Ergodicity breaking and localization

We study two different forms of fluctuation-dissipation processes generating anomalous relaxations to equilibrium of an initial out of equilibrium condition, the former being based on a stationary although very slow correlation function and the latter characterized by the occurrence of crucial events, namely, non-Poisson renewal events, incompatible with the stationary condition. Both forms of regression to equilibrium have the same non-exponential Mittag-Leffler structure. We analyze the single trajectories of the two processes by recording the time distances between two consecutive origin re-crossings and establishing the corresponding waiting time probability density function (PDF), ψ(t). In the former case, with no crucial events, ψ(t) is exponential and in the latter case, with crucial events, ψ(t) is an inverse power law PDF with a diverging first moment. We discuss the consequences that this result is expected to have for the correct interpretation of some anomalous relaxation processes.

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“…[30], see also Ref. [29], agree with the corresponding results of Ref. [27] based on the Hamiltonian formalism of Weiss [26].…”

Section: Discussionsupporting

confidence: 88%

“…The numerical work, as done on the earlier paper of Ref. [29], is realized generating first the free FBM diffusion x(t) using the algorithm of [30] and deriving the FGN ξ(t) from it by time differentiation, ξ(t) ≡ dx/dt. This is equivalent to assigning to the parameter γ in Eq.…”

Section: A Moving From Subdiffusionmentioning

confidence: 99%

Non-Poisson renewal events and memory

2017

Self Cite

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“…In addition, one is also encouraged to enrich the proposed model by incorporating more factors such as, different network behavior for particular types of content extracted from the item itself. More broadly, one is also encouraged to investigate the potential connection between the theoretical approach applied in this paper and the revolution occurring in physics with an increasing interest for renewal processes and ergodicity breaking 40–42 .…”

Section: Discussionmentioning

confidence: 99%

Uncovering and Predicting the Dynamic Process of Collective Attention with Survival Theory

Bao

Zhang

2017

Sci Rep

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The subject of collective attention is in the center of this era of information explosion. It is thus of great interest to understand the fundamental mechanism underlying attention in large populations within a complex evolving system. Moreover, an ability to predict the dynamic process of collective attention for individual items has important implications in an array of areas. In this report, we propose a generative probabilistic model using a self-excited Hawkes process with survival theory to model and predict the process through which individual items gain their attentions. This model explicitly captures three key ingredients: the intrinsic attractiveness of an item, characterizing its inherent competitiveness against other items; a reinforcement mechanism based on sum of each previous attention triggers; and a power-law temporal relaxation function, corresponding to the aging in the ability to attract new attentions. Experiments on two population-scale datasets demonstrate that this model consistently outperforms the state-of-the-art methods.

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“…For 1 > H > 3/4 the behavior of EB(Δ) of FBM is more subtle/complicated: the EB values also depend explicitly on the lag time [rather than only on (Δ/T )-ratio], with EBs being generally significantly larger. In this range of H exponents FBM features a slower approach to ergodicity: we refer to the studies 7,12,14,[21][22][23] for the continuous-and to Refs. 13,18 for the discrete-time calculations of EB for FBM.…”

Section: Introductionmentioning

confidence: 99%

Inertia triggers nonergodicity of fractional Brownian motion

Cherstvy

Wang

Metzler

et al. 2021

Preprint

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How related are the ergodic properties of the over- and underdamped Langevin equations driven by fractional Gaussian noise? We here find that for massive particles performing fractional Brownian motion (FBM) inertial effects not only destroy the stylized fact of the equivalence of the ensemble-averaged mean-squared displacement (MSD) to the time-averaged MSD (TAMSD) of overdamped or massless FBM, but also concurrently dramatically alter the values of the ergodicity breaking parameter (EB). Our theoretical results for the behavior of EB for underdamped ot massive FBM for varying particle mass m, Hurst exponent H, and trace length T are in excellent agreement with the findings of extensive stochastic computer simulations. The current results can be of interest for the experimental community employing various single-particle-tracking techniques and aiming at assessing the degree of nonergodicity for the recorded time series (studying e.g. the behavior of EB versus lag time). To infer FBM as a realizable model of anomalous diffusion for a set single-particle-tracking data when massive particles are being tracked, the EBs from the data should be compared to EBs of massive (rather than massless) FBM.

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Ergodicity breaking and localization

Cited by 11 publications

References 31 publications

Non-Poisson renewal events and memory

Non-Poisson renewal events and memory

Uncovering and Predicting the Dynamic Process of Collective Attention with Survival Theory

Inertia triggers nonergodicity of fractional Brownian motion

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