Continuous Time Random Walks with Reactions Forcing and Trapping

Physical review. E, Statistical, nonlinear, and soft matter physics, 2006

We have revisited the problem of anomalously diffusing species, modeled at the mesoscopic level using continuous time random walks, to include linear reaction dynamics. If a constant proportion of walkers are added or removed instantaneously at the start of each step then the long time asymptotic limit yields a fractional reaction-diffusion equation with a fractional order temporal derivative operating on both the standard diffusion term and a linear reaction kinetics term. If the walkers are added or removed at a constant per capita rate during the waiting time between steps then the long time asymptotic limit has a standard linear reaction kinetics term but a fractional order temporal derivative operating on a nonstandard diffusion term. Results from the above two models are compared with a phenomenological model with standard linear reaction kinetics and a fractional order temporal derivative operating on a standard diffusion term. We have also developed further extensions of the...

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.

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.

Fractional Calculus and Applied Analysis

We consider an ensemble of Ornstein–Uhlenbeck processes featuring a population of relaxation times and a population of noise amplitudes that characterize the heterogeneity of the ensemble. We show that the centre-of-mass like variable corresponding to this ensemble is statistically equivalent to a process driven by a non-autonomous stochastic differential equation with time-dependent drift and a white noise. In particular, the time scaling and the density function of such variable are driven by the population of timescales and of noise amplitudes, respectively. Moreover, we show that this variable is equivalent in distribution to a randomly-scaled Gaussian process, i.e., a process built by the product of a Gaussian process times a non-negative independent random variable. This last result establishes a connection with the so-called generalized grey Brownian motion and suggests application to model fractional anomalous diffusion in biological systems.

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.

Journal of Statistical Physics, 1989

We consider a nonlinear reaction-diffusion model: n Brownian particles move independently in R a and eventually die. The interaction, of binary type, affects only the death rate. The radius of interaction goes to zero as the number of particles increases and we characterize a wide range of speeds at which the radius goes to zero. Within this range we show a law of large numbers for the empirical distributions of the alive particles. The limit is independent of the choice of the speed and it is characterized as the solution of a nonlinear reaction-diffusion equation.

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 "fractional" 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...

Physical Chemistry Chemical Physics, 2014

Anomalous diffusion is frequently described by scaled Brownian motion (SBM), a Gaussian process with a power-law time dependent diffusion coefficient. Its mean squared displacement is x 2 (t) ≃ K (t)t with K (t) ≃ t α−1 for 0 < α < 2. SBM may provide a seemingly adequate description in the case of unbounded diffusion, for which its probability density function coincides with that of fractional Brownian motion. Here we show that free SBM is weakly non-ergodic but does not exhibit a significant amplitude scatter of the time averaged mean squared displacement. More severely, we demonstrate that under confinement, the dynamics encoded by SBM is fundamentally different from both fractional Brownian motion and continuous time random walks. SBM is highly non-stationary and cannot provide a physical description for particles in a thermalised stationary system. Our findings have direct impact on the modelling of single particle tracking experiments, in particular, under confinement inside cellular compartments or when optical tweezers tracking methods are used.

Physical Review E, 2013

We study anomalous diffusion governed by the n-tuple order time-fractional Fokker-Planck equation of Riemann-Liouville type. We find a stochastic representation of this anomalous diffusion; that is, the stochastic process with probability density function evolving according to the considered diffusion equation. The stochastic representation, given in the form of the Brownian motion subordinated by a Lévy process defined in terms of the Lévy exponent, is shown to be an accelerating subdiffusion process. Taking advantage of the stochastic representation, we construct an algorithm for computer simulations of the accelerating-subdiffusion sample paths. Moreover, we obtain a stochastic representation and the simulation algorithm for the case of the anomalous diffusion influenced by an external space-dependent force field.

Physical Review E, 2008

The Fokker-Planck equation for the probability f (r, t) to find a random walker at position r at time t is derived for the case that the the probability to make jumps depends nonlinearly on f (r, t). The result is a generalized form of the classical Fokker-Planck equation where the effects of drift, due to a violation of detailed balance, and of external fields are also considered. It is shown that in the absence of drift and external fields a scaling solution, describing anomalous diffusion, is only possible if the nonlinearity in the jump probability is of the power law type (∼ f η (r, t)), in which case the generalized Fokker-Planck equation reduces to the well-known Porous Media equation. Monte-Carlo simulations are shown to confirm the theoretical results.

Physical Review Letters, 2008

Single particle tracking of mRNA molecules and lipid granules in living cells shows that the time averaged mean squared displacement δ 2 of individual particles remains a random variable while indicating that the particle motion is subdiffusive. We investigate this type of ergodicity breaking within the continuous time random walk model and show that δ 2 differs from the corresponding ensemble average. In particular we derive the distribution for the fluctuations of the random variable δ 2 . Similarly we quantify the response to a constant external field, revealing a generalization of the Einstein relation. Consequences for the interpretation of single molecule tracking data are discussed. 05.40.Fb,87.10.Mn An ensemble of non interacting Brownian particles spreads according to Fick's law as a Gaussian packet. The ensemble averaged mean square displacement (MSD) is x 2 (t) = 2D 1 t where D 1 is the diffusion constant. By an Einstein relation D 1 is expressed in terms of statistical properties of the microscopic jumps according to D 1 = δx 2 /2 τ where τ is the average time between jumps and δx 2 is the variance of the jump lengths. Instead one can analyze the time series x(t) of the particle trajectory and determine the time averaged (TA) MSD

Journal of Computational and Applied Mathematics, 2009

To offer an insight into the rapidly developing theory of fractional diffusion processes we describe in some detail three topics of current interest: (i) the well-scaled passage to the limit from continuous time random walk under power law assumptions to space-time fractional diffusion, (ii) the asymptotic universality of the Mittag-Leffler waiting time law in time-fractional processes, (iii) our method of parametric subordination for generating particle trajectories.

Physical Review E, 2013

We present a Master Equation formulation based on a Markovian random walk model that exhibits sub-diffusion, classical diffusion and super-diffusion as a function of a single parameter. The non-classical diffusive behavior is generated by allowing for interactions between a population of walkers. At the macroscopic level, this gives rise to a nonlinear Fokker-Planck equation. The diffusive behavior is reflected not only in the mean-squared displacement ( r 2 (t) ∼ t γ with 0 < γ ≤ 1.5) but also in the existence of self-similar scaling solutions of the Fokker-Planck equation. We give a physical interpretation of sub-and super-diffusion in terms of the attractive and repulsive interactions between the diffusing particles and we discuss analytically the limiting values of the exponent γ. Simulations based on the Master Equation are shown to be in agreement with the analytical solutions of the nonlinear Fokker-Planck equation in all three diffusion regimes.

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.

Journal of Physics A: Mathematical and Theoretical

Complex systems are known to display anomalous diffusion, whose signature is a space/time scaling x ∼ t δ with δ = 1/2 in the Probability Density Function (PDF). Anomalous diffusion can emerge jointly with both Gaussian, e.g., fractional Brownian motion, and power-law decaying distributions, e.g., Lévy Flights (LFs) or Lévy Walks (LWs). LFs get anomalous scaling, but also infinite position variance and, being jumps of any size allowed even at short times, also infinite energy and discontinuous velocity. LWs are based on random trapping events, resemble a Lévy-type power-law distribution that is truncated in the large displacement range and have finite moments, finite energy and discontinuous velocity. However, both LFs and LWs cannot describe friction-diffusion processes and do not take into account the role of strong heterogeneity in many complex systems, such as biological transport in the crowded cell environment. We propose and discuss a model describing a Heterogeneous Ensemble of Brownian Particles (HEBP) based on a linear Langevin equation. We show that, for proper distributions of relaxation time and velocity diffusivity, the HEBP displays features similar to LWs, in particular power-law decaying PDF, longrange correlations and anomalous diffusion, at the same time keeping finite position moments and finite energy. The main differences between the HEBP model and two LWs are investigated, finding that, even if the PDFs are similar, they differ in three main aspects: (i) LWs are biscaling, while HEBP is monoscaling; (ii) a transition from anomalous (δ = 1/2) to normal (δ = 1/2) diffusion in the long-time regime; (iii) the power-law index of the position PDF and the space/time diffusion scaling are independent in the HEBP, while they both depend on the scaling of the inter-event time PDF in LWs. The HEBP model is derived from a friction-diffusion process, it has finite energy and it satisfies the fluctuation-dissipation theorem.