arXiv subjects

Eli Barkai

Publications and source records attributed to Eli Barkai.

At least 127 records · Page 7Linked to original sources

arXiv2010

Separation of trajectories and its Relation to Entropy for Intermittent Systems with a Zero Lyapunov exponent

One dimensional intermittent maps with stretched exponential separation of nearby trajectories are considered. When time goes infinity the standard Lyapunov exponent is zero. We investigate the distribution of $λ_α= \sum_{i=0}^{t-1} \ln \left| M'(x_i) \right|/t^α$, where $α$ is determined by the nonlinearity of the map in the vicinity of marginally unstable fixed points. The mean of $λ_α$ is determined by the infinite invariant density. Using semi analytical arguments we calculate the infinite invariant density for the Pomeau-Manneville map, and with it obtain excellent agreement between numerical simulation and theory. We show that $α\left< λ_α\right>$ is equal to Krengel's entropy and to the complexity calculated by the Lempel-Ziv compression algorithm. This generalized Pesin's identity shows that $\left< λ_α\right>$ and Krengel's entropy are the natural generalizations of usual Lyapunov exponent and entropy for these systems.

nlin.CD↗

arXiv2010

Infinite invariant densities for anomalous diffusion in optical lattices and other logarithmic potentials

We solve the Fokker-Planck equation for Brownian motion in a logarithmic potential. When the diffusion constant is below a critical value the solution approaches a non-normalizable scaling state, reminiscent of an infinite invariant density. With this non-normalizable density we obtain the phase diagram of anomalous diffusion for this important process. We briefly discuss the consequence for a range of physical systems including atoms in optical lattices and charges in vicinity of long polyelectrolytes. Our work explains in what sense the infinite invariant density and not Boltzmann's equilibrium describes the long time limit of these systems.

cond-mat.stat-mech↗

arXiv2010

Foundation of Fractional Langevin Equation: Harmonization of a Many Body Problem

In this study we derive a single-particle equation of motion, from first-principles, starting out with a microscopic description of a tracer particle in a one-dimensional many-particle system with a general two-body interaction potential. Using a new harmonization technique, we show that the resulting dynamical equation belongs to the class of fractional Langevin equations, a stochastic framework which has been proposed in a large body of works as a means of describing anomalous dynamics. Our work sheds light on the fundamental assumptions of these phenomenological models.

cond-mat.stat-mech↗

arXiv2010

Paradoxes of Subdiffusive Infiltration in Disordered Systems

Infiltration of diffusing particles from one material to another where the diffusion mechanism is either normal or anomalous is a widely observed phenomena. When the diffusion is anomalous we find interesting behaviors: diffusion may lead to an averaged net drift from one material to another even if all particles eventually flow in the opposite direction, or may lead to a flow without drift. Starting with an underlying continuous time random walk model we solve diffusion equations describing this problem. Similar drift against flow is found in the quenched trap model. We argue that such a behavior is a general feature of diffusion in disordered systems.

cond-mat.stat-mech↗

arXiv2009

From non-Brownian Functionals to a Fractional Schrödinger Equation

We derive backward and forward fractional Schrödinger type of equations for the distribution of functionals of the path of a particle undergoing anomalous diffusion. Fractional substantial derivatives introduced by Friedrich and co-workers [PRL {\bf 96}, 230601 (2006)] provide the correct fractional framework for the problem at hand. In the limit of normal diffusion we recover the Feynman-Kac treatment of Brownian functionals. For applications, we calculate the distribution of occupation times in half space and show how statistics of anomalous functionals is related to weak ergodicity breaking.

cond-mat.stat-mech↗

arXiv2008

Ergodic Properties of Fractional Brownian-Langevin Motion

We investigate the time average mean square displacement $\overline{δ^2}(x(t))=\int_0^{t-Δ}[x(t^\prime+Δ)-x(t^\prime)]^2 dt^\prime/(t-Δ)$ for fractional Brownian and Langevin motion. Unlike the previously investigated continuous time random walk model $\overline{δ^2}$ converges to the ensemble average $ \sim t^{2 H}$ in the long measurement time limit. The convergence to ergodic behavior is however slow, and surprisingly the Hurst exponent $H=3/4$ marks the critical point of the speed of convergence. When $H<3/4$, the ergodicity breaking parameter ${EB} = {Var} (\overline{δ^2}) / < \overline{δ^2} >^2\sim k(H) \cdotΔ\cdot t^{-1}$, when $H=3/4$, ${EB} \sim (9/16)(\ln t) \cdotΔ\cdot t^{-1}$, and when $3/4<H <1, {EB} \sim k(H)Δ^{4-4H} t^{4H-4}$. In the ballistic limit $H \to 1$ ergodicity is broken and ${EB} \sim 2$. The critical point $H=3/4$ is marked by the divergence of the coefficient $k(H)$. Fractional Brownian motion as a model for recent experiments of sub-diffusion of mRNA in the cell is briefly discussed and comparison with the continuous time random walk model is made.

physics.data-an↗

arXiv2008

Pesin-Type Identity for Weak Chaos

Pesin's identity provides a profound connection between entropy $h_{KS}$ (statistical mechanics) and the Lyapunov exponent $λ$ (chaos theory). It is well known that many systems exhibit sub-exponential separation of nearby trajectories and then $λ=0$. In many cases such systems are non-ergodic and do not obey usual statistical mechanics. Here we investigate the non-ergodic phase of the Pomeau-Manneville map where separation of nearby trajectories follows $δx_t= δx_0 e^{λ_α t^α}$ with $0<α<1$. The limit distribution of $λ_α$ is the inverse L{é}vy function. The average $< λ_α >$ is related to the infinite invariant density, and most importantly to entropy. Our work gives a generalized Pesin's identity valid for systems with an infinite invariant density.

cond-mat.stat-mech↗

arXiv2008

Weakly non-ergodic Statistical Physics

We find a general formula for the distribution of time averaged observables for weakly non-ergodic systems. Such type of ergodicity breaking is known to describe certain systems which exhibit anomalous fluctuations, e.g. blinking quantum dots and the sub-diffusive continuous time random walk model. When the fluctuations become normal we recover usual ergodic statistical mechanics. Examples of a particle undergoing fractional dynamics in a binding force field are worked out in detail. We briefly discuss possible physical applications in single particle experiments.

cond-mat.stat-mech↗

arXiv2007

Single Molecule Photon Statistics from a Sequence of Laser Pulses

There are many ways of calculating photon statistics in quantum optics in general and single molecule spectroscopy in particular such as the generating function method, the quantum jump approach or time ordering methods. In this paper starting with the optical Bloch equation, within the paths interpretation of Zoller, Marte and Walls we obtain the photon statistics from a sequence of laser pulses expressed by means of quantum trajectories. We find general expressions for Pn(t) - the probability of emitting n photons up to time t, discuss several consequences and show that the interpretation of the quantum trajectories (i) emphasizes contribution to the photon statistics of the coherence paths accumulated in the delay interval between the pulses and (ii) allows simple classification of the terms negligible under certain physical constraints . Applying this method to the concrete example of two square laser pulses we find the probabilities of emitting 0,1 and 2 photons, examine several limiting cases and investigate the upper and lower bounds of P0(t), P1(t) and P2(t) for a sequence of two strong pulses in the limit of long measurement times. Implication to single molecule non-linear spectroscopy and theory of pairs of photons on demand are discussed briefly.

physics.optics↗

arXiv2007

Distribution of Time-Averaged Observables for Weak Ergodicity Breaking

We find a general formula for the distribution of time-averaged observables for systems modeled according to the sub-diffusive continuous time random walk. For Gaussian random walks coupled to a thermal bath we recover ergodicity and Boltzmann's statistics, while for the anomalous subdiffusive case a weakly non-ergodic statistical mechanical framework is constructed, which is based on Lévy's generalized central limit theorem. As an example we calculate the distribution of $\bar{X}$: the time average of the position of the particle, for unbiased and uniformly biased particles, and show that $\bar{X}$ exhibits large fluctuations compared with the ensemble average $ $.

cond-mat.stat-mech↗

arXiv2006

Ergodicity Breaking in a Deterministic Dynamical System

The concept of weak ergodicity breaking is defined and studied in the context of deterministic dynamics. We show that weak ergodicity breaking describes a weakly chaotic dynamical system: a nonlinear map which generates subdiffusion deterministically. In the non-ergodic phase non-trivial distribution of the fraction of occupation times is obtained. The visitation fraction remains uniform even in the non-ergodic phase. In this sense the non-ergodicity is quantified, leading to a statistical mechanical description of the system even though it is not ergodic.

nlin.CD↗

arXiv2006

Strong Correlations Between Fluctuations and Response in Aging Transport

Once the problem of ensemble averaging is removed, correlations between the response of a single molecule to an external driving field $F$, with the history of fluctuations of the particle, become detectable. Exact analytical theory for the continuous time random walk and numerical simulations for the quenched trap model give the behaviors of the correlation between fluctuations of the displacement in the aging period $(0,t_a)$, and the response to bias switched on at time $t_a$. In particular in the dynamical phase where the models exhibit aging we find finite correlations even in the asymptotic limit $t_a \to \infty$, while in the non-aging phase the correlations are zero in the same limit. Linear response theory gives a simple relation between these correlations and the fractional diffusion coefficient.

cond-mat.stat-mech↗

arXiv2006

Power Law Blinking Quantum Dots: Stochastic and Physical Models

We quantify nonergodic and aging behaviors of nanocrystals (or quantum dots) based on stochastic model. Ergodicity breaking is characterized based on time average intensity and time average correlation function, which remain random even in the limit of long measurement time. We argue that certain aspects of nonergodicity can be explained based on a modification of Onsager's diffusion model of an ion pair escaping neutralization. We explain how diffusion models generate nonergodic behavior, namely a simple mechanism is responsible for the breakdown of the standard assumption of statistical mechanics. Data analysis shows that distributions of on and off intervals in the nanocrystal blinking are almost identical, $ψ_{\pm}(τ)\propto A_{\pm}τ^{-(1+α_{\pm})}$ with $A_{+}\approx A_{-}$ and $α_{+}\approxα_{-}=α$ and $α\approx0.8$. The latter exponent indicates that a simple diffusion model with $α=0.5$ neglecting the electron-hole Coulomb interaction and/or tunneling, is not sufficient.

cond-mat.stat-mech↗

arXiv2006

Residence Time Statistics for Normal and Fractional Diffusion in a Force Field

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.

cond-mat.stat-mech↗

arXiv2005

Theory of Single Photon Control from a Two Level System Source

Generation of a single photon or a pair of photons from a single emitter is important for quantum information applications. Using the generating function formalism we investigate the theory of a few photons on demand for the square laser pulse and the rapid adiabatic following method. Exact theory and numerical solutions are used to design control fields for a two level emitter, which yield an optimal single or two photon source, under the constrains of finite laser field strength and finite interaction time. Comparison to experiments of Brunel et al, shows that the experiments were made close to optimal conditions.

quant-ph↗

arXiv2005

A Random Walk to a Non-Ergodic Equilibrium Concept

Random walk models, such as the trap model, continuous time random walks, and comb models exhibit weak ergodicity breaking, when the average waiting time is infinite. The open question is: what statistical mechanical theory replaces the canonical Boltzmann-Gibbs theory for such systems? In this manuscript a non-ergodic equilibrium concept is investigated, for a continuous time random walk model in a potential field. In particular we show that in the non-ergodic phase the distribution of the occupation time of the particle on a given lattice point, approaches U or W shaped distributions related to the arcsin law. We show that when conditions of detailed balance are applied, these distributions depend on the partition function of the problem, thus establishing a relation between the non-ergodic dynamics and canonical statistical mechanics. In the ergodic phase the distribution function of the occupation times approaches a delta function centered on the value predicted based on standard Boltzmann-Gibbs statistics. Relation of our work with single molecule experiments is briefly discussed.

cond-mat.stat-mech↗

arXiv2005

Nonergodisity of a time series obeying Lévy statistics

Time-averaged autocorrelation functions of a dichotomous random process switching between 1 and 0 and governed by wide power law sojourn time distribution are studied. Such a process, called a Lévy walk, describes dynamical behaviors of many physical systems, fluorescence intermittency of semiconductor nanocrystals under continuous laser illumination being one example. When the mean sojourn time diverges the process is non-ergodic. In that case, the time average autocorrelation function is not equal to the ensemble averaged autocorrelation function, instead it remains random even in the limit of long measurement time. Several approximations for the distribution of this random autocorrelation function are obtained for different parameter ranges, and favorably compared to Monte Carlo simulations. Nonergodicity of the power spectrum of the process is briefly discussed, and a nonstationary Wiener-Khintchine theorem, relating the correlation functions and the power spectrum is presented. The considered situation is in full contrast to the usual assumptions of ergodicity and stationarity.

cond-mat.stat-mech↗

arXiv2005

Single Molecule Chemical Reaction: Kramers Approach Revisited

Single molecule chemical reactions yield new insight into fluctuation phenomena which are obscured in measurement of ensemble of molecules. Kramers escape problem is investigated here in a framework suitable for single molecule reactions. In particular we obtain distributions of escape times in simple limiting cases, rather than their mean, and investigate their sensitivity on initial conditions. Rich physical behaviors are observed: sub-Poissonian statistics when the dynamics is only slightly deviating from Newtonian, super-Poissonian behavior when diffusion is dominating, and Poissonian behavior when Kramers original conditions hold. By varying initial conditions escape time distributions can follow a (usual) exponential or a $τ^{-3/2}$ decay, due to regular diffusion. We briefly address experimental results which yield the $τ^{-3/2}$ behavior (with cutoffs) and propose that this behavior is universal.

cond-mat.stat-mech↗