Fractional Fokker-Planck equations for subdiffusion with space- and time-dependent forces

Chaotic Zeeman effect: a fractional diffusion-like approch

It is shown that the chaotic Zeeman effect of a quantum system can be formally viewed as a result of fractional calculus. The fractional calculation brings into the equations the angle θ formed between the internal and the external magnetic field applied to the atom. The further the fractional coefficient α is from the ordinary case corresponding to α = 1 , the more important the chaotic effect is. The case corresponding to α = 1 does not depend on the angle θ , obtaining the nonchaotic situation known in the literature. Non-Gaussian distributions correspond to non-stationary variables. Considering a Lorenzian type distribution, we can make a connection between the fractional formalism and random matrix theory. The connection validates the link between fractional calculus and chaos, and at the same time due to the θ angle, it gives the phenomenon a physical interpretation.

Non-normalizable quasiequilibrium states under fractional dynamics

Particles anomalously diffusing in contact with a thermal bath are initially released from an asymptotically flat potential well. For temperatures that are sufficiently low compared to the potential depth, the dynamical and thermodynamical observables of the system remain almost constant for long times. We show how these stagnated states are characterized as non-normalizable quasiequilibrium (NNQE) states. We use the fractional-time Fokker-Planck equation (FTFPE) and continuous-time random walk approaches to calculate ensemble averages. We obtain analytical estimates of the durations of NNQE states, depending on the fractional order, from approximate theoretical solutions of the FTFPE. We study and compare two types of observables, the mean square displacement typically used to characterize diffusion, and the thermodynamic energy. We show that the typical timescales for transient stagnation depend exponentially on the value of the depth of the potential well, in units of temperature, multiplied by a function of the fractional exponent.

Population heterogeneity in the fractional master equation, ensemble self-reinforcement, and strong memory effects

We formulate a fractional master equation in continuous time with random transition probabilities across the population of random walkers such that the effective underlying random walk exhibits ensemble self-reinforcement. The population heterogeneity generates a random walk with conditional transition probabilities that increase with the number of steps taken previously (self-reinforcement). Through this, we establish the connection between random walks with a heterogeneous ensemble and those with strong memory where the transition probability depends on the entire history of steps. We find the ensemble-averaged solution of the fractional master equation through subordination involving the fractional Poisson process counting the number of steps at a given time and the underlying discrete random walk with self-reinforcement. We also find the exact solution for the variance which exhibits superdiffusion even as the fractional exponent tends to 1.

Fractional Advection-Diffusion-Asymmetry Equation

Fractional kinetic equations employ noninteger calculus to model anomalous relaxation and diffusion in many systems. While this approach is well explored, it so far failed to describe an important class of transport in disordered systems. Motivated by work on contaminant spreading in geological formations, we propose and investigate a fractional advection-diffusion equation describing the biased spreading packet. While usual transport is described by diffusion and drift, we find a third term describing symmetry breaking which is omnipresent for transport in disordered systems. Our work is based on continuous time random walks with a finite mean waiting time and a diverging variance, a case that on the one hand is very common and on the other was missing in the kaleidoscope literature of fractional equations. The fractional space derivatives stem from long trapping times, while previously they were interpreted as a consequence of spatial Lévy flights.

Asymptotic Behavior of the Solution of the Space Dependent Variable Order Fractional Diffusion Equation: Ultraslow Anomalous Aggregation

We find the asymptotic representation of the solution of the variable-order fractional diffusion equation, which remains unsolved since it was proposed by Chechkin, Gorenflo, and Sokolov [J. Phys. A, 38, L679 (2005)JPHAC50305-447010.1088/0305-4470/38/42/L03]. We identify a new advection term that causes ultraslow spatial aggregation of subdiffusive particles due to dominance over the standard advection and diffusion terms in the long-time limit. This uncovers the anomalous mechanism by which nonuniform distributions can occur. We perform Monte Carlo simulations of the underlying anomalous random walk and find excellent agreement with the asymptotic solution.

Subdiffusive continuous-time random walks with stochastic resetting

We analyze two models of subdiffusion with stochastic resetting. Each of them consists of two parts: subdiffusion based on the continuous-time random walk scheme and independent resetting events generated uniformly in time according to the Poisson point process. In the first model the whole process is reset to the initial state, whereas in the second model only the position is subject to resets. The distinction between these two models arises from the non-Markovian character of the subdiffusive process. We derive exact expressions for the two lowest moments of the full propagator, stationary distributions, and first hitting time statistics. We also show, with an example of a constant drift, how these models can be generalized to include external forces. Possible applications to data analysis and modeling of biological systems are also discussed.

Biased continuous-time random walks for ordinary and equilibrium cases: facilitation of diffusion, ergodicity breaking and ageing

We examine renewal processes with power-law waiting time distributions (WTDs) and non-zero drift via computing analytically and by computer simulations their ensemble and time averaged spreading characteristics. All possible values of the scaling exponent α are considered for the WTD ψ(t) ∼ 1/t1+α. We treat continuous-time random walks (CTRWs) with 0 < α < 1 for which the mean waiting time diverges, and investigate the behaviour of the process for both ordinary and equilibrium CTRWs for 1 < α < 2 and α > 2. We demonstrate that in the presence of a drift CTRWs with α < 1 are ageing and non-ergodic in the sense of the non-equivalence of their ensemble and time averaged displacement characteristics in the limit of lag times much shorter than the trajectory length. In the sense of the equivalence of ensemble and time averages, CTRW processes with 1 < α < 2 are ergodic for the equilibrium and non-ergodic for the ordinary situation. Lastly, CTRW renewal processes with α > 2-both for the equilibrium and ordinary situation-are always ergodic. For the situations 1 < α < 2 and α > 2 the variance of the diffusion process, however, depends on the initial ensemble. For biased CTRWs with α > 1 we also investigate the behaviour of the ergodicity breaking parameter. In addition, we demonstrate that for biased CTRWs the Einstein relation is valid on the level of the ensemble and time averaged displacements, in the entire range of the WTD exponent α.

Generalized fractional diffusion equations for subdiffusion in arbitrarily growing domains

The ubiquity of subdiffusive transport in physical and biological systems has led to intensive efforts to provide robust theoretical models for this phenomena. These models often involve fractional derivatives. The important physical extension of this work to processes occurring in growing materials has proven highly nontrivial. Here we derive evolution equations for modeling subdiffusive transport in a growing medium. The derivation is based on a continuous-time random walk. The concise formulation of these evolution equations requires the introduction of a new, comoving, fractional derivative. The implementation of the evolution equation is illustrated with a simple model of subdiffusing proteins in a growing membrane.

Impact of anomalous transport kinetics on the progress of wound healing

This work focuses on the transport kinetics of chemical and cellular species during wound healing. Anomalous transport kinetics, coupling sub- and superdiffusion with chemotaxis, and fractional viscoelasticity of soft tissues are analyzed from a modeling point of view. The paper presents a generalization of well stablished mechano-chemical models of wound contraction (Murphy et al., 2012; Valero et al., 2014) to include the previously mentioned anomalous effects by means of partial differential equations of fractional order. Results show the effect that anomalous dynamics have on the contraction rate and extension and on the distribution of biological species, and indicators of fibroproliferative disorders are identified.

A Fractional Order Recovery SIR Model from a Stochastic Process

Over the past several decades, there has been a proliferation of epidemiological models with ordinary derivatives replaced by fractional derivatives in an ad hoc manner. These models may be mathematically interesting, but their relevance is uncertain. Here we develop an SIR model for an epidemic, including vital dynamics, from an underlying stochastic process. We show how fractional differential operators arise naturally in these models whenever the recovery time from the disease is power-law distributed. This can provide a model for a chronic disease process where individuals who are infected for a long time are unlikely to recover. The fractional order recovery model is shown to be consistent with the Kermack-McKendrick age-structured SIR model, and it reduces to the Hethcote-Tudor integral equation SIR model. The derivation from a stochastic process is extended to discrete time, providing a stable numerical method for solving the model equations. We have carried out simulations of the fractional order recovery model showing convergence to equilibrium states. The number of infecteds in the endemic equilibrium state increases as the fractional order of the derivative tends to zero.

Subdiffusion in an external potential: Anomalous effects hiding behind normal behavior

We propose a model of subdiffusion in which an external force is acting on a particle at all times not only at the moment of jump. The implication of this assumption is the dependence of the random trapping time on the force with the dramatic change of particles behavior compared to the standard continuous time random walk model in the long time limit. Constant force leads to the transition from non-ergodic subdiffusion to ergodic diffusive behavior. However, we show this behavior remains anomalous in a sense that the diffusion coefficient depends on the external force and on the anomalous exponent. For quadratic potential we find that the system remains non-ergodic. The anomalous exponent in this case defines not only the speed of convergence but also the stationary distribution which is different from standard Boltzmann equilibrium.

Nonlinear subdiffusive fractional equations and the aggregation phenomenon

In this article we address the problem of the nonlinear interaction of subdiffusive particles. We introduce the random walk model in which statistical characteristics of a random walker such as escape rate and jump distribution depend on the mean density of particles. We derive a set of nonlinear subdiffusive fractional master equations and consider their diffusion approximations. We show that these equations describe the transition from an intermediate subdiffusive regime to asymptotically normal advection-diffusion transport regime. This transition is governed by nonlinear tempering parameter that generalizes the standard linear tempering. We illustrate the general results through the use of the examples from cell and population biology. We find that a nonuniform anomalous exponent has a strong influence on the aggregation phenomenon.

Continuous-time random walks on networks with vertex- and time-dependent forcing

We have investigated the transport of particles moving as random walks on the vertices of a network, subject to vertex- and time-dependent forcing. We have derived the generalized master equations for this transport using continuous time random walks, characterized by jump and waiting time densities, as the underlying stochastic process. The forcing is incorporated through a vertex- and time-dependent bias in the jump densities governing the random walking particles. As a particular case, we consider particle forcing proportional to the concentration of particles on adjacent vertices, analogous to self-chemotactic attraction in a spatial continuum. Our algebraic and numerical studies of this system reveal an interesting pair-aggregation pattern formation in which the steady state is composed of a high concentration of particles on a small number of isolated pairs of adjacent vertices. The steady states do not exhibit this pair aggregation if the transport is random on the vertices, i.e., without forcing. The manifestation of pair aggregation on a transport network may thus be a signature of self-chemotactic-like forcing.

Kinetic lattice Monte Carlo simulation of viscoelastic subdiffusion

We propose a kinetic Monte Carlo method for the simulation of subdiffusive random walks on a cartesian lattice. The random walkers are subject to viscoelastic forces which we compute from their individual trajectories via the fractional Langevin equation. At every step the walkers move by one lattice unit, which makes them differ essentially from continuous time random walks, where the subdiffusive behavior is induced by random waiting. To enable computationally inexpensive simulations with n-step memories, we use an approximation of the memory and the memory kernel functions with a complexity O(log n). Eventual discretization and approximation artifacts are compensated with numerical adjustments of the memory kernel functions. We verify with a number of analyses that this new method provides binary fractional random walks that are fully consistent with the theory of fractional brownian motion.

FRACTIONAL DYNAMICS AT MULTIPLE TIMES

A continuous time random walk (CTRW) imposes a random waiting time between random particle jumps. CTRW limit densities solve a fractional Fokker-Planck equation, but since the CTRW limit is not Markovian, this is not sufficient to characterize the process. This paper applies continuum renewal theory to restore the Markov property on an expanded state space, and compute the joint CTRW limit density at multiple times.

Weak subordination breaking for the quenched trap model

We map the problem of diffusion in the quenched trap model onto a different stochastic process: Brownian motion that is terminated at the coverage time S(α)=∑(x=-∞)(∞)(n(x))(α), with n(x) being the number of visits to site x. Here 0<α=T/T(g)<1 is a measure of the disorder in the original model. This mapping allows us to treat the intricate correlations in the underlying random walk in the random environment. The operational time S(α) is changed to laboratory time t with a Lévy time transformation. Investigation of Brownian motion stopped at time S(α) yields the diffusion front of the quenched trap model, which is favorably compared with numerical simulations. In the zero-temperature limit of α→0 we recover the renormalization group solution obtained by Monthus [Phys. Rev. E 68, 036114 (2003)]. Our theory surmounts the critical slowing down that is found when α→1. Above the critical dimension 2, mapping the problem to a continuous time random walk becomes feasible, though still not trivial.

Linear response at criticality

We study a set of cooperatively interacting units at criticality, and we prove with analytical and numerical arguments that they generate the same renewal non-Poisson intermittency as that produced by blinking quantum dots, thereby giving a stronger support to the results of earlier investigation. By analyzing how this out-of-equilibrium system responds to harmonic perturbations, we find that the response can be described only using a new form of linear response theory that accounts for aging and the nonergodic behavior of the underlying process. We connect the undamped response of the system at criticality to the decaying response predicted by the recently established nonergodic fluctuation-dissipation theorem for dichotomous processes using information about the second moment of the fluctuations. We demonstrate that over a wide range of perturbation frequencies the response of the cooperative system is greatest when at criticality.

Subdiffusive master equation with space-dependent anomalous exponent and structural instability

We derive the fractional master equation with space-dependent anomalous exponent. We analyze the asymptotic behavior of the corresponding lattice model both analytically and by Monte Carlo simulation. We show that the subdiffusive fractional equations with constant anomalous exponent μ in a bounded domain [0,L] are not structurally stable with respect to the nonhomogeneous variations of parameter μ. In particular, the Gibbs-Boltzmann distribution is no longer the stationary solution of the fractional Fokker-Planck equation whatever the space variation of the exponent might be. We analyze the random distribution of μ in space and find that in the long-time limit, the probability distribution is highly intermediate in space and the behavior is completely dominated by very unlikely events. We show that subdiffusive fractional equations with the nonuniform random distribution of anomalous exponent is an illustration of a "Black Swan," the low probability event of the small value of the anomalous exponent that completely dominates the long-time behavior of subdiffusive systems.

Continuous-time random walks that alter environmental transport properties

We consider continuous-time random walks (CTRWs) in which the walkers have a finite probability to alter the waiting-time and/or step-length transport properties of their environment, resulting in possibly transient anomalous diffusion. We refer to these CTRWs as transmogrifying continuous-time random walks (TCTRWs) to emphasize that they change the form of the transport properties of their environment, and in a possibly strange way. The particular case in which the CTRW waiting-time density has a finite probability to be permanently altered at a given site, following a visitation by a walker, is considered in detail. Master equations for the probability density function of transmogrifying random walkers are derived, and results are compared with Monte Carlo simulations. An interesting finding is that TCTRWs can generate transient subdiffusion or transient superdiffusion without invoking truncated or tempered power law densities for either the waiting times or the step lengths. The transient subdiffusion or transient superdiffusion arises in TCTRWs with Gaussian step-length densities and exponential waiting-time densities when the altered average waiting time is greater than or less than, respectively, the original average waiting time.

Fractional Feynman-Kac equation for weak ergodicity breaking

The continuous-time random walk (CTRW) is a model of anomalous subdiffusion 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 subdiffusion (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 U(t) = 1/t ∫(0)(t) U[x(τ)]dτ is a functional of the path and is described by the well-known Feynman-Kac equation if the motion is Brownian. Here, we derive forward and backward fractional Feynman-Kac equations for functionals of CTRW in a binding potential. We use our equations to study two specific time averages: the fraction of time spent by a particle in half-box, and the time average of the particle's position in a harmonic field. In both cases, we obtain the probability density function of the time averages for t → ∞ and the first two moments. Our results show that both the occupation fraction and the time-averaged position are random variables even for long times, except for α = 1, when they are identical to their ensemble averages. Using our fractional Feynman-Kac equation, we also study the dynamics leading to weak ergodicity breaking, namely the convergence of the fluctuations to their asymptotic values.

Transmission of information between complex systems: 1/f resonance

We study the transport of information between two complex systems with similar properties. Both systems generate non-Poisson renewal fluctuations with a power-law spectrum 1/f(3-μ), the case μ=2 corresponding to ideal 1/f noise. We denote by μ(S) and μ(P) the power-law indexes of the system of interest S and the perturbing system P, respectively. By adopting a generalized fluctuation-dissipation theorem (FDT) we show that the ideal condition of 1/f noise for both systems corresponds to maximal information transport. We prove that to make the system S respond when μ(S)<2 we have to set the condition μ(P)<2. In the latter case, if μ(P)<μ(S), the system S inherits the relaxation properties of the perturbing system. In the case where μ(P)>2, no response and no information transmission occurs in the long-time limit. We consider two possible generalizations of the fluctuation dissipation theorem and show that both lead to maximal information transport in the condition of 1/f noise.

Time transformation for random walks in the quenched trap model

We investigate subdiffusion in the quenched trap model by mapping the problem onto a new stochastic process: Brownian motion stopped at the operational time S(α) = ∑(x=-∞)(∞) (n(x))(α) where n(x) is the visitation number at site x and α is a measure of the disorder. In the limit of zero temperature we recover the renormalization group solution found by Monthus. Our approach is an alternative to the renormalization group and is capable of dealing with any disorder strength.