Sanders Julia, Baldovin Marco, Muratore-Ginanneschi Paolo
Department of Mathematics and Statistics, University of Helsinki, 00014 Helsinki, Finland.
Institute for Complex Systems, CNR, 00185 Rome, Italy.
J Stat Phys. 2024;191(9):117. doi: 10.1007/s10955-024-03320-w. Epub 2024 Sep 17.
Optimal control theory deals with finding protocols to steer a system between assigned initial and final states, such that a trajectory-dependent cost function is minimized. The application of optimal control to stochastic systems is an open and challenging research frontier, with a spectrum of applications ranging from stochastic thermodynamics to biophysics and data science. Among these, the design of nanoscale electronic components motivates the study of underdamped dynamics, leading to practical and conceptual difficulties. In this work, we develop analytic techniques to determine protocols steering finite time transitions at a minimum thermodynamic cost for stochastic underdamped dynamics. As cost functions, we consider two paradigmatic thermodynamic indicators. The first is the Kullback-Leibler divergence between the probability measure of the controlled process and that of a reference process. The corresponding optimization problem is the underdamped version of the Schrödinger diffusion problem that has been widely studied in the overdamped regime. The second is the mean entropy production during the transition, corresponding to the second law of modern stochastic thermodynamics. For transitions between Gaussian states, we show that optimal protocols satisfy a Lyapunov equation, a central tool in stability analysis of dynamical systems. For transitions between states described by general Maxwell-Boltzmann distributions, we introduce an infinite-dimensional version of the Poincaré-Lindstedt multiscale perturbation theory around the overdamped limit. This technique fundamentally improves the standard multiscale expansion. Indeed, it enables the explicit computation of momentum cumulants, whose variation in time is a distinctive trait of underdamped dynamics and is directly accessible to experimental observation. Our results allow us to numerically study cost asymmetries in expansion and compression processes and make predictions for inertial corrections to optimal protocols in the Landauer erasure problem at the nanoscale.
最优控制理论致力于寻找使系统在指定的初始状态和最终状态之间转换的协议，从而使依赖于轨迹的代价函数最小化。将最优控制应用于随机系统是一个开放且具有挑战性的研究前沿领域，其应用范围涵盖从随机热力学到生物物理学和数据科学等多个方面。其中，纳米级电子元件的设计推动了对欠阻尼动力学的研究，这带来了实际和概念上的困难。在这项工作中，我们开发了分析技术，以确定在随机欠阻尼动力学中以最小热力学成本引导有限时间转换的协议。作为代价函数，我们考虑两个典型的热力学指标。第一个是受控过程的概率测度与参考过程的概率测度之间的库尔贝克 - 莱布勒散度。相应的优化问题是在过阻尼状态下已被广泛研究的薛定谔扩散问题的欠阻尼版本。第二个是转换过程中的平均熵产生，这与现代随机热力学的第二定律相对应。对于高斯态之间的转换，我们表明最优协议满足李雅普诺夫方程，这是动力系统稳定性分析中的核心工具。对于由一般麦克斯韦 - 玻尔兹曼分布描述的状态之间的转换，我们围绕过阻尼极限引入了庞加莱 - 林德施泰特多尺度微扰理论的无穷维版本。这项技术从根本上改进了标准的多尺度展开。实际上，它能够明确计算动量累积量，其随时间的变化是欠阻尼动力学的一个显著特征，并且可以直接通过实验观测到。我们的结果使我们能够对膨胀和压缩过程中的成本不对称性进行数值研究，并对纳米尺度下兰道尔擦除问题中最优协议的惯性修正做出预测。
