Ergodic properties of continuous-time random walks: finite-size effects and ensemble dependences

Abstract

The effects of spatial confinements and smooth cutoffs of the waiting time distribution in continuous-time random walks (CTRWs) are studied analytically. We also investigate dependences of ergodic properties on initial ensembles (i.e., distributions of the first waiting time). Here, we consider two ensembles: the equilibrium and a typical non-equilibrium ensembles. For both ensembles, it is shown that the time-averaged mean square displacement (TAMSD) exhibits a crossover from normal to anomalous diffusion due to the spacial confinement and this crossover does not vanish even in the long measurement time limit. Moreover, for the non-equilibrium ensemble, we show that the probability density function of the diffusion constant of TAMSD follows the transient Mittag-Leffler distribution, and that scatter in the TAMSD shows a clear transition from weak ergodicity breaking (an irreproducible regime) to ordinary ergodic behavior (a reproducible regime) as the measurement time increases. This convergence to ordinary ergodicity requires a long measurement time compared to common distributions such as the exponential distribution; in other words, the weak ergodicity breaking persists for a long time. In addition, it is shown that, besides the TAMSD, a class of observables also exhibits this slow convergence to ergodicity. We also point out that, even though the system with the equilibrium initial ensemble shows no aging, its behavior is quite similar to that for the non-equilibrium ensemble.

19 pages, 5 figures

References in corpus (27)

• Modeling the scaling properties of human mobility
• In vivo anomalous diffusion and weak ergodicity breaking of lipid granules
• Random Time-Scale Invariant Diffusion and Transport Coefficients
• Kinetics of protein-DNA interaction: facilitated target location in sequence-dependent potential
• Single particle tracking in systems showing anomalous diffusion: the role of weak ergodicity breaking
• Ergodic Properties of Fractional Brownian-Langevin Motion
• One- and two-particle microrheology
• Viscoelastic subdiffusion: from anomalous to normal
• Aging Continuous Time Random Walks
• Distribution of Time-Averaged Observables for Weak Ergodicity Breaking
• Aging properties of an anomalously diffusing particle
• The long reach of DNA sequence heterogeneity in diffusive processes
• Aging Correlation Functions for Blinking Nano-Crystals, and Other On - Off Stochastic Processes
• Diffusion and Relaxation Controlled by Tempered α-stable Processes
• Ion channel gating: a first passage time analysis of the Kramers type
• Multi-scaling properties of truncated Levy flights
• Weakly non-ergodic Statistical Physics
• Truncation effects in superdiffusive front propagation with Lévy flights
• Full characterization of the fractional Poisson process
• A fractional diffusion equation for two-point probability distributions of a continuous-time random walk
• Generalized Arcsine Law and Stable Law in an Infinite Measure Dynamical System
• Multi-point Distribution Function for the Continuous Time Random Walk
• Ultraslow Convergence to Ergodicity in Transient Subdiffusion
• Separation of trajectories and its Relation to Entropy for Intermittent Systems with a Zero Lyapunov exponent
• Crossover Time in Relative Fluctuations Characterizes the Longest Relaxation Time of Entangled Polymers
• Subordination scenario of anomalous relaxation
• Ergodic transitions in continuous-time random walks

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