AI 中文总结
研究对数时间依赖性老化材料的潜在随机机制，提出特定对数老化过程，其事件时间具乘法性。通过事件计数等统计产生多种定律，区分不同表示，用联合可观测量一致性测试时钟，揭示系综 - 时间不等价等特性。
AI 中文摘要
对数时间依赖性出现在许多老化材料中，但既不是$\ln t$弛豫定律也不是$1/t$事件率能唯一确定潜在的随机机制。我们研究了一个特定的对数老化过程，它由每次事件后迭代年龄条件前向递归定律定义。该规则使事件时间具有乘法性：对数比$U_n = \ln(T_{n + 1}/T_n)$独立同分布，具有明确的非指数密度。因此，事件计数的均值和方差都随$\ln(t/t_0)$线性增长，而第$n$个事件时间的密度在中心区域具有对数正态分布，在固定$n$的远尾具有代数分布。这些时钟统计产生对数漂移和扩散、局部细致平衡下的爱因斯坦关系以及超慢传输和目标生存定律。它们还将轨迹可重复性与系综 - 时间等价性分开：时间平均均方位移的相对散射随$1/\ln(T/t_0)$衰减，尽管其均值不收敛到系综滞后均方位移。我们区分了精确的事件级构造与其扩散极限广义福克 - 普朗克和随机时钟从属表示，以及广义朗之万闭包，后者可以匹配选定的响应和协方差，但不一定能再现事件计数或稀有持续时间统计。因此，所提出的时钟不是通过单个对数曲线来测试，而是通过乘数、计数、传输、首次通过和有限窗口可观测量的联合、无需拟合的一致性来测试。
英文摘要
Logarithmic time dependences occur in many aging materials, but neither a $\ln t$ relaxation law nor a $1/t$ event rate uniquely identifies the underlying stochastic mechanism. We examine a specific log-aging process defined by iterating the age-conditioned forward-recurrence law after every event. This rule makes the event times multiplicative: the logarithmic ratios $U_n=\ln(T_{n+1}/T_n)$ are independent and identically distributed with an explicit non-exponential density. Consequently, both the mean and the variance of the event count grow linearly with $\ln(t/t_0)$, while the density of the $n$th event time has a log-normal central sector and a fixed-$n$ algebraic far tail. These clock statistics generate logarithmic drift and spreading, an Einstein relation under local detailed balance, and ultraslow transit and target-survival laws. They also separate trajectory reproducibility from ensemble--time equivalence: the relative scatter of the time-averaged mean-square displacement decays as $1/\ln(T/t_0)$, although its mean does not converge to the ensemble lag MSD. We distinguish the exact event-level construction from its diffusion-limit generalized Fokker--Planck and random-clock subordination representations, and from a generalized-Langevin closure that can match selected responses and covariances but need not reproduce event counts or rare-duration statistics. The proposed clock is therefore tested not by a single logarithmic curve, but by the joint, no-refitting consistency of multiplier, count, transport, first-passage, and finite-window observables.
Comments86 pages, 12 figures
