Wadia Neha S, Zarcone Ryan V, DeWeese Michael R
Biophysics Graduate Group, University of California at Berkeley, Berkeley, California 94720, USA.
Department of Physics, University of California at Berkeley, Berkeley, California 94720, USA.
Phys Rev E. 2022 Mar;105(3-1):034130. doi: 10.1103/PhysRevE.105.034130.
Considerable progress has recently been made with geometrical approaches to understanding and controlling small out-of-equilibrium systems, but a mathematically rigorous foundation for these methods has been lacking. Towards this end, we develop a perturbative solution to the Fokker-Planck equation for one-dimensional driven Brownian motion in the overdamped limit enabled by the spectral properties of the corresponding single-particle Schrödinger operator. The perturbation theory is in powers of the inverse characteristic timescale of variation of the fastest varying control parameter, measured in units of the system timescale, which is set by the smallest eigenvalue of the corresponding Schrödinger operator. It applies to any Brownian system for which the Schrödinger operator has a confining potential. We use the theory to rigorously derive an exact formula for a Riemannian "thermodynamic" metric in the space of control parameters of the system. We show that up to second-order terms in the perturbation theory, optimal dissipation-minimizing driving protocols minimize the length defined by this metric. We also show that a previously proposed metric is calculable from our exact formula with corrections that are exponentially suppressed in a characteristic length scale. We illustrate our formula using the two-dimensional example of a harmonic oscillator with time-dependent spring constant in a time-dependent electric field. Lastly, we demonstrate that the Riemannian geometric structure of the optimal control problem is emergent; it derives from the form of the perturbative expansion for the probability density and persists to all orders of the expansion.
最近，在理解和控制小型非平衡系统的几何方法方面取得了相当大的进展，但这些方法缺乏数学上严格的基础。为此，我们针对过阻尼极限下的一维驱动布朗运动，通过相应单粒子薛定谔算子的谱性质，开发了福克 - 普朗克方程的微扰解。微扰理论是关于变化最快的控制参数变化的逆特征时间尺度的幂次展开，该时间尺度以系统时间尺度为单位进行测量，系统时间尺度由相应薛定谔算子的最小本征值确定。它适用于任何薛定谔算子具有限制势的布朗系统。我们使用该理论严格推导了系统控制参数空间中黎曼“热力学”度量的精确公式。我们表明，在微扰理论的二阶项以内，使耗散最小化的最优驱动协议会使由该度量定义的长度最小化。我们还表明，先前提出的度量可以从我们的精确公式计算得出，其修正项在一个特征长度尺度上呈指数衰减。我们以在随时间变化的电场中具有随时间变化的弹簧常数的谐振子的二维示例来说明我们的公式。最后，我们证明最优控制问题的黎曼几何结构是涌现的；它源自概率密度微扰展开的形式，并在展开的所有阶数中都存在。
