arXiv subjects

Eli Barkai

Publications and source records attributed to Eli Barkai.

At least 109 records · Page 6Linked to original sources

arXiv2014

The Distribution of the Area under a Bessel Excursion and its Moments

A Bessel excursion is a Bessel process that begins at the origin and first returns there at some given time $T$. We study the distribution of the area under such an excursion, which recently found application in the context of laser cooling. The area $A$ scales with the time as $A \sim T^{3/2}$, independent of the dimension, $d$, but the functional form of the distribution does depend on $d$. We demonstrate that for $d=1$, the distribution reduces as expected to the distribution for the area under a Brownian excursion, known as the Airy distribution, deriving a new expression for the Airy distribution in the process. We show that the distribution is symmetric in $d-2$, with nonanalytic behavior at $d=2$. We calculate the first and second moments of the distribution, as well as a particular fractional moment. We also analyze the analytic continuation from $d<2$ to $d>2$. In the limit where $d\to 4$ from below, this analytically continued distribution is described by a one-sided Lévy $α$-stable distribution with index $2/3$ and a scale factor proportional to $[(4-d)T]^{3/2}$.

math-ph↗

arXiv2014

Occupation times on a comb with ramified teeth

We investigate occupation time statistics for random walks on a comb with ramified teeth. This is achieved through the relation between the occupation time and the first passage times. Statistics of occupation times in half space follows Lamperti's distribution, i.e. the generalized arcsine law holds. Transitions between different behaviors are observed, which are controlled by the size of the backbone and teeth of the comb, as well as bias. Occupation time on a non-simply connected domain is analyzed with a mean-field theory and numerical simulations. In that case, the generalized arcsine law isn't valid.

cond-mat.stat-mech↗

arXiv2014

Numerical algorithms for the forward and backward fractional Feynman-Kac equations

The Feynman-Kac equations are a type of partial differential equations describing the distribution of functionals of diffusive motion. The probability density function (PDF) of Brownian functionals satisfies the Feynman-Kac formula, being a Schrödinger equation in imaginary time. The functionals of no-Brownian motion, or anomalous diffusion, follow the fractional Feynman-Kac equation [J. Stat. Phys. 141, 1071-1092, 2010], where the fractional substantial derivative is involved. Based on recently developed discretized schemes for fractional substantial derivatives [arXiv:1310.3086], this paper focuses on providing algorithms for numerically solving the forward and backward fractional Feynman-Kac equations; since the fractional substantial derivative is non-local time-space coupled operator, new challenges are introduced comparing with the general fractional derivative. Two ways (finite difference and finite element) of discretizing the space derivative are considered. For the backward fractional Feynman-Kac equation, the numerical stability and convergence of the algorithms with first order accuracy are theoretically discussed; and the optimal estimates are obtained. For all the provided schemes, including the first order and high order ones, of both forward and backward Feynman-Kac equations, extensive numerical experiments are performed to show their effectiveness.

physics.comp-ph↗

arXiv2013

Aging renewal theory and application to random walks

The versatility of renewal theory is owed to its abstract formulation. Renewals can be interpreted as steps of a random walk, switching events in two-state models, domain crossings of a random motion, etc. We here discuss a renewal process in which successive events are separated by scale-free waiting time periods. Among other ubiquitous long time properties, this process exhibits aging: events counted initially in a time interval [0,t] statistically strongly differ from those observed at later times [t_a,t_a+t]. In complex, disordered media, processes with scale-free waiting times play a particularly prominent role. We set up a unified analytical foundation for such anomalous dynamics by discussing in detail the distribution of the aging renewal process. We analyze its half-discrete, half-continuous nature and study its aging time evolution. These results are readily used to discuss a scale-free anomalous diffusion process, the continuous time random walk. By this we not only shed light on the profound origins of its characteristic features, such as weak ergodicity breaking. Along the way, we also add an extended discussion on aging effects. In particular, we find that the aging behavior of time and ensemble averages is conceptually very distinct, but their time scaling is identical at high ages. Finally, we show how more complex motion models are readily constructed on the basis of aging renewal dynamics.

cond-mat.stat-mech↗

arXiv2013

Numerical estimate of infinite invariant densities: application to Pesin-type identity

Weakly chaotic maps with unstable fixed points are investigated in the regime where the invariant density is non-normalizable. We propose that the infinite invariant density of these maps can be estimated using as the long time limit of t^(1-alpha) rho(x, t), in agreement with earlier work of Thaler. Here rho(x, t) is the normalizable density of particles. This definition uniquely determines the infinite density and is a valuable tool for numerical estimations. We use this density to estimate the subexponential separation lambda_alpha of nearby trajectories. For a particular map introduced by Thaler we use an analytical expression for the infinite invariant density to calculate lambda_alpha exactly, which perfectly matches simulations without fitting. Misunderstanding which recently appeared in the literature is removed.

cond-mat.stat-mech↗

arXiv2013

Noisy continuous time random walks

Experimental studies of the diffusion of biomolecules in the environment of biological cells are routinely confronted with multiple sources of stochasticity, whose identification renders the detailed data analysis of single molecule trajectories quite intricate. Here we consider subdiffusive continuous time random walks, that represent a seminal model for the anomalous diffusion of tracer particles in complex environments. This motion is characterized by multiple trapping events with infinite mean sojourn time. In real physical situations, however, instead of the full immobilization predicted by the continuous time random walk model, the motion of the tracer particle shows additional jiggling, for instance, due to thermal agitation of the environment. We here present and analyze in detail an extension of the continuous time random walk model. Superimposing the multiple trapping behavior with additive Gaussian noise of variable strength, we demonstrate that the resulting process exhibits a rich variety of apparent dynamic regimes. In particular, such noisy continuous time random walks may appear ergodic while the naked continuous time random walk exhibits weak ergodicity breaking. Detailed knowledge of this behavior will be useful for the truthful physical analysis of experimentally observed subdiffusion.

cond-mat.stat-mech↗

arXiv2013

Transport and the first passage time problem with application to cold atoms in optical traps

Measurements of spatial diffusion of cold atoms in optical lattices have revealed anomalous super-diffusion, which is controlled by the depth of the optical lattice. We use first passage time statistics to derive the diffusion front of the atoms. In particular, the distributions of areas swept under the first passage curve till its first arrival, and of areas under the Bessel excursion are shown to be powerful tools in the analysis of the atomic cloud. A rather general relation between first passage time statistics and diffusivity is discussed, showing that first passage time analysis is a useful tool in the calculation of transport coefficients. A brief introduction to the semi-classical description of Sisyphus cooling is provided which yields a rich phase diagram for the dynamics.

cond-mat.stat-mech↗

arXiv2013

Time averaged Einstein relation and fluctuating diffusivities for the Lévy walk

The Lévy walk model is a stochastic framework of enhanced diffusion with many applications in physics and biology. Here we investigate the time averaged mean squared displacement $\bar{δ^2}$ often used to analyze single particle tracking experiments. The ballistic phase of the motion is non-ergodic and we obtain analytical expressions for the fluctuations of $\bar{δ^2}$. For enhanced sub-ballistic diffusion we observe numerically apparent ergodicity breaking on long time scales. As observed by Akimoto \textit{Phys. Rev. Lett.} \textbf{108}, 164101 (2012) deviations of temporal averages $\bar{δ^2}$ from the ensemble average $< x^2 >$ depend on the initial preparation of the system, and here we quantify this discrepancy from normal diffusive behavior. Time averaged response to a bias is considered and the resultant generalized Einstein relations are discussed.

cond-mat.stat-mech↗

arXiv2012

Aging generates regular motions in weakly chaotic systems

Using intermittent maps with infinite invariant measures, we investigate the universality of time-averaged observables under aging conditions. According to Aaronson-Darling-Kac theorem, in non-aged dynamical systems with infinite invariant measures, the distribution of the normalized time averages of integrable functions converge to the Mittag-Leffler distribution. This well known theorem holds when the start of observations coincides with the start of the dynamical processes. Introducing a concept of the aging limit where the aging time $t_a$ and the total measurement time $t$ goes to infinity while the aging ratio $t_a/t$ is a constant, we obtain a novel distributional limit theorem of time-averaged observables integrable with respect to the infinite invariant density. Applying the theorem to the Lyapunov exponent in intermittent maps, we find that regular motions and a weakly chaotic behavior coexist in the aging limit. This mixed type of dynamics is controlled by the aging ratio and hence is very different from the usual scenario of regular and chaotic motions in Hamiltonian systems. The probability of finding regular motions in non-aged processes is zero, while in the aging regime it is finite and it increases when system ages.

nlin.CD↗

arXiv2012

Fluctuations of $1/f$ noise and the low frequency cutoff paradox

Recent experiments on blinking quantum dots and weak turbulence in liquid crystals reveal the fundamental connection between $1/f$ noise and power law intermittency. The non-stationarity of the process implies that the power spectrum is random -- a manifestation of weak ergodicity breaking. Here we obtain the universal distribution of the power spectrum, which can be used to identify intermittency as the source of the noise. We solve an outstanding paradox on the non integrability of $1/f$ noise and the violation of Parseval's theorem. We explain why there is no physical low frequency cutoff and therefore cannot be found in experiments.

cond-mat.stat-mech↗

arXiv2012

Weak subordination breaking for the quenched trap model

We map the problem of diffusion in the quenched trap model onto a new stochastic process: Brownian motion which is terminated at the coverage "time" ${\cal S}_α=\sum_{x=-\infty} ^\infty (n_x)^α$ 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" ${\cal S}_α$ is changed to laboratory time $t$ with a Lévy time transformation. Investigation of Brownian motion stopped at "time" ${\cal S}_α$ yields the diffusion front of the quenched trap model which is favorably compared with numerical simulations. In the zero temperature limit of $α\to 0$ we recover the renormalization group solution obtained by C. Monthus. Our theory surmounts critical slowing down which is found when $α\to 1$. Above the critical dimension two mapping the problem to a continuous time random walk becomes feasible though still not trivial.

cond-mat.stat-mech↗

arXiv2012

Ageing effects in single particle trajectory averages

We study time averages of single particle trajectories in scale free anomalous diffusion processes, in which the measurement starts at some time t_a>0 after initiation of the process at the time origin, t=0. Using ageing renewal theory we show that for such non-stationary processes a large class of observables are affected by a unique ageing function, which is independent of boundary conditions or the external forces. We quantify the weakly non-ergodic nature of this process in terms of the distribution of time averages and the ergodicity breaking parameter which both explicitly depend on the ageing time t_a. Consequences for the interpretation of single particle tracking data are discussed.

cond-mat.stat-mech↗

arXiv2012

Theory of fractional-Lévy kinetics for cold atoms diffusing in optical lattices

Recently, anomalous superdiffusion of ultra cold 87Rb atoms in an optical lattice has been observed along with a fat-tailed, Lévy type, spatial distribution. The anomalous exponents were found to depend on the depth of the optical potential. We find, within the framework of the semiclassical theory of Sisyphus cooling, three distinct phases of the dynamics, as the optical potential depth is lowered: normal diffusion; Lévy diffusion; and x ~ t^3/2 scaling, the latter related to Obukhov's model (1959) of turbulence. The process can be formulated as a Lévy walk, with strong correlations between the length and duration of the excursions. We derive a fractional diffusion equation describing the atomic cloud, and the corresponding anomalous diffusion coefficient.

cond-mat.stat-mech↗

arXiv2011

A fractional Feynman-Kac equation for weak ergodicity breaking

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 ($ ~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 $\bar{U} = \int_0^t U[x(τ)]dτ/ t$ 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 $t \rightarrow \infty$ 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 the fractional Feynman-Kac equation, we also study the dynamics leading to weak ergodicity breaking, namely the convergence of the fluctuations to their asymptotic values.

cond-mat.stat-mech↗

arXiv2011

Anomalous Infiltration

Infiltration of anomalously diffusing particles from one material to another through a biased interface is studied using continuous time random walk and Levy walk approaches. Subdiffusion in both systems may lead to a net drift from one material to another (e.g. > 0) even if particles eventually flow in the opposite direction (e.g. number of particles in x>0 approaches zero). A weaker paradox is found for a symmetric interface: a flow of particles is observed while the net drift is zero. For a subdiffusive sample coupled to a superdiffusive system we calculate the average occupation fractions and the scaling of the particles distribution. We find a net drift in this system, which is always directed to the superdiffusive material, while the particles flow to the material with smaller sub or superdiffusion exponent. We report the exponents of the first passage times distribution of Levy walks, which are needed for the calculation of anomalous infiltration

cond-mat.stat-mech↗

arXiv2011

In vivo anomalous diffusion and weak ergodicity breaking of lipid granules

Combining extensive single particle tracking microscopy data of endogenous lipid granules in living fission yeast cells with analytical results we show evidence for anomalous diffusion and weak ergodicity breaking. Namely we demonstrate that at short times the granules perform subdiffusion according to the laws of continuous time random walk theory. The associated violation of ergodicity leads to a characteristic turnover between two scaling regimes of the time averaged mean squared displacement. At longer times the granule motion is consistent with fractional Brownian motion.

cond-mat.stat-mech↗

arXiv2010

On distributions of functionals of anomalous diffusion paths

Functionals of Brownian motion have diverse applications in physics, mathematics, and other fields. The probability density function (PDF) of Brownian functionals satisfies the Feynman-Kac formula, which is a Schrodinger equation in imaginary time. In recent years there is a growing interest in particular functionals of non-Brownian motion, or anomalous diffusion, but no equation existed for their PDF. Here, we derive a fractional generalization of the Feynman-Kac equation for functionals of anomalous paths based on sub-diffusive continuous-time random walk. We also derive a backward equation and a generalization to Levy flights. Solutions are presented for a wide number of applications including the occupation time in half space and in an interval, the first passage time, the maximal displacement, and the hitting probability. We briefly discuss other fractional Schrodinger equations that recently appeared in the literature.

cond-mat.stat-mech↗

arXiv2010

Single particle tracking in systems showing anomalous diffusion: the role of weak ergodicity breaking

Anomalous diffusion has been widely observed by single particle tracking microscopy in complex systems such as biological cells. The resulting time series are usually evaluated in terms of time averages. Often anomalous diffusion is connected with non-ergodic behaviour. In such cases the time averages remain random variables and hence irreproducible. Here we present a detailed analysis of the time averaged mean squared displacement for systems governed by anomalous diffusion, considering both unconfined and restricted (corralled) motion. We discuss the behaviour of the time averaged mean squared displacement for two prominent stochastic processes, namely, continuous time random walks and fractional Brownian motion. We also study the distribution of the time averaged mean squared displacement around its ensemble mean, and show that this distribution preserves typical process characteristic even for short time series. Recently, velocity correlation functions were suggested to distinguish between these processes. We here present analytucal expressions for the velocity correlation functions. Knowledge of the results presented here are expected to be relevant for the correct interpretation of single particle trajectory data in complex systems.

cond-mat.stat-mech↗