Dissipative Processes with Infinite Memory

2007

Starting from the model of continuous time random walk that can also be considered as a compound renewal process we focus our interest on random walks in which the probability distributions of the waiting times and jumps have fat tails characterized by power laws with exponent between 0 and 1 for the waiting times, between 0 and 2 for the jumps. By stating the relevant lemmata (of Tauber type) for the distribution functions we need not distinguish between continuous and discrete space and time. We will see that by a well-scaled passage to the diffusion limit diffusion processes fractional in time as well as in space are obtained. The corresponding equation of evolution is a linear partial pseudo-differential equation with fractional derivatives in time and in space, the orders being equal to the above exponents. Such processes are enjoying increasing popularity in applications in physics, chemistry, finance and other fields, and their behaviour can be well approximated and visualized by simulation via various types of random walks. For their explicit solutions there are available integral representations that allow to investigate their detailed structure. For ease of presentation we restrict attention to the spatially one-dimensional symmetric situation.

Physical review. E, 2016

We present a modeling approach for diffusion in a complex medium characterized by a random length scale. The resulting stochastic process shows subdiffusion with a behavior in qualitative agreement with single-particle tracking experiments in living cells, such as ergodicity breaking, p variation, and aging. In particular, this approach recapitulates characteristic features previously described in part by the fractional Brownian motion and in part by the continuous-time random walk. Moreover, for a proper distribution of the length scale, a single parameter controls the ergodic-to-nonergodic transition and, remarkably, also drives the transition of the diffusion equation of the process from nonfractional to fractional, thus demonstrating that fractional kinetics emerges from ergodicity breaking.

Annals of Probability, 2011

In this paper the solutions $u_{\nu}=u_{\nu}(x,t)$ to fractional diffusion equations of order $0<\nu \leq 2$ are analyzed and interpreted as densities of the composition of various types of stochastic processes. For the fractional equations of order $\nu =\frac{1}{2^n}$, $n\geq 1,$ we show that the solutions $u_{{1/2^n}}$ correspond to the distribution of the $n$-times iterated Brownian motion. For these processes the distributions of the maximum and of the sojourn time are explicitly given. The case of fractional equations of order $\nu =\frac{2}{3^n}$, $n\geq 1,$ is also investigated and related to Brownian motion and processes with densities expressed in terms of Airy functions. In the general case we show that $u_{\nu}$ coincides with the distribution of Brownian motion with random time or of different processes with a Brownian time. The interplay between the solutions $u_{\nu}$ and stable distributions is also explored. Interesting cases involving the bilateral exponential distribution are obtained in the limit.

2010

In this paper we revisit the Brownian motion on the basis of {the fractional Langevin equation which turns out to be a particular case of the generalized Langevin equation introduced by Kubo in 1966. The importance of our approach is to model the Brownian motion more realistically than the usual one based on the classical Langevin equation, in that it takes into account also the retarding effects due to hydrodynamic back-flow, i.e. the added mass and the Basset memory drag. We provide the analytical expressions of the correlation functions (both for the random force and the particle velocity) and of the mean squared particle displacement. The random force has been shown to be represented by a superposition of the usual white noise with a &quot;fractional&quot; noise. The velocity correlation function is no longer expressed by a simple exponential but exhibits a slower decay, proportional to t^{-3/2} for long times, which indeed is more realistic. Finally, the mean squared displaceme...

Journal of Functional Analysis, 2008

This paper studies the asymptotic behavior of a one-dimensional directed polymer in a random medium. The latter is represented by a Gaussian field B H on R + × R with fractional Brownian behavior in time (Hurst parameter H ) and arbitrary function-valued behavior in space. The partition function of such a polymer is

Anomalous Transport, 2008

We show the asymptotic long-time equivalence of a generic power law waiting time distribution to the Mittag-Leffler waiting time distribution, characteristic for a time fractional continuous time random walk. This asymptotic equivalence is effected by a combination of "rescaling" time and "respeeding" the relevant renewal process followed by a passage to a limit for which we need a suitable relation between the parameters of rescaling and respeeding. As far as we know such procedure has been first applied in the 1960s by Gnedenko and Kovalenko in their theory of "thinning" a renewal process. Turning our attention to spatially one-dimensional continuous time random walks with a generic power law jump distribution, "rescaling" space can be interpreted as a second kind of "respeeding" which then, again under a proper relation between the relevant parameters leads in the limit to the space-time fractional diffusion equation. Finally, we treat the 'time fractional drift" process as a properly scaled limit of the counting number of a Mittag-Leffler renewal process.

Stochastic Processes and Their Applications, 2009

Fractional diffusion equations of order ν∈(0,2)ν∈(0,2) are examined and solved under different types of boundary conditions. In particular, for the fractional equation on the half-line [0,+∞)[0,+∞) and with an elastic boundary condition at x=0x=0, we are able to provide the general solution in terms of the density of the elastic Brownian motion. This permits us, for equations of order ν=12n, to write the solution as the density of the process obtained by composing the elastic Brownian motion with the (n−1)(n−1)-times iterated Brownian motion. Also the limiting case for n→∞n→∞ is investigated and the explicit form of the solution is expressed in terms of exponentials.Moreover, the fractional diffusion equations on the half-lines [0,+∞)[0,+∞) and (−∞,a](−∞,a] with additional first-order space derivatives are analyzed also under reflecting or absorbing conditions. The solutions in this case lead to composed processes with general form X(|In−1(t)|)X(|In−1(t)|), where only the driving process XX is affected by drift, while the role of time is played by iterated Brownian motion In−1In−1.

2010

Motivated by subdiffusive motion of bio-molecules observed in living cells we study the stochastic properties of a non-Brownian particle whose motion is governed by either fractional Brownian motion or the fractional Langevin equation and restricted to a finite domain. We investigate by analytic calculations and simulations how time-averaged observables (e.g., the time averaged mean squared displacement and displacement correlation) are affected by spatial confinement and dimensionality. In particular we study the degree of weak ergodicity breaking and scatter between different single trajectories for this confined motion in the subdiffusive domain. The general trend is that deviations from ergodicity are decreased with decreasing size of the movement volume, and with increasing dimensionality. We define the displacement correlation function and find that this quantity shows distinct features for fractional Brownian motion, fractional Langevin equation, and continuous time subdiffusion, such that it appears an efficient measure to distinguish these different processes based on single particle trajectory data.

Lecture Notes in Physics, 2003

A physical-mathematical approach to anomalous diffusion may be based on generalized diffusion equations (containing derivatives of fractional order in space or/and time) and related random walk models. By the spacetime fractional diffusion equation we mean an evolution equation obtained from the standard linear diffusion equation by replacing the second-order space derivative with a Riesz-Feller derivative of order α ∈ (0, 2] and skewness θ (|θ| ≤ min {α, 2 − α}), and the first-order time derivative with a Caputo derivative of order β ∈ (0, 1] . The fundamental solution (for the Cauchy problem) of the fractional diffusion equation can be interpreted as a probability density evolving in time of a peculiar self-similar stochastic process. We view it as a generalized diffusion process that we call fractional diffusion process, and present an integral representation of the fundamental solution. A more general approach to anomalous diffusion is however known to be provided by the master equation for a continuous time random walk (CTRW). We show how this equation reduces to our fractional diffusion equation by a properly scaled passage to the limit of compressed waiting times and jump widths. Finally, we describe a method of simulation and display (via graphics) results of a few numerical case studies.

2007

A physical-mathematical approach to anomalous diffusion may be based on generalized diffusion equations (containing derivatives of fractional order in space or/and time) and related random walk models. By the spacetime fractional diffusion equation we mean an evolution equation obtained from the standard linear diffusion equation by replacing the second-order space derivative with a Riesz-Feller derivative of order α ∈ (0, 2] and skewness θ (|θ| ≤ min {α, 2 − α}), and the first-order time derivative with a Caputo derivative of order β ∈ (0, 1] . The fundamental solution (for the Cauchy problem) of the fractional diffusion equation can be interpreted as a probability density evolving in time of a peculiar self-similar stochastic process. We view it as a generalized diffusion process that we call fractional diffusion process, and present an integral representation of the fundamental solution. A more general approach to anomalous diffusion is however known to be provided by the master equation for a continuous time random walk (CTRW). We show how this equation reduces to our fractional diffusion equation by a properly scaled passage to the limit of compressed waiting times and jump widths. Finally, we describe a method of simulation and display (via graphics) results of a few numerical case studies.

Physical Review E

How different are the results of constant-rate resetting of anomalous-diffusion processes in terms of their ensemble-averaged versus time-averaged mean-squared displacements (MSDs versus TAMSDs) and how does stochastic resetting impact nonergodicity? We examine, both analytically and by simulations, the implications of resetting on the MSD-and TAMSD-based spreading dynamics of particles executing fractional Brownian motion (FBM) with a long-time memory, heterogeneous diffusion processes (HDPs) with a power-law space-dependent diffusivity D(x) = D 0 |x| γ and their "combined" process of HDP-FBM. We find, inter alia, that the resetting dynamics of originally ergodic FBM for superdiffusive Hurst exponents develops disparities in scaling and magnitudes of the MSDs and mean TAMSDs indicating weak ergodicity breaking. For subdiffusive HDPs we also quantify the nonequivalence of the MSD and TAMSD and observe a new trimodal form of the probability density function. For reset FBM, HDPs and HDP-FBM we compute analytically and verify by simulations the short-time MSD and TAMSD asymptotes and long-time plateaus reminiscent of those for processes under confinement. We show that certain characteristics of these reset processes are functionally similar despite a different stochastic nature of their nonreset variants. Importantly, we discover nonmonotonicity of the ergodicitybreaking parameter EB as a function of the resetting rate r. For all reset processes studied we unveil a pronounced resetting-induced nonergodicity with a maximum of EB at intermediate r and EB∼(1/r)-decay at large r. Alongside the emerging MSD-versus-TAMSD disparity, this r-dependence of EB can be an experimentally testable prediction. We conclude by discussing some implications to experimental systems featuring resetting dynamics.

Physical Review E, 1997

Anomalous diffusion in which the mean square distance between diffusing quantities increases faster than linearly in ''time'' has been observed in all manner of physical and biological systems from macroscopic surface growth to DNA sequences. Herein we relate the cause of this nondiffusive behavior to the statistical properties of an underlying process using an exact statistical model. This model is a simple two-state process with long-time correlations and is shown to produce a random walk described by an exact fractional diffusion equation. Fractional diffusion equations describe anomalous transport and are shown to have exact solutions in terms of Fox functions, including Lévy ␣-stable processes in the superdiffusive domain (1/2ϽHϽ1).

Journal of Statistical Physics, 2006

We investigate statistics of occupation times for an over-damped Brownian particle in an external force field. A backward Fokker-Planck equation introduced by Majumdar and Comtet describing the distribution of occupation times is solved. The solution gives a general relation between occupation time statistics and probability currents which are found from solutions of the corresponding problem of first passage time. This general relationship between occupation times and first passage times, is valid for normal Markovian diffusion and for non-Markovian sub-diffusion, the latter modeled using the fractional Fokker-Planck equation. For binding potential fields we find in the long time limit ergodic behavior for normal diffusion, while for the fractional framework weak ergodicity breaking is found, in agreement with previous results of Bel and Barkai on the continuous time random walk on a lattice. For non-binding potential rich physical behaviors are obtained, and classification of occupation time statistics is made possible according to whether or not the underlying random walk is recurrent and the averaged first return time to the origin is finite. Our work establishes a link between fractional calculus and ergodicity breaking.

Physical Review E, 2011

Fractional Brownian motion is a Gaussian process x(t) with zero mean and two-time correlations x(t1)x(t2) = D t 2H 1 + t 2H 2 − |t1 − t2| 2H , where H, with 0 < H < 1 is called the Hurst exponent. For H = 1/2, x(t) is a Brownian motion, while for H = 1/2, x(t) is a non-Markovian process. Here we study x(t) in presence of an absorbing boundary at the origin and focus on the probability density P+(x, t) for the process to arrive at x at time t, starting near the origin at time 0, given that it has never crossed the origin. It has a scaling form P+(x, t) ∼ t −H R+(x/t H). Our objective is to compute the scaling function R+(y), which up to now was only known for the Markov case H = 1/2. We develop a systematic perturbation theory around this limit, setting H = 1/2 + , to calculate the scaling function R+(y) to first order in. We find that R+(y) behaves as R+(y) ∼ y φ as y → 0 (near the absorbing boundary), while R+(y) ∼ y γ exp(−y 2 /2) as y → ∞, with φ = 1 − 4 + O(2) and γ = 1 − 2 + O(2). Our-expansion result confirms the scaling relation φ = (1 − H)/H proposed in Ref. [29]. We verify our findings via numerical simulations for H = 2/3. The tools developed here are versatile, powerful, and adaptable to different situations.

Physics Reports, 2000

What can fractional equations do, what can they do better, and why should one care at all? 1 1.2. What is the scope of this report? 5 2. Introduction 5 2.1. Anomalous dynamics in complex systems 5 2.2. Historical remarks 7 2.3. Anomalous di!usion: experiments and models 9 3. From continuous time random walk to fractional di!usion equations 13 3.1. Revisiting the realm of Brownian motion 14 3.2. The continuous time random walk model 15 3.3. Back to Brownian motion 17 3.4. Long rests: a fractional di!usion equation describing subdi!usion 18 3.5. Long jumps: LeH vy #ights 25 3.6. The competition between long rests and long jumps 29 3.7. What's the course, helmsman? 30 4. Fractional di!usion}advection equations 31 4.1. The Galilei invariant fractional di!usion}advection equation 32 4.2. The Gallilei variant fractional di!usion}advection equation 4.3. Alternative approaches for LeH vy #ights 5. The fractional Fokker}Planck equation: anomalous di!usion in an external force "eld 5.1. The Fokker}Planck equation 5.2. The fractional Fokker}Planck equation 5.3. Separation of variables and the fractional Ornstein}Uhlenbeck process 5.4. The connection between the fractional solution and its Brownian counterpart 5.5. The fractional analogue of Kramers escape theory from a potential well 5.6. The derivation of the fractional Fokker}Planck equation 5.7. A fractional Fokker}Planck equation for LeH vy #ights 5.8. A generalised Kramers}Moyal expansion 6. From the Langevin equation to fractional di!usion: microscopic foundation of dispersive transport close to thermal equilibrium 6.1. Langevin dynamics and the three stages to subdi!usion