Fractional Kramers Equation †

Physical Review E, 2010

We consider different Markovian embedding schemes of non-Markovian stochastic processes that are described by generalized Langevin equations (GLE) and obey thermal detailed balance under equilibrium conditions. At thermal equilibrium superdiffusive behavior can emerge if the total integral of the memory kernel vanishes. Such a situation of vanishing static friction is caused by a super-Ohmic thermal bath. One of the simplest models of ballistic superdiffusion is determined by a bi-exponential memory kernel that was proposed by Bao [J.-D. Bao, J. Stat. Phys. 114, 503 (2004)]. We show that this non-Markovian model has infinitely many different 4-dimensional Markovian embeddings. Implementing numerically the simplest one, we demonstrate that (i) the presence of a periodic potential with arbitrarily low barriers changes the asymptotic large time behavior from free ballistic superdiffusion into normal diffusion; (ii) an additional biasing force renders the asymptotic dynamics superdiffusive again. The development of transients that display a qualitatively different behavior compared to the true large-time asymptotics presents a general feature of this non-Markovian dynamics. These transients though may be extremely long. As a consequence, they can be even mistaken as the true asymptotics. We find that such intermediate asymptotics exhibit a giant enhancement of superdiffusion in tilted washboard potentials and it is accompanied by a giant transient superballistic current growing proportional to t α eff with an exponent α eff that can exceed the ballistic value of two.

Physical Review Letters, 2010

We study a minimal non-Markovian model of superdiffusion which originates from long-range velocity correlations within the generalized Langevin equation (GLE) approach. The model allows for a three-dimensional Markovian embedding. The emergence of a transient hyperdiffusion, ∆x 2 (t) ∝ t 2+λ , with λ ∼ 1 − 3 is detected in tilted washboard potentials before it ends up in a ballistic asymptotic regime. We relate this phenomenon to a transient heating of particles T kin (t) ∝ t λ from the thermal bath temperature T to some maximal kinetic temperature Tmax. This hyperdiffusive transient regime ceases when the particles arrive at the maximal kinetic temperature.

Physical Review E, 2006

Random processes are generally described by Gaussian statistics as formulated by the central limit theorem. However, there exists a large number of exceptions to this rule that can be found in a variety of fields. Diffusion processes are often analyzed by the scaling law ͗r 2 ͘ϳt 2␤ , where the second moment of the diffusion propagator or molecular mean square displacement, ͗r 2 ͘, in the case of Gaussian diffusion is proportional to t, i.e., ␤ =1/2. A deviation from Gaussian behavior may be either superdiffusion ͑␤ Ͼ 1/2͒ or subdiffusion ͑␤ Ͻ 1/2͒. In this paper we demonstrate that all three diffusion regimes may be observed for the surfactant self-diffusion, on the length scale of 10 −6 m and the time scale of 0.02-0.8 s. in a system of wormlike micelles, depending on small variations in the sample composition. The self-diffusion is followed by pulsed gradient NMR where one not only measures the second moment of the diffusion propagator, but actually measures the Fourier transform of the full diffusion propagator itself. A generalized diffusion equation in terms of fractional time derivatives provides a general description of all the different diffusion regimes, and where 1/␤ can be interpreted as a dynamic fractal dimension. Experimentally, we find ␤ =1/4 and 3/4, in the regimes of sub-and superdiffusion, respectively. The physical interpretation of the subdiffusion behavior is that the dominating diffusion mechanism corresponds to a lateral diffusion along the contour of the wormlike micelles. Superdiffusion is obtained near the overlap concentration where the average micellar size is smaller so that the center of mass diffusion of the micelles contributes to the transport of surfactant molecules.

The Journal of Chemical Physics, 2005

We study time series produced by the blinking quantum dots, by means of an aging experiment, and we examine the results of this experiment in the light of two distinct approaches to complexity, renewal and slow modulation. We find that the renewal approach fits the result of the aging experiment, while the slow modulation perspective does not. We make also an attempt at establishing the existence of an intermediate condition.

Physical Review E, 2003

Lévy walks are random processes with an underlying spatiotemporal coupling. This coupling penalizes long jumps, and therefore Lévy walks give a proper stochastic description for a particle's motion with broad jump length distribution. We derive a generalized dynamical formulation for Lévy walks in which the fractional equivalent of the material derivative occurs. Our approach will be useful for the dynamical formulation of Lévy walks in an external force field or in phase space for which the description in terms of the continuous time random walk or its corresponding generalized master equation are less well suited.

The Journal of Chemical Physics, 2002

The Debye theory of dielectric relaxation of an assembly of polar molecules is reformulated using a fractional noninertial Fokker-Planck equation for the purpose of extending that theory to explain anomalous dielectric relaxation. The fractional Fourier-Planck equation in question is a generalization of the Smoluchowski equation for the dynamics of Brownian particles in configuration space to include anomalous relaxation. It is shown that this model can reproduce nonexponential Cole-Cole-type anomalous dielectric relaxation behavior and that it reduces to the classical Debye model of rotational diffusion when the anomalous exponent is unity.

Physical Review E, 2012

Anomalous diffusion of a particle in a cosine periodic potential is treated using fractional diffusion equations in both phase and configuration space. Exact solutions of two distinct forms of the fractional Klein-Kramers (Fokker-Planck) equation for the distribution function in phase space are obtained via matrix continued fractions yielding the average velocity, the velocity autocorrelation function, its spectrum, etc. In the overdamped limit, the results yielded by both equations agree with those from a fractional probability density diffusion equation in configuration space. A simple analytic solution for the spectrum of the velocity correlation function is also given using the effective eigenvalue approximation. The results represent generalizations of the conventional solutions for the normal diffusion of a Brownian particle in a cosine potential to fractional dynamics (giving rise to anomalous diffusion).

Physical Review E, 2003

The fractional Klein-Kramers ͑Fokker-Planck͒ equation describing the fractal time dynamics of an assembly of fixed axis dipoles rotating in an N-fold cosine potential representing the internal field due to neighboring molecules is solved using matrix continued fractions. The result can be considered as a generalization of the solution for the normal Brownian motion in a cosine periodic potential to fractional dynamics ͑giving rise to anomalous diffusion͒ and also represents a generalization of Fröhlich's model of relaxation over a potential barrier. The solution includes both inertial and strong internal field effects, which in combination produce a strong resonance peak ͑Poley absorption͒ at high frequencies due to librations of the dipoles in the potential, an anomalous relaxation band at low frequencies mainly arising from overbarrier relaxation, and a weaker relaxation band at midfrequencies due to the fast intrawell modes. The high-frequency behavior is controlled by the inertia of the dipole, so that the Gordon sum rule for dipolar absorption is satisfied, ensuring a return to optical transparency at very high frequencies. Application of the model to the broadband ͑0-THz͒ dielectric loss spectrum of a dilute solution of the probe dipolar molecule CH 2 Cl 2 in glassy decalin is demonstrated.

Physical Review E, 2006

The fractional translational diffusion of a particle in a double-well potential ͑excluding inertial effects͒ is considered. The position correlation function and its spectrum are evaluated using a fractional probability density diffusion equation ͑based on the diffusion limit of a fractal time random walk͒. Exact and approximate solutions for the dynamic susceptibility describing the position response to a small external field are obtained. The exact solution is given by matrix continued fractions while the approximate solution relies on the exponential separation of the time scales of the fast "intrawell" and low overbarrier relaxation processes associated with the bistable potential. It is shown that knowledge of the characteristic relaxation times for normal diffusion allows one to predict accurately the anomalous relaxation behavior of the system for all relevant time scales.

The Journal of Chemical Physics, 2008

Nuclear magnetic resonance ͑NMR͒ has proven to be the most valuable tool for investigating internal dynamics of proteins. In this perspective, the interpretation of NMR relaxation data eventually relies on a model of the motions. In this article, we propose to compare two radically different approaches that aim at describing internal dynamics in proteins. It is shown that the correlation functions predicted by a network of coupled rotators can be interpreted in terms of a heuristic approach based on fractional Brownian dynamics for each of the vectors in the network. Our results are interpreted in terms of the probability distributions of relaxation modes in both processes, the median of which turns out to be the relevant quantity for the comparison of both models.

Communications in Theoretical Physics, 2014

Linear theory of stationary response in thermal systems subjected to external perturbations requires to find equilibrium correlation function of the responding system variable in the absence of external perturbations. Studies of the response of the systems exhibiting anomalously slow dynamics are often based on the continuous time random walk description (CTRW) with divergent mean waiting times. The bulk of the literature on anomalous response contains linear response functions like one by Cole-Cole calculated from such a CTRW theory and applied to thermal systems. Here we show within a fairly simple and general model that for the systems with divergent mean waiting times the stationary response is absent, in accordance with some recent studies. The absence of stationary response at thermal equilibrium (or dying to zero non-stationary response in aging experiments) would confirm CTRW with divergent mean waiting times as underlying physical relaxation mechanism, but reject it otherwise. We show that the absence of stationary response is closely related to the breaking of ergodicity of the corresponding dynamical variable. As an important new result, we derive a generalized Cole-Cole response within ergodic CTRW dynamics with finite waiting time. Moreover, we provide a physically reasonable explanation of the origin and wide presence of 1/f noise in condensed systems for ergodic dynamics close to normal, rather than mostly deviating.

The Journal of Chemical Physics, 2003

We investigate aging continuous time random walks (ACTRW), introduced by Monthus and Bouchaud [J. Phys. A 29, 3847 (1996)]. Statistical behaviors of the displacement of the random walker r = r(t) − r(0) in the time interval (0, t) are obtained, after aging the random walk in the time interval (−ta, 0). In ACTRW formalism, the Green function P (r, ta, t) depends on the age of the random walk ta and the forward time t. We derive a generalized Montroll-Weiss equation, which yields an exact expression for the Fourier double Laplace transform of the ACTRW Green function. Asymptotic long times ta and t behaviors of the Green function are investigated in detail. In the limit of t ≫ ta, we recover the standard non-equilibrium CTRW behaviors, while the important regimes t ≪ ta and t ≃ ta exhibit interesting aging effects. Convergence of the ACTRW results towards the CTRW behavior, becomes extremely slow when the diffusion exponent becomes small. In the context of biased ACTRW, we investigate an aging Einstein relation. We briefly discuss aging in Scher-Montroll type of transport in disordered materials. PACS numbers: 05.45.-a, 74.40+k, 05.40.Fb

Physical Review E, 2008

The dynamical phase diagram of the fractional Langevin equation is investigated for harmonically bound particle. It is shown that critical exponents mark dynamical transitions in the behavior of the system. Four different critical exponents are found. (i) αc = 0.402 ± 0.002 marks a transition to a non-monotonic under-damped phase, (ii) αR = 0.441... marks a transition to a resonance phase when an external oscillating field drives the system, (iii) αχ 1 = 0.527... and (iv) αχ 2 = 0.707... marks transition to a double peak phase of the "loss" when such an oscillating field present. As a physical explanation we present a cage effect, where the medium induces an elastic type of friction. Phase diagrams describing over-damped, under-damped regimes, motion and resonances, show behaviors different from normal.

Physical Review E, 2009

We investigate the time average mean square displacement δ 2 (x(t)) =

Physical Review E, 2011

Continuous-time random walk (CTRW) is a model of anomalous sub-diffusion in which particles are immobilized for random times between successive jumps. A power-law distribution of the waiting times, ψ(τ ) ∼ τ −(1+α) , leads to sub-diffusion ( x 2 ∼ t α ) for 0 < α < 1. In closed systems, the long stagnation periods cause time-averages to divert from the corresponding ensemble averages, which is a manifestation of weak ergodicity breaking. The time-average of a general observable

Physical Review Letters, 2008

We investigate the dynamical phase diagram of the fractional Langevin equation and show that critical exponents mark dynamical transitions in the behavior of the system. For a free and harmonically bound particle the critical exponent αc = 0.402 ± 0.002 marks a transition to a non-monotonic under-damped phase. The critical exponent αR = 0.441... marks a transition to a resonance phase, when an external oscillating field drives the system. Physically, we explain these behaviors using a cage effect, where the medium induces an elastic type of friction. Phase diagrams describing the under-damped, the over-damped and critical frequencies of the fractional oscillator, recently used to model single protein experiments, show behaviors vastly different from normal.

Nonlinear Systems and Complexity, 2013

This book chapter introduces to the concept of weak chaos, aspects of its ergodic theory description, and properties of the anomalous dynamics associated with it. In the first half of the chapter we study simple one-dimensional deterministic maps, in the second half basic stochastic models and eventually an experiment. We start by reminding the reader of fundamental chaos quantities and their relation to each other, exemplified by the paradigmatic Bernoulli shift. Using the intermittent Pomeau-Manneville map the problem of weak chaos and infinite ergodic theory is outlined, defining a very recent mathematical field of research. Considering a spatially extended version of the Pomeau-Manneville map leads us to the phenomenon of anomalous diffusion. This problem will be discussed by applying stochastic continuous time random walk theory and by deriving a fractional diffusion equation. Another important topic within modern nonequilibrium statistical physics are fluctuation relations, which we investigate for anomalous dynamics. The chapter concludes by showing the importance of anomalous dynamics for understanding experimental results on biological cell migration.

Physical Review E, 2012

We calculate the survival probability of an immobile target surrounded by a sea of uncorrelated diffusive or subdiffusive evanescent traps (i.e., traps that disappear in the course of their motion). Our calculation is based on a fractional reaction-subdiffusion equation derived from a continuous time random walk model of the system. Contrary to an earlier method valid only in one dimension (d = 1), the equation is applicable in any Euclidean dimension d and elucidates the interplay between anomalous subdiffusive transport, the irreversible evanescence reaction, and the dimension in which both the traps and the target are embedded. Explicit results for the survival probability of the target are obtained for a density ρ(t) of traps which decays (i) exponentially and (ii) as a power law. In the former case, the target has a finite asymptotic survival probability in all integer dimensions, whereas in the latter case there are several regimes where the values of the decay exponent for ρ(t) and the anomalous diffusion exponent of the traps determine whether or not the target has a chance of eternal survival in one, two, and three dimensions.

The European Physical Journal B

The linear Boltzmann equation approach is generalized to describe fractional superdiffusive transport of the Lévy walk type in external force fields. The time distribution between scattering events is assumed to have a finite mean value and infinite variance. It is completely characterized by the two scattering rates, one fractional and a normal one, which defines also the mean scattering rate. We formulate a general fractional linear Boltzmann equation approach and exemplify it with a particularly simple case of the Bohm and Gross scattering integral leading to a fractional generalization of the Bhatnagar, Gross and Krook kinetic equation. Here, at each scattering event the particle velocity is completely randomized and takes a value from equilibrium Maxwell distribution at a given fixed temperature. We show that the retardation effects are indispensable even in the limit of infinite mean scattering rate and argue that this novel fractional kinetic equation provides a viable alternative to the fractional Kramers-Fokker-Planck (KFP) equation by Barkai and Silbey and its generalization by Friedrich et al. based on the picture of divergent mean time between scattering events. The case of divergent mean time is also discussed at length and compared with the earlier results obtained within the fractional KFP.

Frontiers in Physics

The generalized diffusion equations with fractional order derivatives have shown be quite efficient to describe the diffusion in complex systems, with the advantage of producing exact expressions for the underlying diffusive properties. Recently, researchers have proposed different fractionaltime operators (namely: the Caputo-Fabrizio and Atangana-Baleanu) which, differently from the well-known Riemann-Liouville operator, are defined by non-singular memory kernels. Here we proposed to use these new operators to generalize the usual diffusion equation. By analyzing the corresponding fractional diffusion equations within the continuous time random walk framework, we obtained waiting time distributions characterized by exponential, stretched exponential, and powerlaw functions, as well as a crossover between two behaviors. For the mean square displacement, we found crossovers between usual and confined diffusion, and between usual and sub-diffusion. We obtained the exact expressions for the probability distributions, where non-Gaussian and stationary distributions emerged. This former feature is remarkable because the fractional diffusion equation is solved without external forces and subjected to the free diffusion boundary conditions. We have further shown that these new fractional diffusion equations are related to diffusive processes with stochastic resetting, and to fractional diffusion equations with derivatives of distributed order. Thus, our results show that these new operators are a simple and efficient way for incorporating different structural aspects into the system, opening new possibilities for modeling and investigating anomalous diffusive processes.

Physical Review E, 2007

The asymptotic survival probability of a spherical target in the presence of a single subdiffusive trap or surrounded by a sea of subdiffusive traps in a continuous Euclidean medium is calculated. In one and two dimensions the survival probability of the target in the presence of a single trap decays to zero as a power law and as a power law with logarithmic correction, respectively. The target is thus reached with certainty, but it takes the trap an infinite time on average to do so. In dimensions higher than two a single trap may never reach the target and so the survival probability is finite and, in fact, does not depend on whether the traps move diffusively or subdiffusively. When the target is surrounded by a sea of traps, on the other hand, its survival probability decays as a stretched exponential in all dimensions (with a logarithmic correction in the exponent for d = 2). A trap will therefore reach the target with certainty, and will do so in a finite time. These results may be directly related to enzyme binding kinetics on DNA in the crowded cellular environment.

Physics of Plasmas, 2002

We propose fractional Fokker-Planck equation for the kinetic description of relaxation and superdiffusion processes in constant magnetic and random electric fields. We assume that the random electric field acting on a test charged particle is isotropic and possesses non-Gaussian Levy stable statistics. These assumptions provide us with a straightforward possibility to consider formation of anomalous stationary states and superdiffusion processes, both properties are inherent to strongly non-equilibrium plasmas of solar systems and thermonuclear devices. We solve fractional kinetic equations, study the properties of the solution, and compare analytical results with those of numerical simulation based on the solution of the Langevin equations with the noise source having Levy stable probability density. We found, in particular, that the stationary states are essentially non-Maxwellian ones and, at the diffusion stage of relaxation, the characteristic displacement of a particle grows superdiffusively with time and is inversely proportional to the magnetic field.

Fluctuation Relations and Beyond, 2013

We study Fluctuation Relations (FRs) for dynamics that are anomalous, in the sense that the diffusive properties strongly deviate from the ones of standard Brownian motion. We first briefly review the concept of transient work FRs for stochastic dynamics modeled by the ordinary Langevin equation. We then introduce three generic types of dynamics generating anomalous diffusion: Lévy flights, long-time correlated Gaussian stochastic processes and time-fractional kinetics. By combining Langevin and kinetic approaches we calculate the work probability distributions in the simple nonequilibrium situation of a particle subject to a constant force. This allows us to check the transient FR for anomalous dynamics. We find a new form of FRs, which is intimately related to the validity of fluctuation-dissipation relations. Analogous results are obtained for a particle in a harmonic potential dragged by a constant force. We argue that these findings are important for understanding fluctuations in experimentally accessible systems. As an example, we discuss the anomalous dynamics of biological cell migration both in equilibrium and in nonequilibrium under chemical gradients.

Physical Review E, 2005

In this article we give a general prescription for incorporating memory effects in phase space kinetic equation, and consider in particular the generalized "fractional" relaxation time model equation. We solve this for small-signal charge carriers undergoing scattering, trapping, and detrapping in a time-of-flight experimental arrangement in two ways: ͑i͒ approximately via the Chapman-Enskog scheme for the weak gradient, hydrodynamic regime, from which the fractional form of Fick's law and diffusion equation follow; and ͑ii͒ exactly, without any limitations on gradients. The latter yields complete and exact expressions in terms of generalized Mittag-Lefler functions for experimentally observable quantities. These expressions enable us to examine in detail the transition from the nonhydrodynamic stage to the hydrodynamic regime, and thereby establish the limits of validity of Fick's law and the corresponding fractional diffusion equation.

Reviews of Modern Physics, 2015

Random walk is a fundamental concept with applications ranging from quantum physics to econometrics. Remarkably, one specific model of random walks appears to be ubiquitous across many fields as a tool to analyze transport phenomena in which the dispersal process is faster than dictated by Brownian diffusion. The Lévy walk model combines two key features, the ability to generate anomalously fast diffusion and a finite velocity of a random walker. Recent results in optics, Hamiltonian chaos, cold atom dynamics, bio-physics, and behavioral science demonstrate that this particular type of random walks provides significant insight into complex transport phenomena. This review provides a self-consistent introduction to Lévy walks, surveys their existing applications, including latest advances, and outlines further perspectives.

The European Physical Journal Special Topics, 2013

Methods for the exact solution of fractional Fokker-Planck equations for anomalous diffusion in an external potential are discussed using both ordinary and matrix continued fractions, whereby the scalar multi-term recurrence relations generated by such fractional diffusion equations are reduced to three-term matrix ones. The procedure is illustrated by solving various problems concerning the anomalous translational diffusion in both periodic and double-well potentials.

The Journal of Chemical Physics, 2009

We address the problem of establishing which is the proper form of quantum master equation generating a survival probability identical to that corresponding to the nonergodic sequence of "light on" and "light off" fluorescence fluctuations in blinking quantum dots. We adopt a theoretical perspective based on the assumption that the abrupt transitions from the light on to light off state are the results of many collisions between system and environment, properly described by the Lindblad equation, and that between two consecutive collisions the system dynamics are frozen. This generates a quantum master equation belonging to the recently proposed class of generalized Lindblad equations, with a time convoluted structure, involving in the specific case of this paper both the unitary and the nonunitary contribution of the Lindblad equation. This is the property that under the low-frequency condition makes the new class of generalized Lindblad equation generates the required survival probability. We make the conjecture that this equation corresponds to the cooperative dynamics of many units that, in isolation, are described by the ordinary Lindblad equation. When the time scale of the unitary term of the Lindblad equation is shorter than the dephasing time, the cooperation generates a surprisingly extended macroscopic coherence.

Physical Review E, 2004

We study the random growth of surfaces from within the perspective of a single column, namely, the fluctuation of the column height around the mean value, y(t) ≡ h(t) − h(t) , which is depicted as being subordinated to a standard fluctuation-dissipation process with friction γ. We argue that the main properties of Kardar-Parisi-Zhang theory, in one dimension, are derived by identifying the distribution of return times to y(0) = 0, which is a truncated inverse power law, with the distribution of subordination times. The agreement of the theoretical prediction with the numerical treatment of the 1 + 1 dimensional model of ballistic deposition is remarkably good, in spite of the finite size effects affecting this model.

Physical review. E, 2017

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 relaxat...

Physical Review E, 2011

In this paper we extend the subdiffusive Klein-Kramers model, in which the waiting times are modeled by the α-stable laws, to the case of waiting times belonging to the class of tempered α-stable distributions. We introduce a generalized version of the Klein-Kramers equation, in which the fractional Riemman-Liouville derivative is replaced with a more general integro-differential operator. This allows a transition from the initial subdiffusive character of motion to the standard diffusion for long times to be modeled. Taking advantage of the corresponding Langevin equation, we study some properties of the tempered dynamics, in particular, we approximate solutions of the tempered Klein-Kramers equation via Monte Carlo methods. Also, we study the distribution of the escape time from the potential well and compare it to the classical results in the Kramers escape theory. Finally, we derive the analytical formula for the first-passage-time distribution for the case of free particles. We show that the well-known Sparre Andersen scaling holds also for the tempered subdiffusion.

Physical Review E, 2007

Subdiffusion in the presence of an external force field can be described in phase space by the fractional Klein-Kramers equation. In this paper, we explore the stochastic structure of this equation. Using a subordination method, we define a random process whose probability density function is a solution of the fractional Klein-Kramers equation. The structure of the introduced process agrees with the two-stage scenario underlying the anomalous diffusion mechanism, in which trapping events are superimposed on the Langevin dynamics. We develop an efficient computer algorithm for visualization of fractional Klein-Kramers dynamics and present some simulation results based on Monte Carlo techniques.

Journal of Statistical Physics, 2018

In this article, anomalous transport models of energetic particles in space plasmas are developed by deriving the fractional force-less Fokker-Planck equation and the fractional diffusion-advection equation from the Klein-Kramers equation. Analytical solutions of the space-time fractional equations are obtained by use of the Caputo and the Riesz fractional derivatives with the Laplace-Fourier technique. The solutions are given in terms of Fox's H-function and the profiles of the particles densities are discussed in each case for different values of fractional orders.

Physical Review E, 2008

Ivanov-Anderson (IA) model (and an earlier treatment by Kubo) envisages a decay of the orientational correlation by random but large amplitude molecular jumps, as opposed to infinitesimal small jumps assumed in Brownian diffusion. Recent computer simulation studies on water and supercooled liquids have shown that large amplitude motions may indeed be more of a rule than exception. Existing theoretical studies on jump diffusion mostly assume an exponential (Poissonian) waiting time distribution for jumps, thereby again leading to an exponential decay. Here we extend the existing formalism of Ivanov and Anderson to include an algebraic waiting time distribution between two jumps. As a result, the first (ℓ = 1) and second (ℓ = 2) rank orientational time correlation functions show the same long time power law, but their short time decay behavior is quite different. The predicted Cole-Cole plot of dielectric relaxation reproduces various features of non-Debye behaviour observed experimentally. We also developed a theory where both unrestricted small jumps and large angular jumps coexist simultaneously. The small jumps are shown to have a large effect on the long time decay, particularly in mitigating the effects of algebraic waiting time distribution, and in giving rise to an exponential-like decay, with a time constant, surprisingly, less than the time constant that arises from small amplitude decay alone.

Physical Review E, 2016

The integro-differential wave equation for the probability density function for a classical onedimensional Lévy walk with continuous sample paths has been derived. This equation involves a classical wave operator together with memory integrals describing the spatio-temporal coupling of the Lévy walk. It is valid for any running time PDF of a walker and it does not involve any longtime large-scale approximations. It generalizes the well-known telegraph equation obtained from the persistent random walk. Several non-Markovian cases are considered when the particle's velocity alternates at the gamma and power-law distributed random times.

Physical Review E, 2014

We investigate the dynamical phase diagram of the generalized Langevin equation of the free particle driven by a Mittag-Leffler noise and show critical curves and a critical value of the exponent parameter of the Mittag-Leffler function that mark different dynamical regimes. By considering that the modeling of a Mittag-Leffer memory kernel corresponds to a power-law second-order memory kernel, we show that the generalized Langevin equation of the velocity autocorrelation function (VACF) is transformed in a fractional Langevin equation. In the superdiffusive case our results exhibit oscillations and negative correlations of the VACF that are not provided by the usual power-law noise model.

The Journal of chemical physics, 2014

The paper presents a rigorous derivation of the velocity autocorrelation function for an anomalously diffusing slow solute particle in a bath of fast solvent molecules. The result is obtained within the framework of the generalized Langevin equation and uses only scaling arguments and identities which are based on asymptotic analysis. It agrees with the velocity autocorrelation function of an anomalously diffusing Rayleigh particle whose dynamics is described by a fractional Ornstein-Uhlenbeck process in velocity space. A simple semi-analytical example illustrates under which conditions the latter model is appropriate.

Physical review. E, Statistical, nonlinear, and soft matter physics, 2001

We consider a Lévy flyer of order alpha that starts from a point x(0) on an interval [O,L] with absorbing boundaries. We find a closed-form expression for the average number of flights the flyer takes and the total length of the flights it travels before it is absorbed. These two quantities are equivalent to the mean first passage times for Lévy flights and Lévy walks, respectively. Using fractional differential equations with a Riesz kernel, we find exact analytical expressions for both quantities in the continuous limit. We show that numerical solutions for the discrete Lévy processes converge to the continuous approximations in all cases except the case of alpha--&gt;2, and the cases of x(0)--&gt;0 and x(0)--&gt;L. For alpha&gt;2, when the second moment of the flight length distribution exists, our result is replaced by known results of classical diffusion. We show that if x(0) is placed in the vicinity of absorbing boundaries, the average total length has a minimum at alpha=1, c...