to provide a concise overview of these well-established derivation techniques and subsequently obtain a local condition for the time-reversal asymmetry that is specifically valid for the bridge. We are specifically interested in cases in which detailed balance is not satisfied and aim to eventually quantify the bridge asymmetry and understand how to use it to derive useful information about the underlying out-of-equilibrium dynamics. To this end, we derived a necessary condition for time-reversal symmetry, expressed in terms of the current velocity of the original stochastic process and a quantity linked to detailed balance. As expected, this formulation demonstrates that the bridge is symmetric when detailed balance holds, a sufficient condition that was already known. However, it also suggests that a bridge can exhibit symmetry even when the underlying process violates detailed balance. While we did not identify a specific instance of complete symmetry under broken detailed balance, we present an example of partial symmetry. In this case, some, but not all, components of the bridge display time-reversal symmetry. This example is drawn from a minimal non-equilibrium model, namely Brownian Gyrators, that are linear stochastic processes. We examined non-equilibrium systems driven by a "mechanical" force, specifically those in which the linear drift cannot be expressed as the gradient of a potential. While Gaussian processes like Brownian Gyrators offer valuable insights, it is known that they can be overly simplistic, even in their time-reversal properties. Therefore, we transformed the model into polar coordinates, obtaining a non-Gaussian process representing the squared modulus of the original process. Despite this increased complexity and the violation of detailed balance in the full process, we demonstrate through exact calculations that the bridge of the squared modulus in the isotropic case, constrained to start and end at the origin, exhibits perfect time-reversal symmetry.
我们研究随机过程的桥问题，即分析在时间间隔τ内被约束在固定位置开始和终止的轨迹的统计特性。我们主要关注这些轨迹的时间反演对称性：在哪些条件下，统计特性在变换t→τ - t下保持不变？为了解决这个问题，我们将通过杜布变换或随机最优控制等效推导得到的描述桥的随机微分方程与时间反转桥的相应方程进行比较。我们旨在对这些成熟的推导技术进行简要概述，并随后获得一个专门适用于桥的时间反演不对称性的局部条件。我们特别感兴趣的是不满足细致平衡的情况，并旨在最终量化桥的不对称性，并了解如何利用它来推导有关潜在非平衡动力学的有用信息。为此，我们推导了一个时间反演对称性的必要条件，该条件用原始随机过程的当前速度和与细致平衡相关的一个量来表示。正如预期的那样，这种表述表明当细致平衡成立时桥是对称的，这是一个已知的充分条件。然而，它也表明即使基础过程违反细致平衡，桥也可能表现出对称性。虽然我们没有确定在细致平衡被打破时完全对称的具体实例，但我们给出了一个部分对称的例子。在这种情况下，桥的一些但不是所有分量显示出时间反演对称性。这个例子取自一个最小的非平衡模型，即布朗回转器，它是线性随机过程。我们研究了由“机械”力驱动的非平衡系统，特别是那些线性漂移不能表示为势的梯度的系统。虽然像布朗回转器这样的高斯过程提供了有价值的见解，但众所周知，即使在它们的时间反演特性方面，它们也可能过于简单。因此，我们将模型转换为极坐标，得到一个表示原始过程平方模的非高斯过程。尽管在整个过程中复杂性增加且违反了细致平衡，但我们通过精确计算表明，在各向同性情况下，约束在原点开始和结束的平方模的桥表现出完美的时间反演对称性。
