Microscopic Origin of the Logarithmic Time Evolution of Aging Processes in Complex Systems

Abstract

There exists compelling experimental evidence in numerous systems for logarithmically slow time evolution, yet its full theoretical understanding remains elusive. We here introduce and study a generic transition process in complex systems, based on nonrenewal, aging waiting times. Each state n of the system follows a local clock initiated at t = 0. The random time τ between clock ticks follows the waiting time density ψ(τ). Transitions between states occur only at local clock ticks and are hence triggered by the local forward waiting time, rather than by ψ(τ). For power-law forms ψ(τ) ≃ τ(-1-α) (0 < α < 1) we obtain a logarithmic time evolution of the state number ⟨n(t)⟩ ≃ log(t/t(0)), while for α > 2 the process becomes normal in the sense that ⟨n(t)⟩ ≃ t. In the intermediate range 1 < α < 2 we find the power-law growth ⟨n(t)⟩ ≃ t(α-1). Our model provides a universal description for transition dynamics between aging and nonaging states.

Transient aging in fractional Brownian and Langevin-equation motion

“…Such weakly nonergodic dynamics in the absence and presence of confinement was indeed observed experimentally in both the walls and the bulk volume of living biological cells [12][13][14]. Interestingly, a similar weak ergodicity breaking is obtained for time-and spacecorrelated CTRWs [15] and superaging systems [16] as well as for Markovian diffusion with scaling forms of the position dependence of the diffusion coefficient [17]. In superdiffusive systems the ergodic violation becomes ultraweak in the sense that x 2 ( ) and δ 2 ( ) differ only by a constant factor [18].…”

Non-Ergodicity and Ageing in Anomalous Diffusion

“…96 For α > 2 the process is normal and statistically equivalent to a Poisson update, which is equivalent to the above scenario with finite length for the vacancy diffusion leading to the linear time dependence n(t) t. However, similar to our observations above, the case with a finite characteristic update time τ but diverging variance of waiting times with 1 < α < 2 displays the power-law anomaly n(t) t α−1 . 96 Interestingly, the time average over the time series n(t), 96 way can be viewed as the α → 0 behaviour of the power-law relation in Eq.…”

“…(6.1), can be shown to be governed by the limiting distribution for the product of independent random variables, the log-normal distribution. 96 This approach may thus be of relevance to a large range of applications in which this distribution is identified. 103 In the regular, renewal subdiffusive CTRW ageing affects the statistics of the first jump, given in terms of the forward waiting time t 1 .…”

“…In a modified CTRW model, in which every jump is dominated by the forward waiting time (4.9), the dynamics of the process is significantly slowed down, giving rise to logarithmic time evolutions. 96 These can be connected to single file systems in which each particle separately becomes trapped with a scale-free distribution of trapping times ψ(t). 97 …”

Slow relaxation of cascade-induced defects in Fe

“…On the other hand, the Gibbs model [12] and its variants [13][14][15] argue that the system initially possesses a distribution of relaxation events with a near-constant density as a function of activation barrier, or rates described by a multiplicative stochastic process [2], which leads to logarithmic relaxation. A newly proposed model links logarithmic timeevolution to the system moving from one local state to another, where the waiting time of each state is defined by a power law and where all states evolve simultaneously [16]. Due to a lack of atomistic evidence, the validity of many of these proposed models remains ambiguous and the identity of the drivers to logarithmic relaxation remains elusive.…”