Trace of anomalous diffusion in a biased quenched trap model

Disorder-Induced Anomalous Mobility Enhancement in Confined Geometries

Strong, scale-free disorder disrupts typical transport properties like the Stokes-Einstein relation and linear response, leading to anomalous diffusion observed in amorphous materials, glasses, living cells, and other systems. Our study reveals that the combination of scale-free quenched disorder and geometrical constraints induces unconventional single-particle mobility behavior. Specifically, in a two-dimensional channel with width w, under external drive, tighter geometrical constraints (smaller w) enhance mobility. We derive an explicit form of the response to an external force by utilizing the double-subordination approach for the quenched trap model. The observed mobility enhancement occurs in the low-temperature regime where the distribution of localization times is scale-free.

Sample-to-sample fluctuations of transport coefficients in the totally asymmetric simple exclusion process with quenched disorder

We consider the totally asymmetric simple exclusion processes on quenched random energy landscapes. We show that the current and the diffusion coefficient differ from those for homogeneous environments. Using the mean-field approximation, we analytically obtain the site density when the particle density is low or high. As a result, the current and the diffusion coefficient are described by the dilute limit of particles or holes, respectively. However, in the intermediate regime, due to the many-body effect, the current and the diffusion coefficient differ from those for single-particle dynamics. The current is almost constant and becomes the maximal value in the intermediate regime. Moreover, the diffusion coefficient decreases with the particle density in the intermediate regime. We obtain analytical expressions for the maximal current and the diffusion coefficient based on the renewal theory. The deepest energy depth plays a central role in determining the maximal current and the diffusion coefficient. As a result, the maximal current and the diffusion coefficient depend crucially on the disorder, i.e., non-self-averaging. Based on the extreme value theory, we find that sample-to-sample fluctuations of the maximal current and diffusion coefficient are characterized by the Weibull distribution. We show that the disorder averages of the maximal current and the diffusion coefficient converge to zero as the system size is increased and quantify the degree of the non-self-averaging effect for the maximal current and the diffusion coefficient.

Non-self-averaging of current in a totally asymmetric simple exclusion process with quenched disorder

We investigate the current properties in the totally asymmetric simple exclusion process (TASEP) on a quenched random energy landscape. In low- and high-density regimes, the properties are characterized by single-particle dynamics. In the intermediate one, the current becomes constant and is maximized. Based on the renewal theory, we derive accurate results for the maximum current. The maximum current significantly depends on a disorder realization, i.e., non-self-averaging (SA). We demonstrate that the disorder average of the maximum current decreases with the system size, and the sample-to-sample fluctuations of the maximum current exceed those of current in the low- and high-density regimes. We find a significant difference between single-particle dynamics and the TASEP. In particular, the non-SA behavior of the maximum current is always observed, whereas the transition from non-SA to SA for current in single-particle dynamics exists.

Time-averaging and nonergodicity of reset geometric Brownian motion with drift

How do near-bankruptcy events in the past affect the dynamics of stock-market prices in the future? Specifically, what are the long-time properties of a time-local exponential growth of stock-market prices under the influence of stochastically occurring economic crashes? Here, we derive the ensemble- and time-averaged properties of the respective "economic" or geometric Brownian motion (GBM) with a nonzero drift exposed to a Poissonian constant-rate price-restarting process of "resetting." We examine-based both on thorough analytical calculations and on findings from systematic stochastic computer simulations-the general situation of reset GBM with a nonzero [positive] drift and for all special cases emerging for varying parameters of drift, volatility, and reset rate in the model. We derive and summarize all short- and long-time dependencies for the mean-squared displacement (MSD), the variance, and the mean time-averaged MSD (TAMSD) of the process of Poisson-reset GBM under the conditions of both rare and frequent resetting. We consider three main regions of model parameters and categorize the crossovers between different functional behaviors of the statistical quantifiers of this process. The analytical relations are fully supported by the results of computer simulations. In particular, we obtain that Poisson-reset GBM is a nonergodic stochastic process, with generally MSD(Δ)≠TAMSD(Δ) and Variance(Δ)≠TAMSD(Δ) at short lag times Δ and for long trajectory lengths T. We investigate the behavior of the ergodicity-breaking parameter in each of the three regions of parameters and examine its dependence on the rate of reset at Δ/T≪1. Applications of these theoretical results to the analysis of prices of reset-containing options are pertinent.

Correlated continuous-time random walk in the velocity field: the role of velocity and weak asymptotics

Within the framework of a space-time correlated continuous-time random walk model, anomalous diffusion of particles moving in the velocity field is studied in this paper. The weak asymptotic form ω(t) ∼ t^-(1+α), 1 < α < 2 for large t, is considered to be the waiting time distribution. The analytical results reveal that the diffusion in the velocity field, i.e., the mean squared displacement, can display a multi-fractional form caused by dispersive bias and space-time correlation. The numerical results indicate that the multi-fractional diffusion leads to a crossover phenomenon in-between the process at an intermediate timescale, followed by a steady state which is always determined by the largest diffusion exponent term. In addition, the role of velocity and weak asymptotics is discussed. The extremely small fluid velocity can characterize the diffusion by a diffusion coefficient instead of diffusion exponent, which is distinctly different from the former definition. In particular, for the waiting time displaying a weak asymptotic property, if the anomalous part is suppressed by the normal part, a second crossover phenomenon appears at an intermediate timescale, followed by a steady normal diffusion, which implies that the anomalies underlying the process are smoothed out at large timescales. Moreover, we discuss that the consideration of bias and correlation could help to avoid a possible not readily noticeable mistake in studying the topic concerned in this paper, which may be helpful in the relevant experimental research.

Correlated continuous-time random walk in a velocity field: Anomalous bifractional crossover

The diffusion of space-time correlated continuous-time random walk moving in the velocity field, which includes the fluid flowing freely and the fluid flowing through porous media, is investigated in this paper. Results reveal that it presents anomalous diffusion merely caused by space-time correlation in the freely flowing fluid, and the bias from the velocity field only supplies a standard advection, which is verified by the corresponding generalized diffusion equation which includes a standard advection term. However, the diffusion in the fluid flowing through porous media, i.e., the mean squared displacement, can display a bifractional form of which one originates from space-time correlation and the other one originates from dispersive bias caused by sticking of the porous media. The fractional advection term emerging in the corresponding generalized diffusion equation confirms the results. Moreover, the coexistence of correlation and dispersive bias result in crossover phenomenon in-between the diffusive process at an intermediate timescale, but just as the definition of diffusion, the one owning the largest diffusion exponent always prevails at large timescales. However, since the two fractional diffusions originate from a different mechanism, even if it owns the smaller diffusion exponent, that one can dominate the diffusion if it fluctuates much stronger than the other one, which no longer obeys the previous conclusion.