Wen Bao, Li Ming-Gen, Liu Jian, Bao Jing-Dong
Institutes of Science and Development, Chinese Academy of Sciences, Beijing 100190, China.
School of Public Policy and Management, University of Chinese Academy of Sciences, Beijing 100049, China.
Entropy (Basel). 2023 Jun 30;25(7):1012. doi: 10.3390/e25071012.
In statistical mechanics, the ergodic hypothesis (i.e., the long-time average is the same as the ensemble average) accompanying anomalous diffusion has become a continuous topic of research, being closely related to irreversibility and increasing entropy. While measurement time is finite for a given process, the time average of an observable quantity might be a random variable, whose distribution width narrows with time, and one wonders how long it takes for the convergence rate to become a constant. This is also the premise of ergodic establishment, because the ensemble average is always equal to the constant. We focus on the time-dependent fluctuation width for the time average of both the velocity and kinetic energy of a force-free particle described by the generalized Langevin equation, where the stationary velocity autocorrelation function is considered. Subsequently, the shortest time scale can be estimated for a system transferring from a stationary state to an effective ergodic state. Moreover, a logarithmic spatial potential is used to modulate the processes associated with free ballistic diffusion and the control of diffusion, as well as the minimal realization of the whole power-law regime. The results presented suggest that non-ergodicity mimics the sparseness of the medium and reveals the unique role of logarithmic potential in modulating diffusion behavior.
在统计力学中，伴随反常扩散的遍历性假设（即长时间平均值与系综平均值相同）已成为一个持续的研究课题，与不可逆性和熵增加密切相关。虽然给定过程的测量时间是有限的，但可观测量的时间平均值可能是一个随机变量，其分布宽度随时间变窄，人们会好奇收敛速率变为常数需要多长时间。这也是遍历性确立的前提，因为系综平均值始终等于常数。我们关注由广义朗之万方程描述的无外力粒子的速度和动能的时间平均值的时间相关涨落宽度，其中考虑了平稳速度自相关函数。随后，可以估计系统从稳态转变为有效遍历态的最短时间尺度。此外，使用对数空间势来调制与自由弹道扩散和扩散控制相关的过程，以及整个幂律 regime 的最小实现。给出的结果表明，非遍历性模拟了介质的稀疏性，并揭示了对数势在调制扩散行为中的独特作用。
