School of Mathematics and Statistics, University of Sheffield, Sheffield, S3 7RH, United Kingdom.
Department of Ecology and Evolution, University of Chicago, Chicago, Illinois 60637, USA.
Phys Rev E. 2016 Jun;93(6):062127. doi: 10.1103/PhysRevE.93.062127. Epub 2016 Jun 20.
We investigate the geometric structure of a nonequilibrium process and its geodesic solutions. By employing an exactly solvable model of a driven dissipative system (generalized nonautonomous Ornstein-Uhlenbeck process), we compute the time-dependent probability density functions (PDFs) and investigate the evolution of this system in a statistical metric space where the distance between two points (the so-called information length) quantifies the change in information along a trajectory of the PDFs. In this metric space, we find a geodesic for which the information propagates at constant speed, and demonstrate its utility as an optimal path to reduce the total time and total dissipated energy. In particular, through examples of physical realizations of such geodesic solutions satisfying boundary conditions, we present a resonance phenomenon in the geodesic solution and the discretization into cyclic geodesic solutions. Implications for controlling population growth are further discussed in a stochastic logistic model, where a periodic modulation of the diffusion coefficient and the deterministic force by a small amount is shown to have a significant controlling effect.
我们研究了非平衡过程的几何结构及其测地线解。通过采用一个可精确求解的驱动耗散系统模型（广义非自治 Ornstein-Uhlenbeck 过程），我们计算了随时间变化的概率密度函数（PDF），并在一个统计度量空间中研究了该系统的演化，其中两点之间的距离（所谓的信息长度）量化了 PDF 轨迹上信息的变化。在这个度量空间中，我们找到了一条测地线，信息沿着这条测地线以恒定速度传播，并展示了它作为一种最优路径的实用性，可以减少总时间和总耗散能量。特别是，通过满足边界条件的这种测地线解的物理实现的例子，我们展示了测地线解中的共振现象和循环测地线解的离散化。在一个随机 logistic 模型中进一步讨论了控制种群增长的意义，其中通过小量的周期性调制扩散系数和确定性力被证明具有显著的控制效果。
