Logarithmic Aging Diffusion from a Multiplicative Event Clock: Rare Event Statistics, Ultraslow Transport, and Ensemble-Time Inequivalence

Abstract

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.

86 pages, 12 figures