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Institute
The relaxation of a dissipative system to its equilibrium state often shows a multiexponential pattern with relaxation rates, which are typically considered to be independent of the initial condition. The rates follow from the spectrum of a Hermitian operator obtained by a similarity transformation of the initial Fokker-Planck operator. However, some initial conditions are mapped by this similarity transformation to functions which growat infinity. These cannot be expanded in terms of the eigenfunctions of a Hermitian operator, and show different relaxation patterns. Considering the exactly solvable examples of Gaussian and generalized Levy Ornstein-Uhlenbeck processes (OUPs) we show that the relaxation rates belong to the Hermitian spectrum only if the initial condition belongs to the domain of attraction of the stable distribution defining the noise. While for an ordinary OUP initial conditions leading to nonspectral relaxation can be considered exotic, for generalized OUPs driven by Levy noise, such initial conditions are the rule. DOI: 10.1103/PhysRevLett.110.150602
We derive. the ensemble-and time-averaged mean-squared displacements (MSD, TAMSD) for Poisson-reset geometric Brownian motion (GBM), in agreement with simulations. We find MSD and TAMSD saturation for frequent resetting, quantify the spread of TAMSDs via the ergodicity-breaking parameter and compute distributions of prices. General MSD-TAMSD nonequivalence proves reset GBM nonergodic.
We investigate an intermittent stochastic process in which diffusive motion with a time-dependent diffusion coefficient, D(t)∼tα−1, α>0 (scaled Brownian motion), is stochastically reset to its initial position and starts anew. The resetting follows a renewal process with either an exponential or a power-law distribution of the waiting times between successive renewals. The resetting events, however, do not affect the time dependence of the diffusion coefficient, so that the whole process appears to be a nonrenewal one. We discuss the mean squared displacement of a particle and the probability density function of its positions in this process. We show that scaled Brownian motion with resetting demonstrates rich behavior whose properties essentially depend on the interplay of the parameters of the resetting process and the particle's displacement infree motion. The motion of particles can remain almost unaffected by resetting but can also get slowed down or even be completely suppressed. Especially interesting are the nonstationary situations in which the mean squared displacement stagnates but the distribution of positions does not tend to any steady state. This behavior is compared to the situation [discussed in the companion paper; A. S. Bodrova et al., Phys. Rev. E 100, 012120 (2019)] in which the memory of the value of the diffusion coefficient at a resetting time is erased, so that the whole process is a fully renewal one. We show that the properties of the probability densities in such processes (erasing or retaining the memory on the diffusion coefficient) are vastly different.
We examine the non-ergodic properties of scaled Brownian motion (SBM), a non-stationary stochastic process with a time dependent diffusivity of the form D(t) similar or equal to t(alpha-1). We compute the ergodicity breaking parameter EB in the entire range of scaling exponents a, both analytically and via extensive computer simulations of the stochastic Langevin equation. We demonstrate that in the limit of long trajectory lengths T and short lag times Delta the EB parameter as function of the scaling exponent a has no divergence at alpha - 1/2 and present the asymptotes for EB in different limits. We generalize the analytical and simulations results for the time averaged and ergodic properties of SBM in the presence of ageing, that is, when the observation of the system starts only a finite time span after its initiation. The approach developed here for the calculation of the higher time averaged moments of the particle displacement can be applied to derive the ergodic properties of other stochastic processes such as fractional Brownian motion.
Random search with resetting
(2018)
We provide a unified renewal approach to the problem of random search for several targets under resetting. This framework does not rely on specific properties of the search process and resetting procedure, allows for simpler derivation of known results, and leads to new ones. Concentrating on minimizing the mean hitting time, we show that resetting at a constant pace is the best possible option if resetting helps at all, and derive the equation for the optimal resetting pace. No resetting may be a better strategy if without resetting the probability of not finding a target decays with time to zero exponentially or faster. We also calculate splitting probabilities between the targets, and define the limits in which these can be manipulated by changing the resetting procedure. We moreover show that the number of moments of the hitting time distribution under resetting is not less than the sum of the numbers of moments of the resetting time distribution and the hitting time distribution without resetting.
Generalized (non-Markovian) diffusion equations with different memory kernels and subordination schemes based on random time change in the Brownian diffusion process are popular mathematical tools for description of a variety of non-Fickian diffusion processes in physics, biology, and earth sciences. Some of such processes (notably, the fluid limits of continuous time random walks) allow for either kind of description, but other ones do not. In the present work we discuss the conditions under which a generalized diffusion equation does correspond to a subordination scheme, and the conditions under which a subordination scheme does possess the corresponding generalized diffusion equation. Moreover, we discuss examples of random processes for which only one, or both kinds of description are applicable.
How do different reset protocols affect ergodicity of a diffusion process in single-particle-tracking experiments? We here address the problem of resetting of an arbitrary stochastic anomalous-diffusion process (ADP) from the general mathematical points of view and assess ergodicity of such reset ADPs for an arbitrary resetting protocol. The process of stochastic resetting describes the events of the instantaneous restart of a particle’s motion via randomly distributed returns to a preset initial position (or a set of those). The waiting times of such resetting events obey the Poissonian, Gamma, or more generic distributions with specified conditions regarding the existence of moments. Within these general approaches, we derive general analytical results and support them by computer simulations for the behavior of the reset mean-squared displacement (MSD), the new reset increment-MSD (iMSD), and the mean reset time-averaged MSD (TAMSD). For parental nonreset ADPs with the MSD(t)∝ tμ we find a generic behavior and a switch of the short-time growth of the reset iMSD and mean reset TAMSDs from ∝ _μ for subdiffusive to ∝ _1 for superdiffusive reset ADPs. The critical condition for a reset ADP that recovers its ergodicity is found to be more general than that for the nonequilibrium stationary state, where obviously the iMSD and the mean TAMSD are equal. The consideration of the new statistical quantifier, the iMSD—as compared to the standard MSD—restores the ergodicity of an arbitrary reset ADP in all situations when the μth moment of the waiting-time distribution of resetting events is finite. Potential applications of these new resetting results are, inter alia, in the area of biophysical and soft-matter systems.
How do different reset protocols affect ergodicity of a diffusion process in single-particle-tracking experiments? We here address the problem of resetting of an arbitrary stochastic anomalous-diffusion process (ADP) from the general mathematical points of view and assess ergodicity of such reset ADPs for an arbitrary resetting protocol. The process of stochastic resetting describes the events of the instantaneous restart of a particle’s motion via randomly distributed returns to a preset initial position (or a set of those). The waiting times of such resetting events obey the Poissonian, Gamma, or more generic distributions with specified conditions regarding the existence of moments. Within these general approaches, we derive general analytical results and support them by computer simulations for the behavior of the reset mean-squared displacement (MSD), the new reset increment-MSD (iMSD), and the mean reset time-averaged MSD (TAMSD). For parental nonreset ADPs with the MSD(t)∝ tμ we find a generic behavior and a switch of the short-time growth of the reset iMSD and mean reset TAMSDs from ∝ _μ for subdiffusive to ∝ _1 for superdiffusive reset ADPs. The critical condition for a reset ADP that recovers its ergodicity is found to be more general than that for the nonequilibrium stationary state, where obviously the iMSD and the mean TAMSD are equal. The consideration of the new statistical quantifier, the iMSD—as compared to the standard MSD—restores the ergodicity of an arbitrary reset ADP in all situations when the μth moment of the waiting-time distribution of resetting events is finite. Potential applications of these new resetting results are, inter alia, in the area of biophysical and soft-matter systems.
We investigate an intermittent stochastic process in which the diffusive motion with time-dependent diffusion coefficient D(t)∼tα−1 with α>0 (scaled Brownian motion) is stochastically reset to its initial position, and starts anew. In the present work we discuss the situation in which the memory on the value of the diffusion coefficient at a resetting time is erased, so that the whole process is a fully renewal one. The situation when the resetting of the coordinate does not affect the diffusion coefficient's time dependence is considered in the other work of this series [A. S. Bodrova et al., Phys. Rev. E 100, 012119 (2019)]. We show that the properties of the probability densities in such processes (erasing or retaining the memory on the diffusion coefficient) are vastly different. In addition we discuss the first-passage properties of the scaled Brownian motion with renewal resetting and consider the dependence of the efficiency of search on the parameters of the process.
Various mathematical Black-Scholes-Merton-like models of option pricing employ the paradigmatic stochastic process of geometric Brownian motion (GBM). The innate property of such models and of real stock-market prices is the roughly exponential growth of prices with time [on average, in crisis-free times]. We here explore the ensemble- and time averages of a multiplicative-noise stochastic process with power-law-like time-dependent volatility, sigma(t) similar to t(alpha), named scaled GBM (SGBM). For SGBM, the mean-squared displacement (MSD) computed for an ensemble of statistically equivalent trajectories can grow faster than exponentially in time, while the time-averaged MSD (TAMSD)-based on a sliding-window averaging along a single trajectory-is always linear at short lag times Delta. The proportionality factor between these the two averages of the time series is Delta/T at short lag times, where T is the trajectory length, similarly to GBM. This discrepancy of the scaling relations and pronounced nonequivalence of the MSD and TAMSD at Delta/T << 1 is a manifestation of weak ergodicity breaking for standard GBM and for SGBM with s (t)-modulation, the main focus of our analysis. The analytical predictions for the MSD and mean TAMSD for SGBM are in quantitative agreement with the results of stochastic computer simulations.