Waiting Time Probability Density Function Research Articles

Four-compartment epidemic model with retarded transition rates.

We study an epidemic model for a constant population by taking into account four compartments of the individuals characterizing their states of health. Each individual is in one of the following compartments: susceptible S; incubated, i.e., infected yet not infectious, C; infected and infectious I; and recovered, i.e., immune, R. An infection is visible only when an individual is in state I. Upon infection, an individual performs the transition pathway S→C→I→R→S, remaining in compartments C, I, and R for a certain random waiting time t_{C}, t_{I}, and t_{R}, respectively. The waiting times for each compartment are independent and drawn from specific probability density functions (PDFs) introducing memory into the model. The first part of the paper is devoted to the macroscopic S-C-I-R-S model. We derive memory evolution equationsinvolving convolutions (time derivatives of general fractional type). We consider several cases. The memoryless case is represented by exponentially distributed waiting times. Cases of long waiting times with fat-tailed waiting-time distributions are considered as well where the S-C-I-R-S evolution equationstake the form of time-fractional ordinary differential equations. We obtain formulas for the endemic equilibrium and a condition of its existence for cases when the waiting-time PDFs have existing means. We analyze the stability of healthy and endemic equilibria and derive conditions for which the endemic state becomes oscillatory (Hopf) unstable. In the second part, we implement a simple multiple-random-walker approach (microscopic model of Brownian motion of Z independent walkers) with random S-C-I-R-S waiting times in computer simulations. Infections occur with a certain probability by collisions of walkers in compartments I and S. We compare the endemic states predicted in the macroscopic model with the numerical results of the simulations and find accordance of high accuracy. We conclude that a simple random-walker approach offers an appropriate microscopic description for the macroscopic model. The S-C-I-R-S-type models open a wide field of applications allowing the identification of pertinent parameters governing the phenomenology of epidemic dynamics such as extinction, convergence to a stable endemic equilibrium, or persistent oscillatory behavior.

Generalized diffusion and random search processes

We consider a one-dimensional Brownian search in the presence of trapping. The diffusion equation of the particle is represented by a memory kernel that enters the general waiting time probability density function. We find the general form of the first arrival time density, search reliability and efficiency and analyze several special cases of the memory kernel. We also analyze the Lévy search in the presence of trapping in cases of single and multiple targets, as well as combined Lévy–Brownian search strategies in case of a single target. The presented results are general and could be of interest for further investigation of different optimal search strategies, as well as in the animal foraging or spreading of contamination particles in the environment.

Inhomogeneous parametric scaling and variable-order fractional diffusion equations

We discuss the derivation and the solutions of integrodifferential equations (variable-order time-fractional diffusion equations) following as continuous limits for lattice continuous time random walk schemes with power-law waiting-time probability density functions whose parameters are position-dependent. We concentrate on subdiffusive cases and discuss two situations as examples: A system consisting of two parts with different exponents of subdiffusion, and a system in which the subdiffusion exponent changes linearly from one end of the interval to another one. In both cases we compare the numerical solutions of generalized master equations describing the process on the lattice to the corresponding solutions of the continuous equations, which follow by exact solution of the corresponding equations in the Laplace domain with subsequent numerical inversion using the Gaver-Stehfest algorithm.

Nonparametric Estimation of the Size and Waiting Time Distributions of Pulsar Glitches

Abstract Glitch size and waiting time probability density functions (PDFs) are estimated for the five pulsars that have glitched the most using the nonparametric kernel density estimator. Two objects exhibit decreasing size and waiting time PDFs. Their activity is Poisson-like, and their size statistics are approximately scale-invariant. Three objects exhibit a statistically significant local maximum in the PDFs, including one (PSR J1341−6220), which was classified as Poisson-like in previous analyses. Their activity is quasiperiodic, although the dispersion in waiting times is relatively broad. The classification is robust: it is preserved across a wide range of bandwidth choices. There is no compelling evidence for multimodality, but this issue should be revisited when more data become available. The implications for superfluid vortex avalanche models of pulsar glitches are explored briefly.

Models for characterizing the transition among anomalous diffusions with different diffusion exponents

Based on the theory of continuous time random walks (CTRW), we build the models of characterizing the transitions among anomalous diffusions with different diffusion exponents, often observed in the natural world. In the CTRW framework, we take the waiting time probability density function (PDF) as an infinite series in three parameter Mittag–Leffler functions. According to the models, the mean squared displacement of the process is analytically obtained and numerically verified, in particular, the trend of its transition is shown; furthermore the stochastic representation of the process is presented and the positiveness of the PDF of the position of the particles is strictly proved. Finally, the fractional moments of the model are calculated, and the analytical solutions of the model with external harmonic potential are obtained and some applications are proposed.

From continuous time random walks to the generalized diffusion equation

Abstract We obtain a generalized diffusion equation in modified or Riemann-Liouville form from continuous time random walk theory. The waiting time probability density function and mean squared displacement for different forms of the equation are explicitly calculated. We show examples of generalized diffusion equations in normal or Caputo form that encode the same probability distribution functions as those obtained from the generalized diffusion equation in modified form. The obtained equations are general and many known fractional diffusion equations are included as special cases.

Non-Poisson renewal events and memory.

We study two different forms of fluctuation-dissipation processes generating anomalous relaxations to equilibrium of an initial out-of-equilibrium condition, the former being based on a stationary although very slow correlation function and the latter characterized by the occurrence of crucial events, namely, non-Poisson renewal events, incompatible with the stationary condition. Both forms of regression to equilibrium have the same nonexponential Mittag-Leffler structure. We analyze the single trajectories of the two processes by recording the time distances between two consecutive origin recrossings and establishing the corresponding waiting time probability density function (PDF), ψ(t). In the former case, with no crucial events, ψ(t) is an exponential, and in the latter case, with crucial events, ψ(t) is an inverse power law PDF with a diverging first moment. We discuss the consequences that this result is expected to have for the correct interpretation of some anomalous relaxation processes.

Continuous-time random-walk model for anomalous diffusion in expanding media

Expanding media are typical in many different fields, e.g., in biology and cosmology. In general, a medium expansion (contraction) brings about dramatic changes in the behavior of diffusive transport properties such as the set of positional moments and the Green's function. Here, we focus on the characterization of such effects when the diffusion process is described by the continuous-time random-walk (CTRW) model. As is well known, when the medium is static this model yields anomalous diffusion for a proper choice of the probability density function (pdf) for the jump length and the waiting time, but the behavior may change drastically if a medium expansion is superimposed on the intrinsic random motion of the diffusing particle. For the case where the jump length and the waiting time pdfs are long-tailed, we derive a general bifractional diffusion equation which reduces to a normal diffusion equation in the appropriate limit. We then study some particular cases of interest, including Lévy flights and subdiffusive CTRWs. In the former case, we find an analytical exact solution for the Green's function (propagator). When the expansion is sufficiently fast, the contribution of the diffusive transport becomes irrelevant at long times and the propagator tends to a stationary profile in the comoving reference frame. In contrast, for a contracting medium a competition between the spreading effect of diffusion and the concentrating effect of contraction arises. In the specific case of a subdiffusive CTRW in an exponentially contracting medium, the latter effect prevails for sufficiently long times, and all the particles are eventually localized at a single point in physical space. This "big crunch" effect, totally absent in the case of normal diffusion, stems from inefficient particle spreading due to subdiffusion. We also derive a hierarchy of differential equations for the moments of the transport process described by the subdiffusive CTRW model in an expanding medium. From this hierarchy, the full time evolution of the second-order moment is obtained for some specific types of expansion. In the case of an exponential expansion, exact recurrence relations for the Laplace-transformed moments are obtained, whence the long-time behavior of moments of arbitrary order is subsequently inferred. Our analytical and numerical results for both Lévy flights and subdiffusive CTRWs confirm the intuitive expectation that the medium expansion hinders the mixing of diffusive particles occupying separate regions. In the case of Lévy flights, we quantify this effect by means of the so-called "Lévy horizon."

Effects of the Tempered Aging and the Corresponding Fokker–Planck Equation

In the renewal processes, if the waiting time probability density function is a tempered power-law distribution, then the process displays a transition dynamics; and the transition time depends on the parameter $\lambda$ of the exponential cutoff. In this paper, we discuss the aging effects of the renewal process with the tempered power-law waiting time distribution. By using the aging renewal theory, the $p$-th moment of the number of renewal events $n_a(t_a, t)$ in the interval $(t_a, t_a+t)$ is obtained for both the weakly and strongly aged systems; and the corresponding surviving probabilities are also investigated. We then further analyze the tempered aging continuous time random walk and its Einstein relation, and the mean square displacement is attained. Moreover, the tempered aging diffusion equation is derived.

Characterization of stationary states in random walks with stochastic resetting.

It is known that introducing a stochastic resetting in a random-walk process can lead to the emergence of a stationary state. Here we study this point from a general perspective through the derivation and analysis of mesoscopic (continuous-time random walk) equations for both jump and velocity models with stochastic resetting. In the case of jump models it is shown that stationary states emerge for any shape of the waiting-time and jump length distributions. The existence of such state entails the saturation of the mean square displacement to an universal value that depends on the second moment of the jump distribution and the resetting probability. The transient dynamics towards the stationary state depends on how the waiting time probability density function decays with time. If the moments of the jump distribution are finite then the tail of the stationary distributions is universally exponential, but for Lévy flights these tails decay as a power law whose exponent coincides with that from the jump distribution. For velocity models we observe that the stationary state emerges only if the distribution of flight durations has finite moments of lower order; otherwise, as occurs for Lévy walks, the stationary state does not exist, and the mean square displacement grows ballistically or superdiffusively, depending on the specific shape of the distribution of movement durations.

Ageing first passage time density in continuous time random walks and quenched energy landscapes

We study the first passage dynamics of an ageing stochastic process in the continuous time random walk (CTRW) framework. In such CTRW processes the test particle performs a random walk, in which successive steps are separated by random waiting times distributed in terms of the waiting time probability density function (). An ageing stochastic process is defined by the explicit dependence of its dynamic quantities on the ageing time ta, the time elapsed between its preparation and the start of the observation. Subdiffusive ageing CTRWs with describe systems such as charge carriers in amorphous semiconducters, tracer dispersion in geological and biological systems, or the dynamics of blinking quantum dots. We derive the exact forms of the first passage time density for an ageing subdiffusive CTRW in the semi-infinite, confined, and biased case, finding different scaling regimes for weakly, intermediately, and strongly aged systems: these regimes, with different scaling laws, are also found when the scaling exponent is in the range , for sufficiently long ta. We compare our results with the ageing motion of a test particle in a quenched energy landscape. We test our theoretical results in the quenched landscape against simulations: only when the bias is strong enough, the correlations from returning to previously visited sites become insignificant and the results approach the ageing CTRW results. With small bias or without bias, the ageing effects disappear and a change in the exponent compared to the case of a completely annealed landscape can be found, reflecting the build-up of correlations in the quenched landscape.

Correlation function induced by a generalized diffusion equation with the presence of a harmonic potential

An integro-differential diffusion equation with linear force, based on the continuous time random walk model, is considered. The equation generalizes the ordinary and fractional diffusion equations, which includes short, intermediate and long-time memory effects described by the waiting time probability density function. Analytical expression for the correlation function is obtained and analyzed, which can be used to describe, for instance, internal motions of proteins. The result shows that the generalized diffusion equation has a broad application and it may be used to describe different kinds of systems.

Uncoupled continuous-time random walk model: analytical and numerical solutions.

Solutions for the continuous-time random walk (CTRW) model are known in few cases. In this work, the uncoupled CTRW model is investigated analytically and numerically. In particular, the probability density function (PDF) and n-moment are obtained and analyzed. Exponential and Gaussian functions are used for the jump length PDF, whereas the Mittag-Leffler function and a combination of exponential and power-laws function is used for the waiting time PDF. The exponential and Gaussian jump length PDFs have finite jump length variances and they give the same second moment; however, their distribution functions present different behaviors near the origin. The combination of exponential and power-law function for the waiting time PDF can generate a crossover from anomalous regime to normal regime. Moreover, the parameter of the exponential jump length PDF does not change the behavior of the n-moment for all time intervals, and for the Gaussian jump length PDF the n-moment also indicates a similar behavior.

INTEGRO-DIFFERENTIAL EQUATIONS ASSOCIATED WITH CONTINUOUS-TIME RANDOM WALK

The continuous-time random walk (CTRW) model is a useful tool for the description of diffusion in nonequilibrium systems, which is broadly applied in nature and life sciences, e.g., from biophysics to geosciences. In particular, the integro-differential equations for diffusion and diffusion-advection are derived asymptotically from the decoupled CTRW model and a generalized Chapmann–Kolmogorov equation, with generic waiting time probability density function (PDF) and external force. The advantage of the integro-differential equations is that they can be used to investigate the entire diffusion process i.e., covering initial-, intermediate- and long-time ranges of the process. Therefore, this method can distinguish the evolution detail for a system having the same behavior in the long-time limit but with different initial- and intermediate-time behaviors. An integro-differential equation for diffusion-advection is also presented for the description of the subdiffusive and superdiffusive regime. Moreover, the methods of solving the integro-differential equations are developed, and the analytic solutions for PDFs are obtained for the cases of force-free and linear force.

Integro-differential equation for joint probability density in phase space associated with continuous-time random walk

We derive an integro-differential equation for the joint probability density function in phase space associated with the continuous-time random walk, with generic waiting time probability density function and external force. This equation permits us to investigate whole diffusion processes covering initial-, intermediate-, and long-time ranges, which can distinguish the evolution details for systems having the same behavior in the long-time limit with different initial- and intermediate-time behaviors. Moreover, we obtained analytic solutions for probability density functions both in velocity and phase spaces, and interesting dynamic behaviors are discovered.

Continuous-time random walk: exact solutions for the probability density function and first two moments

We consider the decoupled continuous-time random walk model with a finite characteristic waiting time and approximate jump length variance. We take the waiting time probability density function (PDF) given by a combination of the exponential and the Mittag-Leffler function. Using this waiting time PDF, we investigate the diffusion behavior for all times. We obtain exact solutions for the first two moments and the PDF for the force-free and linear force cases. Due to the finite characteristic waiting time and jump length variance, the model presents, for the force-free case, normal diffusive behavior in the long-time limit. Further, the model can describe anomalous behavior at intermediate times.

Asymptotic front behavior in anA+B→2Areaction under subdiffusion

We discuss the front propagation in the $A+B\rightarrow 2A$ reaction under subdiffusion which is described by continuous time random walks with a heavy-tailed power law waiting time probability density function. Using a crossover argument, we discuss the two scaling regimes of the front propagation: an intermediate asymptotic regime given by the front solution of the corresponding continuous equation, and the final asymptotics, which is fluctuation-dominated and therefore lays out of reach of the continuous scheme. We moreover show that the continuous reaction subdiffusion equation indeed possesses a front solution that decelerates and becomes narrow in the course of time. This continuous description breaks down for larger times when the front gets atomically sharp. We show that the velocity of such fronts decays in time faster than in the continuous regime.

Continuous time random walk: Galilei invariance and relation for the nth moment

We consider a decoupled continuous time random walk model with a generic waiting time probability density function (PDF). For the force-free case we derive an integro-differential diffusion equation which is related to the Galilei invariance for the probability density. We also derive a general relation which connects the nth moment in the presence of any external force to the second moment without external force, i.e. it is valid for any waiting time PDF. This general relation includes the generalized second Einstein relation, which connects the first moment in the presence of any external force to the second moment without any external force. These expressions for the first two moments are verified by using several kinds of the waiting time PDF. Moreover, we present new anomalous diffusion behaviours for a waiting time PDF given by a product of power-law and exponential function.

Continuous time random walk with generic waiting time and external force

We derive an integrodifferential diffusion equation for decoupled continuous time random walk that is valid for a generic waiting time probability density function and external force. Using this equation we also study diffusion behaviors for a couple of specific waiting time probability density functions such as exponential, a combination of power law and generalized Mittag-Leffler function and a sum of exponentials under the influence of a harmonic trap. We show that first two waiting time probability density functions can reproduce the results of the ordinary and fractional diffusion equations for all the time regions from small to large times. But the third one shows a much more complicated pattern. Furthermore, from the integrodifferential diffusion equation we show that the second Einstein relation can hold for any waiting time probability density function.

Integrodifferential diffusion equation for continuous-time random walk

In this paper, we present an integrodifferential diffusion equation for continuous-time random walk that is valid for a generic waiting time probability density function. Using this equation, we also study diffusion behaviors for a couple of specific waiting time probability density functions such as exponential and a combination of power law and generalized Mittag-Leffler function. We show that for the case of the exponential waiting time probability density function, a normal diffusion is generated and the probability density function is Gaussian distribution. In the case of the combination of a power law and generalized Mittag-Leffler waiting probability density function, we obtain the subdiffusive behavior for all the time regions from small to large times and probability density function is non-Gaussian distribution.