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The detrended fluctuation analysis (DFA) is extensively useful in stochastic processes to unveil the long-term correlation. Here, we apply the DFA to point processes that mimick earthquake data. The point processes are synthesized by a model similar to the Epidemic-Type Aftershock Sequence model, and we apply the DFA to time series $N(t)$ of the point processes, where $N(t)$ is the cumulative number of events up to time $t$. Crossover phenomena are found in the DFA for these time series, and extensive numerical simulations suggest that the crossover phenomena are signatures of non-stationarity in the time series. We also find that the crossover time represents a characteristic time scale of the non-stationary process embedded in the time series. Therefore, the DFA for point processes is especially useful in extracting information of non-stationary processes when time series are superpositions of stationary and non-stationary signals. Furthermore, we apply the DFA to the cumulative number $N(t)$ of real earthquakes in Japan, and we find a crossover phenomenon similar to that found for the synthesized data., Comment: 9 pages, 5 figures

We report on a fundamental role of a non-normalized formal steady state, i.e., an infinite invariant density, in a semi-Markov process where the state is determined by the inter-event time of successive renewals. The state describes certain observables found in models of anomalous diffusion, e.g., the velocity in the generalized L\'evy walk model and the energy of a particle in the trap model. In our model, the inter-event-time distribution follows a fat-tailed distribution, which makes the state value more likely to be zero because long inter-event times imply small state values. We find two scaling laws describing the density for the state value, which accumulates in the vicinity of zero in the long-time limit. These laws provide universal behaviors in the accumulation process and give the exact expression of the infinite invariant density. Moreover, we provide two distributional limit theorems for time-averaged observables in these non-stationary processes. We show that the infinite invariant density plays an important role in determining the distribution of time averages., Comment: 16 pages, 7 figures

We investigate Brownian motion with diffusivity alternately fluctuating between fast and slow states. We assume that sojourn-time distributions of these two states are given by exponential or power-law distributions. We develop a theory of alternating renewal processes to study a relaxation function which is expressed with an integral of the diffusivity over time. This relaxation function can be related to a position correlation function if the particle is in a harmonic potential, and to the self-intermediate scattering function if the potential force is absent. It is theoretically shown that, at short times, the exponential relaxation or the stretched-exponential relaxation are observed depending on the power law index of the sojourn-time distributions. In contrast, at long times, a power law decay with an exponential cutoff is observed. The dependencies on the initial ensembles (i.e., equilibrium or non-equilibrium initial ensembles) are also elucidated. These theoretical results are consistent with numerical simulations., Comment: 15 pages, 5 figures

Tracking tracer particles in heterogeneous environments plays an important role in unraveling the material properties. These heterogeneous structures are often static and depend on the sample realizations. Sample-to-sample fluctuations of such disorder realizations sometimes become considerably large. When we investigate the sample-to-sample fluctuations, fundamental averaging procedures are a thermal average for a single disorder realization and the disorder average for different disorder realizations. Here, we report on non-self-averaging phenomena in quenched trap models with finite system sizes, where we consider the periodic and the reflecting boundary conditions. Sample-to-sample fluctuations of diffusivity greatly exceeds trajectory-to-trajectory fluctuations of diffusivity in the corresponding annealed model. For a single disorder realization, the time-averaged mean square displacement and position-dependent observables converge to constants with the aid of the existence of the equilibrium distribution. This is a manifestation of ergodicity. As a result, the time-averaged quantities do not depend on the initial condition nor on the thermal histories but depend crucially on the disorder realization., Comment: 12 pages, 7 figures

We propose a method to detect alternating diffusive states undergoing a free diffusive state and a trapped state described by the Ornstein-Uhlenbeck process. Using a stochastic model with alternating diffusive states, a phenomenological model of glassy dynamics, we show that control parameters in the method may be determined by the mean square displacement and the non-Gaussianity parameter. Our method works when diffusivities for the two states are clearly distinct and all the states last longer than a specified relaxation time. Applying our method to molecular dynamics simulation data of supercooled liquids, we show that trapped states last for a long time and the sojourn-time distribution for trapped states becomes a power-law form as the temperature approaches the glass temperature.

We study a relaxation behavior of an Ornstein-Uhlenbeck (OU) process with a time-dependent and fluctuating diffusivity. In this process, the dynamics of a position vector is modeled by the Langevin equation with a linear restoring force and a fluctuating diffusivity (FD). This process can be interpreted as a simple model of the relaxational dynamics with internal degrees of freedom or in a heterogeneous environment. By utilizing the functional integral expression and the transfer matrix method, we show that the relaxation function can be expressed in terms of the eigenvalues and eigenfunctions of the transfer matrix, for general FD processes. We apply our general theory to two simple FD processes, where the FD is described by the Markovian two-state model or an OU type process. We show analytic expressions of the relaxation functions in these models, and their asymptotic forms. We also show that the relaxation behavior of the OU process with an FD is qualitatively different from those obtained from conventional models such as the generalized Langevin equation., 22 pages, 3 figures, to appear in Phys. Rev. E

In heterogeneous environments, the diffusivity is not constant but changes with time. It is important to detect changes in the diffusivity from single-particle-tracking trajectories in experiments. Here, we devise a novel method for detecting the transition times of the diffusivity from trajectory data. A key idea of this method is the introduction of a characteristic time scale of the diffusive states, which is obtained by a fluctuation analysis of the time-averaged mean square displacements. We test our method in silico by using the Langevin equation with a fluctuating diffusivity. We show that our method can successfully detect the transition times of diffusive states and obtain the diffusion coefficient as a function of time. This method will provide a quantitative description of the fluctuating diffusivity in heterogeneous environments and can be applied to time series with transitions of states., Comment: 11 pages, 4 figures

We investigate two types of random walks with a fluctuating probability (bias) in which the random walker jumps to the right. One is a ‘time-quenched framework’ using bias time series such as periodic, quasi-periodic, and chaotic time series (chaotically driven bias). The other is a ‘time-annealed framework’ using the fluctuating bias generated by a stochastic process, which is not quenched in time. We show that the diffusive properties in the time-quenched framework can be characterised by the ensemble average of the time-averaged variance (ETVAR), whereas the ensemble average of the time-averaged mean square displacement (ETMSD) fails to capture the diffusion, even when the total bias is zero. We demonstrate that the ETVAR increases linearly with time, and the diffusion coefficient can be estimated by the time average of the local diffusion coefficient. In the time-annealed framework, we analytically and numerically show normal diffusion and superdiffusion, similar to the Lévy walk. Our findings will lead to new developments in information and communication technologies, such as efficient energy transfer for information propagation and quick solution searching.

We consider Langevin equation with dichotomously fluctuating diffusivity, where the diffusion coefficient changes dichotomously in time, in order to study fluctuations of time-averaged observables in temporary heterogeneous diffusion process. We find that occupation time statistics is a powerful tool for calculating the time-averaged mean square displacement in the model. We show that the time-averaged diffusion coefficients are intrinsically random when the mean sojourn time for one of the states diverges. Our model provides anomalous fluctuations of time-averaged diffusivity, which have relevance to large fluctuations of the diffusion coefficient in single-particle-tracking experiments., Comment: 6 pages, 3 figures

Local diffusion coefficients in disordered systems such as spin glass systems and living cells are highly heterogeneous and may change over time. Such a time-dependent and spatially heterogeneous environment results in irreproducibility of single-particle-tracking measurements. Irreproducibility of time-averaged observables has been theoretically studied in the context of weak ergodicity breaking in stochastic processes. Here, we provide rigorous descriptions of equilibrium and non-equilibrium diffusion processes for the annealed transit time model, which is a heterogeneous diffusion model in living cells. We give analytical solutions for the mean square displacement (MSD) and the relative standard deviation of the time-averaged MSD for equilibrium and non-equilibrium situations. We find that the time-averaged MSD grows linearly with time and that the diffusion coefficients are intrinsically random in non-equilibrium situations. Our findings pave the way for a theoretical understanding of distributional behavior of the diffusion coefficients in disordered systems., 13 pages, 5 figures

Recently, anomalous subdiffusion, aging, and scatter of the diffusion coefficient have been reported in many single-particle-tracking experiments, though origins of these behaviors are still elusive. Here, as a model to describe such phenomena, we investigate a Langevin equation with diffusivity fluctuating between a fast and a slow state. We assume that the sojourn time distributions of these two states are given by power laws. It is shown that, for a non-equilibrium ensemble, the ensemble-averaged mean square displacement (MSD) shows transient subdiffusion. In contrast, the time-averaged MSD shows normal diffusion, but an effective diffusion coefficient transiently shows aging behavior. The propagator is non-Gaussian for short time, and converges to a Gaussian distribution in a long time limit; this convergence to Gaussian is extremely slow for some parameter values. For equilibrium ensembles, both ensemble-averaged and time-averaged MSDs show only normal diffusion, and thus we cannot detect any traces of the fluctuating diffusivity with these MSDs. Therefore, as an alternative approach to characterize the fluctuating diffusivity, the relative standard deviation (RSD) of the time-averaged MSD is utilized, and it is shown that the RSD exhibits slow relaxation as a signature of the long-time correlation in the fluctuating diffusivity. Furthermore, it is shown that the RSD is related to a non-Gaussian parameter of the propagator. To obtain these theoretical results, we develop a two-state renewal theory as an analytical tool., 23 pages, 10 figures

Local diffusion coefficients in disordered materials such as living cells are highly heterogeneous. Quenched disorder is utilized substantially to study such complex systems, whereas its analytical treatment is difficult to handle. We consider finite systems with quenched disorder in order to investigate the effects of sample disorder fluctuations and confinement on single-particle diffusivity. While the system is ergodic in a single disorder realization, the time-averaged mean squared displacement depends on the disorder, i.e., the system is ergodic but non-self-averaging. We find that the inverse Levy distribution is a universal distribution for diffusivity in the sense that it can be applied for arbitrary dimensions. Quantifying the degree of the non-self-averaging effect, we show that fluctuations of single-particle diffusivity far exceed the corresponding annealed theory and also find confinement effects. The relevance for experimental situations is also discussed., Comment: 5 pages, 3 figures

The mean-square displacement (MSD) is widely utilized to study the dynamical properties of stochastic processes. The time-averaged MSD (TAMSD) provides some information on the dynamics which cannot be extracted from the ensemble-averaged MSD. In particular, the relative standard deviation (RSD) of the TAMSD can be utilized to study the long time relaxation behavior. In this work, we consider a class of Langevin equations which are multiplicatively coupled to time-dependent and fluctuating diffusivities. Various interesting dynamics models such as entangled polymers and supercooled liquids can be interpreted as the Langevin equations with time-dependent and fluctuating diffusivities. We derive a general formula for the RSD of the TAMSD for the Langevin equation with the time-dependent and fluctuating diffusivity. We show that the RSD can be expressed in terms of the correlation function of the diffusivity. The RSD exhibits the crossover at the long time region. The crossover time is related to a weighted average relaxation time for the diffusivity. Thus the crossover time gives some information on the relaxation time of fluctuating diffusivity which cannot be extracted from the ensemble-averaged MSD. We discuss the universality and possible applications of the formula via some simple examples., 28 pages, 4 figures, submitted to Phys. Rev. E

Tracking the sign of fluctuations governed by the $(1+1)$-dimensional Kardar-Parisi-Zhang (KPZ) universality class, we show, both experimentally and numerically, that its evolution has an unexpected link to a simple stochastic model called the renewal process, studied in the context of aging and ergodicity breaking. Although KPZ and the renewal process are fundamentally different in many aspects, we find remarkable agreement in some of the time correlation properties, such as the recurrence time distributions and the persistence probability, while the two systems can be different in other properties. Moreover, we find inequivalence between long-time and ensemble averages in the fraction of time occupied by a specific sign of the KPZ-class fluctuations. The distribution of its long-time average converges to nontrivial broad functions, which are found to differ significantly from that of the renewal process, but instead be characteristic of KPZ. Thus, we obtain a new type of ergodicity breaking for such systems with many-body interactions. Our analysis also detects qualitative differences in time-correlation properties of circular and flat KPZ-class interfaces, which were suggested from previous experiments and simulations but still remain theoretically unexplained., 14 pages, 8 figures; major change from v1 in the analysis of generalized persistence probability. J. Stat. Phys. (published online, 2016)

The statistical features of the large fluctuations in the model of the aftershock phenomenon are studied in terms of the stationary-nonstationary chaos transition process. We demonstrate that the cross-over phenomena from Weibull to Log-Weibull distributions for the waiting time of the largest deviation are clearly observed in the logarithmic scaling regime.

Dynamical aspects of the transition process between stationary and nonstaionary chaos are numerically studied using the modified Bernoulli map. It was shown in a previous paper that the mean path of those transient processes reveals a universal logarithmic scaling of the renewal function, though there appear very large fluctuations around the mean path. First, we demonstrate the universal features of fluctuating transient paths. Next, we propose a new statistical quantity to describe the maximum fluctuation, and characterize the detailed structure of the logarithmic scaling in terms of the distribution of these quantities. The main point of the present paper is to report that large fluctuations obey two statistical laws, the Weibull and Log-Weibull distributions, and that the cross-over of both distributions is the universal phenomenon in the modified Bernoulli systems independent of the details of the mechanism which induces the stationary-nonstationary transition. We also discuss a seismological law in relation to the universality of the large fluctuations in the stationarynonstationary chaos transition.

Statistical features of the transition process from stationary to nonstationary chaos are studied using modified Bernoulli maps. The stationary-nonstationary chaos transition process can be generated by varying the value of the parameter that controls the intensity of the intermittency. The measure-theoretical structures of the chaos transition process are significantly different from the time-independent case, and new statistical phenomena appear, even when the value of the system parameter is changed continuously in time. The temporal behavior of the transition process is generally separated into three characteristic phases. The first phase appears in the initial stage of the transition, and the third one in the last stage, where the transition is almost finished, but the most interesting phase is the second one, which continues for long period. The entire transition process is analyzed using the renewal function, which describes the temporal behavior of the mean accumulated number of intermittent jumps. The logarithmic scaling relation appearing in the second phase is studied in detail with finite-range statistics. Finally, the statistical laws of the stationary-nonstationary chaos transition are discussed from the viewpoint of the modeling of seismological phenomena, and it is shown that seismological data are accounted for quite well by a metaphor model in terms of the stationary-nonstationary chaos transition.

Models with mixed origins of anomalous subdiffusion have been considered important for understanding transport in biological systems. Here, one such mixed model, the quenched trap model (QTM) on fractal lattices, is investigated. It is shown that both ensemble- and time-averaged mean square displacements (MSDs) show subdiffusion with different scaling exponents, i.e., this system shows weak ergodicity breaking. Moreover, time-averaged MSD exhibits aging and converges to a random variable following the modified Mittag--Leffler distribution. It is also shown that the QTM on a fractal lattice can not be reduced to the continuous-time random walks, if the spectral dimension of the fractal lattice is less than 2., 5 pages, 2 figures

A novel route to the exponential trapping-time distribution within a solidlike state in water clusters is described. We propose a simple homogeneous network (SHN) model to investigate dynamics on the potential energy networks of water clusters. In this model, it is shown that the trapping-time distribution in a solidlike state follows the exponential distribution, whereas the trapping-time distribution in local potential minima within the solidlike state is not exponential. To confirm the exponential trapping-time distribution in a solidlike state, we investigate water clusters, $($H${}_2$O$)_6$ and $($H${}_2$O$)_{12}$, by molecular dynamics simulations. These clusters change dynamically from solidlike to liquidlike state and vice versa. We find that the probability density functions of trapping times in a solidlike state are described by the exponential distribution whereas those of interevent times of large fluctuations in potential energy within the solidlike state follow the Weibull distributions. The results provide a clear evidence that transition dynamics between solidlike and liquidlike states in water clusters are well described by the SHN model, suggesting that the exponential trapping-time distribution within a solidlike state originates from the homogeneous connectivity in the potential energy network., Comment: 9 pages, 8 figures

In entangled polymer systems, there are several characteristic time scales, such as the entanglement time and the disengagement time. In molecular simulations, the longest relaxation time (the disengagement time) can be determined by the mean square displacement (MSD) of a segment or by the shear relaxation modulus. Here, we propose the relative fluctuation analysis method, which is originally developed for characterizing large fluctuations, to determine the longest relaxation time from the center of mass trajectories of polymer chains (the time-averaged MSDs). Applying the method to simulation data of entangled polymers (by the slip-spring model and the simple reptation model), we provide a clear evidence that the longest relaxation time is estimated as the crossover time in the relative fluctuations., Comment: 17 pages, 9 figures, to appear in J. Chem. Phys

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., Comment: 19 pages, 5 figures

We report on a novel response to biases in deterministic superdiffusion. For its reduced map, we show using infinite ergodic theory that the time-averaged velocity (TAV) is intrinsically random and its distribution obeys the generalized arc-sine distribution. A distributional limit theorem indicates that the TAV response to a bias appears in the distribution, which is an example of what we term a distributional response induced by a bias. Although this response in single trajectories is intrinsically random, the ensemble-averaged TAV response is linear., Comment: 13 pages, 5 figures

We investigate continuous time random walks with truncated $\alpha$-stable trapping times. We prove distributional ergodicity for a class of observables; namely, the time-averaged observables follow the probability density function called the Mittag--Leffler distribution. This distributional ergodic behavior persists for a long time, and thus the convergence to the ordinary ergodicity is considerably slower than in the case in which the trapping-time distribution is given by common distributions. We also find a crossover from the distributional ergodic behavior to the ordinary ergodic behavior., Comment: 4 pages, 3 figures

We characterize dynamical instability of weak chaos as subexponential instability. We show that a one-dimensional, conservative, ergodic measure preserving map with subexponential instability has an infinite invariant measure, and then we present a generalized Lyapunov exponent to characterize subexponential instability.

We construct a one-dimensional piecewise linear intermittent map from the interevent time distribution for a given renewal process. Then, we characterize intermittency by the asymptotic behavior near the indifferent fixed point in the piecewise linear intermittent map. Thus, we provide a new framework to understand a unified characterization of intermittency, and also present the Lyapunov exponent of renewal processes. This method is applied to the occurrence of earthquakes using the Japan Meteorological Agency (JMA) catalog. We demonstrate that interevent times are not independent and identically distributed random variables by analyzing the return map of interevent times, but that there is a systematic change in conditional probability distribution functions of interevent times., 12 pages, 6 figures

We propose a definition of equilibrium and non-equilibrium states in dynamical systems on the basis of the time average. We show numerically that there exists a non-equilibrium non-stationary state in the coupled modified Bernoulli map lattice., Comment: 4 pages, 2 figures

How do lipid molecules in membranes perform a flip-flop? The flip-flops of lipid molecules play a crucial role in the formation and flexibility of membranes. However, little has been determined about the behavior of flip-flops, either experimentally, or in molecular dynamics simulations. Here, we provide numerical results of the flip-flops of model lipid molecules in a model membrane and investigate the statistical properties, using millisecond-order coarse-grained molecular simulations (dissipative particle dynamics). We find that there are three different ways of flip-flops, which can be clearly characterized by their paths on the free energy surface. Furthermore, we found that the probability of the number of the flip-flops is well fitted by the Poisson distribution, and the probability density function for the inter-occurrence times of flip-flops coincides with that of the forward recurrence times. These results indicate that the occurrence of flip-flops is a Poisson process, which will play an important role in the flexibilities of membranes.

The logarithmic scaling regime is often generated in the dynamical processes of the stationary-nonstationary chaos transition, where the breakdown of the adiabaticity plays an essential role. We obtained the analytical criterion for the onset time of the non-adiabaticity carrying out with a typical non-hyperbolic chaotic system.

Phase diagram based on the mean square displacement (MSD) and the distribution of diffusion coefficients of the time-averaged MSD for the stored-energy-driven L\'evy flight (SEDLF) is presented. In the SEDLF, a random walker cannot move while storing energy, and it jumps by the stored energy. The SEDLF shows a whole spectrum of anomalous diffusions including subdiffusion and superdiffusion, depending on the coupling parameter between storing time (trapping time) and stored energy. This stochastic process can be investigated analytically with the aid of renewal theory. Here, we consider two different renewal processes, i.e., ordinary renewal process and equilibrium renewal process, when the mean trapping time does not diverge. We analytically show the phase diagram according to the coupling parameter and the power exponent in the trapping-time distribution. In particular, we find that distributional behavior of time-averaged MSD intrinsically appears in superdiffusive as well as normal diffusive regime even when the mean trapping time does not diverge., Comment: 9 pages, 3 figures