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Eli Barkai

Publications and source records attributed to Eli Barkai.

At least 91 records · Page 5Linked to original sources
arXiv2017

Using Hilbert transform and classical chains to simulate quantum walks

We propose a simulation strategy which uses a classical device of linearly coupled chain of springs to simulate quantum dynamics, in particular the quantum walks. Through this strategy, we obtain the quantum wave function from classical evolution. Specially, this goal is achieved with the classical momenta of the particles on the chain and their Hilbert transform, from which we construct the many-body momentum and Hilbert transformed momentum pair correlation functions yielding the real and imaginary parts of the wave function, respectively. With such wave function, we show that the classical chain's energy and heat spreading densities can be related to the wave function's modulus square. This relation indicates a concept of "phonon random walks", and thus it provides a new perspective to understand ballistic heat transport. The results here may give a definite answer to Feynman's idea of using a classical device to simulate quantum physics.

cond-mat.stat-mech↗
arXiv2017

Scale invariant Green-Kubo relation for time averaged diffusivity

In recent years it was shown both theoretically and experimentally that in certain systems exhibiting anomalous diffusion the time and ensemble average mean squared displacement are remarkably different. The ensemble average diffusivity is obtained from a scaling Green-Kubo relation, which connects the scale invariant non-stationary velocity correlation function with the transport coefficient. Here we obtain the relation between time averaged diffusivity, usually recorded in single particle tracking experiments, and the underlying scale invariant velocity correlation function. The time averaged mean squared displacement is given by $\overline{δ^2} \sim 2 D_νt^βΔ^{ν-β}$ where $t$ is the total measurement time and $Δ$ the lag time. Here $ν>1$ is the anomalous diffusion exponent obtained from ensemble averaged measurements $\langle x^2 \rangle \sim t^ν$ while $β\ge -1$ marks the growth or decline of the kinetic energy $\langle v^2 \rangle \sim t^β$. Thus we establish a connection between exponents which can be read off the asymptotic properties of the velocity correlation function and similarly for the transport constant $D_ν$. We demonstrate our results with non-stationary scale invariant stochastic and deterministic models, thereby highlighting that systems with equivalent behavior in the ensemble average can differ strongly in their time average. This is the case, for example, if averaged kinetic energy is finite, i.e. $β=0$, where $\langle \overline{δ^2}\rangle \neq \langle x^2\rangle$.

cond-mat.stat-mech↗
arXiv2017

$1/f^β$ noise for scale-invariant processes: How long you wait matters

We study the power spectrum which is estimated from a nonstationary signal. In particular we examine the case when the signal is observed in a measurement time window $[t_w,t_w+t_m]$, namely the observation started after a waiting time $t_w$, and $t_m$ is the measurement duration. We introduce a generalized aging Wiener-Khinchin theorem which relates between the spectrum and the time- and ensemble-averaged correlation function for arbitrary $t_m$ and $t_w$. Furthermore we provide a general relation between the non-analytical behavior of the scale-invariant correlation function and the aging $1/f^β$ noise. We illustrate our general results with two-state renewal models with sojourn times' distributions having a broad tail.

cond-mat.stat-mech↗
arXiv2017

Lévy flights versus Lévy walks in bounded domains

Lévy flights and Lévy walks serve as two paradigms of random walks resembling common features but also bearing fundamental differences. One of the main dissimilarities are discontinuity versus continuity of their trajectories and infinite versus finite propagation velocity. In consequence, well developed theory of Lévy flights is associated with their pathological physical properties, which in turn are resolved by the concept of Lévy walks. Here, we explore Lévy flights and Lévy walks models on bounded domains examining their differences and analogies. We investigate analytically and numerically whether and under which conditions both approaches yield similar results in terms of selected statistical observables characterizing the motion: the survival probability, mean first passage time and stationary PDFs. It is demonstrated that similarity of models is affected by the type of boundary conditions and value of the stability index defining asymptotics of the jump length distribution.

cond-mat.stat-mech↗
arXiv2017

Large-deviations for spatial diffusion of cold atoms

Large-deviations theory deals with tails of probability distributions and the rare events of random processes, for example spreading packets of particles. Mathematically, it concerns the exponential fall-of of the density of thin-tailed systems. Here we investigate the spatial density $P_t(x)$ of laser cooled atoms, where at intermediate length scales the shape is fat-tailed. We focus on the rare events beyond this range, which dominate important statistical properties of the system. Through a novel friction mechanism induced by the laser fields, the density is explored with the recently proposed nonnormalized infinite-covariant density approach. The small and large fluctuations give rise to a bi-fractal nature of the spreading packet.

cond-mat.stat-mech↗
arXiv2017

Limit theorems for Lévy walks in $d$ dimensions: rare and bulk fluctuations

We consider super-diffusive Lévy walks in $d \geqslant 2$ dimensions when the duration of a single step, i.e., a ballistic motion performed by a walker, is governed by a power-law tailed distribution of infinite variance and finite mean. We demonstrate that the probability density function (PDF) of the coordinate of the random walker has two different scaling limits at large times. One limit describes the bulk of the PDF. It is the $d-$dimensional generalization of the one-dimensional Lévy distribution and is the counterpart of central limit theorem (CLT) for random walks with finite dispersion. In contrast with the one-dimensional Lévy distribution and the CLT this distribution does not have universal shape. The PDF reflects anisotropy of the single-step statistics however large the time is. The other scaling limit, the so-called 'infinite density', describes the tail of the PDF which determines second (dispersion) and higher moments of the PDF. This limit repeats the angular structure of PDF of velocity in one step. Typical realization of the walk consists of anomalous diffusive motion (described by anisotropic $d-$dimensional Lévy distribution) intermitted by long ballistic flights (described by infinite density). The long flights are rare but due to them the coordinate increases so much that their contribution determines the dispersion. We illustrate the concept by considering two types of Lévy walks, with isotropic and anisotropic distributions of velocities. Furthermore, we show that for isotropic but otherwise arbitrary velocity distribution the $d-$dimensional process can be reduced to one-dimensional Lévy walk.

cond-mat.stat-mech↗
arXiv2016

Quantum walks: the first detected passage time problem

Even after decades of research the problem of first passage time statistics for quantum dynamics remains a challenging topic of fundamental and practical importance. Using a projective measurement approach, with a sampling time $τ$, we obtain the statistics of first detection events for quantum dynamics on a lattice, with the detector located at the origin. A quantum renewal equation for a first detection wave function, in terms of which the first detection probability can be calculated, is derived. This formula gives the relation between first detection statistics and the solution of the corresponding Schrödinger equation in the absence of measurement. We demonstrate our results with tight binding quantum walk models. We examine a closed system, i.e. a ring, and reveal the intricate influence of the sampling time $τ$ on the statistics of detection, discussing the quantum Zeno effect, half dark states, revivals and optimal detection. The initial condition modifies the statistics of a quantum walk on a finite ring in surprising ways. In some cases the average detection time is independent of the sampling time while in others the average exhibits multiple divergences as the sampling time is modified. For an unbounded one dimensional quantum walk the probability of first detection decays like $(\mbox{time})^{(-3)}$ with superimposed oscillations, with exceptional behavior when the sampling period $τ$ times the tunnelling rate $γ$ is a multiple of $π/2$. The amplitude of the power law decay is suppressed as $τ\to 0$ due to the Zeno effect. Our work presented here, is an extended version of Friedman et al. arXiv:1603.02046 [cond-mat.stat-mech], and it predicts rich physical behaviors compared with classical Brownian motion, for which the first passage probability density decays monotonically like $(\mbox{time})^{-3/2}$, as elucidated by Schrödinger in $1915$.

cond-mat.stat-mech↗
arXiv2016

Universal Fluctuations of Single-Particle Diffusivity in Quenched Environment

Local diffusion coefficients in disordered materials such as living cells are highly heterogeneous. Quenched disorder is utilized substantially to study such complex systems, whereas its analytical treatment is difficult to handle. We consider finite systems with quenched disorder in order to investigate the effects of sample disorder fluctuations and confinement on single-particle diffusivity. While the system is ergodic in a single disorder realization, the time-averaged mean squared displacement depends on the disorder, i.e., the system is ergodic but non-self-averaging. We find that the inverse Levy distribution is a universal distribution for diffusivity in the sense that it can be applied for arbitrary dimensions. Quantifying the degree of the non-self-averaging effect, we show that fluctuations of single-particle diffusivity far exceed the corresponding annealed theory and also find confinement effects. The relevance for experimental situations is also discussed.

cond-mat.stat-mech↗
arXiv2016

Heavy-tailed phase-space distributions beyond Boltzmann-Gibbs and equipartition: Statistics of confined cold atoms

The Boltzmann-Gibbs density, a central result of equilibrium statistical mechanics, relates the energy of a system in contact with a thermal bath to its equilibrium statistics. This relation is lost for non-thermal systems such as cold atoms in optical lattices, where the heat bath is replaced by the laser beams of the lattice. We investigate in detail the stationary phase-space probability for Sisyphus cooling under harmonic confinement. In particular, we elucidate whether the total energy of the system still describes its stationary state statistics. We find that this is true for the center part of the phase-space density for deep lattices, where the Boltzmann-Gibbs density provides an approximate description. The relation between energy and statistics also persists for strong confinement and in the limit of high energies, where the system becomes underdamped. However, the phase-space density now exhibits heavy power-law tails. In all three cases we find expressions for the leading order phase-space density and corrections which break the equivalence of probability and energy and violate energy equipartition. The non-equilibrium nature of the steady state is confounded by explicit violations of detailed balance. We complement these analytical results with numerical simulations to map out the intricate structure of the phase-space density.

cond-mat.stat-mech↗
arXiv2016

Tempered Fractional Feynman-Kac Equation

Functionals of Brownian/non-Brownian motions have diverse applications and attracted a lot of interest of scientists. This paper focuses on deriving the forward and backward fractional Feynman-Kac equations describing the distribution of the functionals of the space and time tempered anomalous diffusion, belonging to the continuous time random walk class. Several examples of the functionals are explicitly treated, including the occupation time in half-space, the first passage time, the maximal displacement, the fluctuations of the occupation fraction, and the fluctuations of the time-averaged position.

physics.data-an↗
arXiv2015

Fluctuations around equilibrium laws in ergodic continuous-time random walks

We study occupation time statistics in ergodic continuous-time random walks. Under thermal detailed balance conditions, the average occupation time is given by the Boltzmann-Gibbs canonical law. But close to the non-ergodic phase, the finite-time fluctuations around this mean are large and nontrivial. They exhibit dual time scaling and distribution laws: the infinite density of large fluctuations complements the Lévy-stable density of bulk fluctuations. Neither of the two should be interpreted as a stand-alone limiting law, as each has its own deficiency: the infinite density has an infinite norm (despite particle conservation), while the stable distribution has an infinite variance (although occupation times are bounded). These unphysical divergences are remedied by consistent use and interpretation of both formulas. Interestingly, while the system's canonical equilibrium laws naturally determine the mean occupation time of the ergodic motion, they also control the infinite and Lévy-stable densities of fluctuations. The duality of stable and infinite densities is in fact ubiquitous for these dynamics, as it concerns the time averages of general physical observables.

cond-mat.stat-mech↗
arXiv2015

Fluctuations of ring polymers

We present an exact solution for the distribution of sample averaged monomer to monomer distance of ring polymers. For non-interacting and weakly-interacting models these distributions correspond to the distribution of the area under the reflected Bessel bridge and the Bessel excursion respectively, and are shown to be identical in dimension d greater or equal 2. A symmetry of the problem reveals that dimension d and 4 minus d are equivalent, thus the celebrated Airy distribution describing the areal distribution of the one dimensional Brownian excursion describes also a polymer in three dimensions. For a self-avoiding polymer in dimension d we find numerically that the fluctuations of the scaled averaged distance are nearly identical in dimensions 2 and 3, and are well described to a first approximation by the non-interacting excursion model in dimension 5.

cond-mat.soft↗
arXiv2014

Fractional Edgeworth Expansion: Corrections to the Gaussian-Lévy Central Limit Theorem

In this article we generalize the classical Edgeworth expansion for the probability density function (PDF) of sums of a finite number of symmetric independent identically distributed random variables with a finite variance to sums of variables with an infinite variance which converge by the generalized central limit theorem to a Lévy $α$-stable density function. Our correction may be written by means of a series of fractional derivatives of the Lévy and the conjugate Lévy PDFs. This series expansion is general and applies also to the Gaussian regime. To describe the terms in the series expansion, we introduce a new family of special functions and briefly discuss their properties. We implement our generalization to the distribution of the momentum for atoms undergoing Sisyphus cooling, and show the improvement of our leading order approximation compared to previous approximations. In vicinity of the transition between Lévy and Gauss behaviors, convergence to asymptotic results slows down.

cond-mat.stat-mech↗
arXiv2014

Deviations from Boltzmann-Gibbs equilibrium in confined optical lattices

Cold atoms in dissipative optical lattices have long been known to exhibit anomalous kinetics due to an effective nonlinear friction force. Here we show that confining the spatial motion of the atoms will lead to an anomalous non-Boltzmann-Gibbs equilibrium state, with a power law tail at large energies. Only in the limit of deep optical lattices, do we regain the Boltzmann-Gibbs state. For strong confinement relative to the damping, we find an explicit expression for the equilibrium phase-space distribution, which generally differs from the canonical Boltzmann-Gibbs state at all energies. Both in the low and high energy limits, the equilibrium distribution is a function of the system's Hamiltonian. At intermediate energies, however, the distribution is not a function of energy only and equipartition is violated.

cond-mat.stat-mech↗
arXiv2014

Infinite densities for Lévy walks

Motion of particles in many systems exhibits a mixture between periods of random diffusive like events and ballistic like motion. In many cases, such systems exhibit strong anomalous diffusion, where low order moments $< |x(t)|^q >$ with $q$ below a critical value $q_c$ exhibit diffusive scaling while for $q>q_c$ a ballistic scaling emerges. The mixed dynamics constitutes a theoretical challenge since it does not fall into a unique category of motion, e.g., the known diffusion equations and central limit theorems fail to describe both aspects. In this paper we resolve this problem by resorting to the concept of infinite density. Using the widely applicable Lévy walk model, we find a general expression for the corresponding non-normalized density which is fully determined by the particles velocity distribution, the anomalous diffusion exponent $α$ and the diffusion coefficient $K_α$. We explain how infinite densities play a central role in the description of dynamics of a large class of physical processes and discuss how they can be evaluated from experimental or numerical data.

cond-mat.stat-mech↗
arXiv2014

1/f noise for intermittent quantum dots exhibits non-stationarity and critical exponents

The power spectrum of quantum dot fluorescence exhibits $1/f^β$ noise, related to the intermittency of these nanosystems. As in other systems exhibiting $1/f$ noise, this power spectrum is not integrable at low frequencies, which appears to imply infinite total power. We report measurements of individual quantum dots that address this long-standing paradox. We find that the level of $1/f^β$ noise decays with the observation time. The change of the spectrum with time places a bound on the total power. These observations are in stark contrast with most measurements of noise in macroscopic systems which do not exhibit any evidence for non-stationarity. We show that the traditional description of the power spectrum with a single exponent $β$ is incomplete and three additional critical exponents characterize the dependence on experimental time.

cond-mat.stat-mech↗
arXiv2014

Localization and universal fluctuations in ultraslow diffusion processes

We study ultraslow diffusion processes with logarithmic mean squared displacement (MSD) $\langle x^2(t)\rangle\simeq\log^γt$. Comparison of annealed continuous time random walks (CTRWs) with logarithmic waiting time distribution $ψ(τ)\simeq1/(τ\log^{1+γ}τ)$ and Sinai diffusion in quenched random landscapes shows striking similarities, despite their very different physical nature. In particular, they exhibit a weakly non-ergodic disparity of the time and ensemble averaged MSDs. Remarkably, for the CTRW we observe that the fluctuations of time averages become universal with an exponential suppression of mobile trajectories. We discuss the fundamental connection between the Golosov localization effect and non-ergodicity in the sense of the disparity between ensemble and time averaged MSD.

cond-mat.stat-mech↗