Strong anomalous diffusive behaviors of the two-state random walk process

Generalized two-state random walk model: Nontrivial anomalous diffusion, aging, and ergodicity breaking

The intermittent stochastic motion is a dichotomous process that alternates between two distinct states. This phenomenon, observed across various physical and biological systems, is attracting increasing interest and highlighting the need for comprehensive theories to describe it. In this paper, we introduce a generalized intermittent random walk model based on a renewal process that alternates between the continuous time random walk (CTRW) state and the generalized Lévy walk (gLW) state. Notably, the nonlinear space-time coupling inherent in the gLW state allows this generalized model to encompass a variety of random walk models and makes it applicable to diverse systems. By deriving the velocity correlation function and utilizing the scaling Green-Kubo relation, the ensemble-averaged and time-averaged mean-squared displacement (MSD) is calculated, and the anomalous diffusive behavior, aging effect, and ergodic property of the model are further analyzed and discussed. The results reveal that, due to the intermittent nature, there are two diffusive terms in the expression of the MSD, and the diffusion can be intermediately characterized by the diffusive term with the largest diffusion coefficient instead of the diffusive term with the largest diffusion exponent, which is significantly different from single-state stochastic process. We demonstrate that, due to the power-law distribution of sojourn times, nonlinear space-time coupling, and intermittent characteristics, both ergodicity and nonergodicity can coexist in intermittent stochastic processes.

Simulation of the continuous time random walk using subordination schemes

The continuous time random walk model has been widely applied in various fields, including physics, biology, chemistry, finance, social phenomena, etc. In this work, we present an algorithm that utilizes a subordinate formula to generate data of the continuous time random walk in the long time limit. The algorithm has been validated using commonly employed observables, such as typical fluctuations of the positional distribution, rare fluctuations, the mean and the variance of the position, and breakthrough curves with time-dependent bias, demonstrating a perfect match.

Anomalous diffusion, nonergodicity, non-Gaussianity, and aging of fractional Brownian motion with nonlinear clocks

How do nonlinear clocks in time and/or space affect the fundamental properties of a stochastic process? Specifically, how precisely may ergodic processes such as fractional Brownian motion (FBM) acquire predictable nonergodic and aging features being subjected to such conditions? We address these questions in the current study. To describe different types of non-Brownian motion of particles-including power-law anomalous, ultraslow or logarithmic, as well as superfast or exponential diffusion-we here develop and analyze a generalized stochastic process of scaled-fractional Brownian motion (SFBM). The time- and space-SFBM processes are, respectively, constructed based on FBM running with nonlinear time and space clocks. The fundamental statistical characteristics such as non-Gaussianity of particle displacements, nonergodicity, as well as aging are quantified for time- and space-SFBM by selecting different clocks. The latter parametrize power-law anomalous, ultraslow, and superfast diffusion. The results of our computer simulations are fully consistent with the analytical predictions for several functional forms of clocks. We thoroughly examine the behaviors of the probability-density function, the mean-squared displacement, the time-averaged mean-squared displacement, as well as the aging factor. Our results are applicable for rationalizing the impact of nonlinear time and space properties superimposed onto the FBM-type dynamics. SFBM offers a general framework for a universal and more precise model-based description of anomalous, nonergodic, non-Gaussian, and aging diffusion in single-molecule-tracking observations.

Coexistence of ergodicity and nonergodicity in the aging two-state random walks

The two-state stochastic phenomenon is observed in various systems and is attracting more interest, and it can be described by the two-state random walk (TSRW) model. The TSRW model is a typical two-state renewal process alternating between the continuous-time random walk state and the Lévy walk state, in both of which the sojourn time distributions follow a power law. In this paper, by discussing the statistical properties and calculating the ensemble averaged and time averaged mean squared displacement, the ergodic property and the ultimate diffusive behavior of the aging TSRW is studied. Results reveal that because of the two-state intermittent feature, ergodicity and nonergodicity can coexist in the aging TSRW, which behave as the time scalings of the time averages and ensemble averages not being identically equal. In addition, we find that the unique state occupation mechanism caused by the diverging mean of the sojourn times of one state, determines the aging TSRW's ultimate diffusive behavior at extremely large timescales, i.e., instead of the term with the larger diffusion exponent, the diffusion is surprisingly characterized by the term with the smaller one, which is distinctly different from previous conclusions and known results. At last, we note that the Lévy walk with rests model which also displays aging and ergodicity breaking, can be generalized by the TSRW model.

Anomalous diffusion, aging, and nonergodicity of scaled Brownian motion with fractional Gaussian noise: overview of related experimental observations and models

How does a systematic time-dependence of the diffusion coefficient $D (t)$ affect the ergodic and statistical characteristics of fractional Brownian motion (FBM)? Here, we examine how the behavior of the...