Anomalous transport in fluid fleld with random waiting time depending on the preceding jump length

Anomalous(or non-Fickian) transport behaviors of particles have been widely observed in complex porous media.To capture the energy-dependent characteristics of non-Fickian transport of a particle in flow fields,in the present paper a generalized continuous time random walk model whose waiting time probability distribution depends on the preceding jump length is introduced,and the corresponding master equation in Fourier-Laplace space for the distribution of particles is derived.As examples,two generalized advection-dispersion equations for Gaussian distribution and levy flight with the probability density function of waiting time being quadratic dependent on the preceding jump length are obtained by applying the derived master equation.     

Anomalous transport in fluid field with random waiting time depending on the preceding jump length

Anomalous (or non-Fickian) transport behaviors of particles have been widely observed in complex porous media. To capture the energy-dependent characteristics of non-Fickian transport of a particle in flow fields, in the present paper a generalized continuous time random walk model whose waiting time probability distribution depends on the preceding jump length is introduced, and the corresponding master equation in Fourier-Laplace space for the distribution of particles is derived. As examples, two generalized advection-dispersion equations for Gaussian distribution and lévy flight with the probability density function of waiting time being quadratic dependent on the preceding jump length are obtained by applying the derived master equation.     

Continuous time random walk with jump length correlated with waiting time

A coupled continuous time random walk (CTRW) model is proposed, in which the jump length of a walker is correlated with waiting time. The power law distribution is chosen as the probability density function of waiting time and the Gaussian-like distribution as the probability density function of jump length. Normal diffusion, subdiffusion and superdiffusion can be realized within the present model. It is shown that the competition between long-tailed distribution and correlation of jump length and waiting time will lead to different diffusive behavior.     

Extremal behavior of a coupled continuous time random walk

Coupled continuous time random walks (CTRWs) model normal and anomalous diffusion of random walkers by taking the sum of random jump lengths dependent on the random waiting times immediately preceding each jump. They are used to simulate diffusion-like processes in econophysics such as stock market fluctuations, where jumps represent financial market microstructure like log returns. In this and many other applications, the magnitude of the largest observations (e.g. a stock market crash) is of considerable importance in quantifying risk. We use a stochastic process called a coupled continuous time random maxima (CTRM) to determine the density governing the maximum jump length of a particle undergoing a CTRW. CTRM are similar to continuous time random walks but track maxima instead of sums. The many ways in which observations can depend on waiting times can produce an equally large number of CTRM governing density shapes. We compare densities governing coupled CTRM with their uncoupled counterparts for three simple observation/wait dependence structures.     

Space–time fractional diffusion equations and asymptotic behaviors of a coupled continuous time random walk model

In this paper, we consider a type of continuous time random walk model where the jump length is correlated with the waiting time. The asymptotic behaviors of the coupled jump probability density function in the Fourier–Laplace domain are discussed. The corresponding fractional diffusion equations are derived from the given asymptotic behaviors. Corresponding to the asymptotic behaviors of the joint probability density function in the Fourier–Laplace space, the asymptotic behaviors of the waiting time probability density and the conditional probability density for jump length are also discussed.     

Langevin Picture of Lévy Walks and Their Extensions

In this paper we derive Langevin picture of Lévy walks. Applying recent advances in the theory of coupled continuous time random walks we find a limiting process of the properly scaled Lévy walk. Next, we introduce extensions of Levy walks, in which jump sizes are some functions of waiting times. We prove that under proper scaling conditions, such generalized Lévy walks converge in distribution to the appropriate limiting processes. We also derive the corresponding fractional diffusion equations and investigate behavior of the mean square displacements of the limiting processes, showing that different coupling functions lead to various types of anomalous diffusion.     

Random walks with infinite spatial and temporal moments

The continuous-time random walk of Montroll and Weiss has been modified by Scher and Lax to include a coupled spatial-temporal memory. We treat novel cases for the random walk and the corresponding generalized master equation when combinations of both spatial, and temporal moments of the memory are infinite. The asymptotic properties of the probability distribution for being at any lattice site as a function of time and its variance are calculated. The resulting behavior includes localized, diffusive, wavelike, and Levy's stable laws for the appropriate scaled variable. We show that an infinite mean waiting time can lead to long time diffusive behavior, while a finite mean waiting time is not sufficient to ensure the same.     

Subdiffusion and long-time anticorrelations in a stochastic single file

The subdiffusion of a stochastic single file is interpreted as a jumping process. Contrary to the current continuous time random walk models, its statistics is characterized by finite averages of the jumping times and square displacements. Subdiffusion is then related to a persistent anticorrelation of the jump sequences. In continuous time representation, this corresponds to negative power-law velocity autocorrelations, attributable to the restricted geometry of the file diffusion.     

Fractional diffusion with two time scales

Moving particles that rest along their trajectory lead to time-fractional diffusion equations for the scaling limit distributions. For power law waiting times with infinite mean, the equation contains a fractional time derivative of order between 0 and 1. For finite mean waiting times, the most revealing approach is to employ two time scales, one for the mean and another for deviations from the mean. For finite mean power law waiting times, the resulting equation contains a first derivative as well as a derivative of order between 1 and 2. Finite variance waiting times lead to a second-order partial differential equation in time. In this article we investigate the various solutions with regard to moment growth and scaling properties, and show that even infinite mean waiting times do not necessarily induce subdiffusion, but can lead to super-diffusion if the jump distribution has non-zero mean.     

A computer simulation study of ionic conductivity in polymer electrolytes

In this paper we present a computer simulation study of ionic conductivity in solid polymeric electrolytes. The multiphase nature of the material is taken into account. The polymer is represented by a regular lattice whose sites represent either crystalline or amorphous regions with the charge carrier performing a random walk. Different waiting times are assigned to sites corresponding to the different phases. A random walk (RW) is used to calculate the conductivity through the Nernst-Einstein relation. Our walk algorithm takes into account the reorganization of the different phases over time scales comparable to time scales for the conduction process. This is a characteristic feature of the polymer network. The qualitative nature of the variation of conductivity with salt concentration agrees with the experimental values for PEO-NH₄I and PEO-NH₄SCN. The average jump distance estimated from our work is consistent with the reported bond lengths for such polymers.     

Stochastic Modeling of Indoor Air Temperature

Temperature is one of the main parameters describing thermal comfort and indoor air quality. In this paper we propose an approach, based on a modification of the continuous time random walk, to model the indoor air temperature. We perform a statistical analysis of the recorded time series, that allows us to point out the main statistical properties of the recorded variable. The obtained conclusions about the nature of the process lead to a continuous time random walk, that in contrast to the classical approach, models time dependence of the jumps distribution. Moreover, we show that the waiting times can be modeled by a tempered stable distribution, which yields a subdiffusive behavior in short times and diffusive behavior in longer times. Finally, by conducting a simulation study we illustrate possible applications of the presented approach in the thermal comfort monitoring and forecasting.     

Environment-dependent continuous time random walk

A generalized continuous time random walk model which is dependent on environmental damping is proposed in which the two key parameters of the usual random walk theory: the jumping distance and the waiting time, are replaced by two new ones: the pulse velocity and the flight time. The anomalous diffusion of a free particle which is characterized by the asymptotical mean square displacement <x²(t)>～t^α is realized numerically and analysed theoretically, where the value of the power index α is in a region of 0 < α < 2. Particularly, the damping leads to a sub-diffusion when the impact velocities are drawn from a Gaussian density function and the super-diffusive effect is related to statistical extremes, which are called rare-though-dominant events.     

From classical dynamics to continuous time random walks

The migration of a classical dynamical system between regions of configuration space can be treated as a continuous time random walk between these regions. Derivation of a classical analog of the quantum mechanical generalized master equation provides expressions for the waiting time distribution in terms of transition memory functions. A short memory approximation to these memory functions is equivalent to the well-known transition state method. An example is discussed for which this approximation seems reasonable but is entirely wrong.     

On the Wang–Uhlenbeck Problem in Discrete Velocity Space

The arguably simplest model for dynamics in phase space is the one where the velocity can jump between only two discrete values, ±v with rate constant k. For this model, which is the continuous-space version of a persistent random walk, analytic expressions are found for the first passage time distributions to the origin. Since the evolution equation of this model can be regarded as the two-state finite-difference approximation in velocity space of the Kramers–Klein equation, this work constitutes a solution of the simplest version of the Wang–Uhlenbeck problem. Formal solution (in Laplace space) of generalizations where the velocity can assume an arbitrary number of discrete states that mimic the Maxwell distribution is also provided.     

Topological description of the aging dynamics in simple glasses

We numerically investigate the aging dynamics of a monatomic Lennard-Jones glass, focusing on the topology of the potential energy landscape which, to this aim, has been partitioned in basins of attraction of stationary points (saddles and minima). The analysis of the stationary points visited during the aging dynamics shows the existence of two distinct regimes: (i) at short times the system visits basins of saddles whose energies and orders decrease with t; (ii) at long times the system mainly lies in basins pertaining to minima of slowly decreasing energy. The long time dynamics can be represented by a simple random walk on a network of minima with a jump probability proportional to the inverse of the waiting time.     

Quantum walk under coherence non-generating channels

We investigate the probability distribution of the quantum walk under coherence non-generating channels. We definea model called generalized classical walk with memory. Under certain conditions, generalized classical random walk withmemory can degrade into classical random walk and classical random walk with memory. Based on its various spreadingspeed, the model may be a useful tool for building algorithms. Furthermore, the model may be useful for measuring thequantumness of quantum walk. The probability distributions of quantum walks are generalized classical random walkswith memory under a class of coherence non-generating channels. Therefore, we can simulate classical random walkand classical random walk with memory by coherence non-generating channels. Also, we find that for another class ofcoherence non-generating channels, the probability distributions are influenced by the coherence in the initial state of thecoin. Nevertheless, the influence degrades as the number of steps increases. Our results could be helpful to explore therelationship between coherence and quantum walk.     

Some properties of the asymptotic solutions of the Montroll-Weiss equation

Curves of asymptotic probability densities appropriate to the continuous time random walk model of Montroll and Weiss are presented and are calculated numerically using the fast Fourier transform. The behavior of the moments is briefly discussed and it is shown that the Einstein formula relating the diffusion and mobility coefficients can be generalized to include the case where the mean waiting time between hops is infinite.     

Fractal walk and walk on fractals

The one-dimensional walk of a particle executing instantaneous jumps between the randomly distributed “atoms” at which it resides for a random time is considered. The random distances between the neighboring atoms and the time intervals between jumps are mutually independent. The asymptotic (t → ∞) behavior of this process is studied in connection with the problem of interpretation of the generalized fractional diffusion equation (FDE). It is shown that the interpretation of the FDE as the equation describing the walk (diffusion) in a fractal medium is incorrect in the model problem considered. The reason is that the FDE implies that the consecutive jumps (fractal walk) are independent, whereas they are correlated in the case under consideration: a particle leaving an atom in the direction opposite to the preceding direction traverses the same path until arriving at the atom.