Ageing first passage time density in continuous time random walks and quenched energy landscapes

Abstract

We study the first passage dynamics of an ageing stochastic process in the continuous time random walk (CTRW) framework. In such CTRW processes the test particle performs a random walk, in which successive steps are separated by random waiting times distributed in terms of the waiting time probability density function ψ(t) ≃ t −1−α (0 ≤ α ≤ 2). An ageing stochastic process is defined by the explicit dependence of its dynamic quantities on the ageing time t a , the time elapsed between its preparation and the start of the observation. Subdiffusive ageing CTRWs describe systems such as charge carriers in amorphous semiconductors, tracer dispersion in geological and biological systems, or the dynamics of blinking quantum dots. We derive the exact forms of the first passage time density for an ageing subdiffusive CTRW in the semi-infinite, confined, and biased case, finding different scaling regimes for weakly, intermediately, and strongly aged systems: these regimes, with different scaling laws, are also found when the scaling exponent is in the range 1 < α < 2, for sufficiently long t a . We compare our results with the ageing motion of a test particle in a quenched energy landscape. We test our theoretical results in the quenched landscape against simulations: only when the bias is strong enough, the correlations from returning to previously visited sites become insignificant and the results approach the aging CTRW results. With small or without bias, the ageing effects disappear and a change in the exponent compared to the case of a completely annealed landscape can be found, reflecting the build-up of correlations in the quenched landscape.

Fluctuations in interacting particle systems with memory

“…However, the situation is more complicated for many-particle systems, or those with a dependence on multiple currents, and it may be practically difficult to obtain explicit forms for the relevant waiting time distributions. There is a vast body of work on CTRWs with identically distributed, typically power law, waiting times (see [33,34] for just a couple of recent examples, discussing different scaling regimes and the effects of bias) as well as on more general time-homogeneous semi-Markov processes [35] and applications [36]. Helpful explanations of the connections between different commonly-employed formulations can be found in [37] and [38].…”

Manifestations of Projection-Induced Memory: General Theory and the Tilted Single File

“…The boundary conditions in Eq. (27) restrict the domain to a hypercone x 0 ∈ Ξ such that x 0,i ≤ x 0,i+1 for i = 1, . .…”

“…, k N ) the N -tuple of all single-state indices k i one can show by direct substitution that the many-body eigenvalues are given by λ k = N i=1 λ k i and the corresponding orthonormal many-body eigenfunctions that obey the non-crossing internal boundary conditions Eq. (27) have the form…”

“…Over the past decades the field of anomalous diffusion and non-Markovian dynamics grew to a mainstream physical topic [1,2,3,4,5,6,7,8,9,10] backed up by a surge of experimental observations [11,12,13,14,15,16] (the list of works is anything but exhaustive). From a theoretical point of view the description of anomalous and non-Markovian phenomena is not universal [1] and can be roughly (and judiciously) classified according to the underlying phenomenology: (i) renewal continuous-time random walk and fractional Fokker-Planck approaches [1,2,17,3,18], (ii) diffusion in disordered media [19,20,21,22,23,24,25,26,27], (iii) generalized Langevin equation descriptions [28,29,30,31,32,33,34,35,36], (iv) spatially heterogeneous diffusion [37,38,39,40,41,…”

Rate equations, spatial moments, and concentration profiles for mobile-immobile models with power-law and mixed waiting time distributions

“…By choosing a Mittag-Leffler (ML) waiting time PDF our model contains the bi-fractional solute transport models in [45,71] in the long-time limit, including a power-law decay of the total mobile mass, while retaining a finite value of the memory function in the zero-time limit, γ ML (0). From a physical perspective the accumulation of immobile particles is similar to particles diffusing in an energy landscape scattered with energetic traps with power-law trapping times [75][76][77]. We note that while many studies focus on BTCs, some work has been reported regarding the spatial tracer plumes [20,45,78,79].…”