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Eli Barkai

Publications and source records attributed to Eli Barkai.

At least 73 records · Page 4Linked to original sources
arXiv2020

Infinite invariant density in a semi-Markov process with continuous state variables

We report on a fundamental role of a non-normalized formal steady state, i.e., an infinite invariant density, in a semi-Markov process where the state is determined by the inter-event time of successive renewals. The state describes certain observables found in models of anomalous diffusion, e.g., the velocity in the generalized Lévy walk model and the energy of a particle in the trap model. In our model, the inter-event-time distribution follows a fat-tailed distribution, which makes the state value more likely to be zero because long inter-event times imply small state values. We find two scaling laws describing the density for the state value, which accumulates in the vicinity of zero in the long-time limit. These laws provide universal behaviors in the accumulation process and give the exact expression of the infinite invariant density. Moreover, we provide two distributional limit theorems for time-averaged observables in these non-stationary processes. We show that the infinite invariant density plays an important role in determining the distribution of time averages.

cond-mat.stat-mech↗
arXiv2020

Regularized Boltzmann-Gibbs statistics for a Brownian particle in a non-confining field

We consider an overdamped Brownian particle subject to an asymptotically flat potential with a trap of depth $U_0$ around the origin. When the temperature is small compared to the trap depth ($ξ=k_B T/U_0 \ll 1$), there exists a range of timescales over which physical observables remain practically constant. This range can be very long, of the order of the Arrhenius factor ${\rm e}^{1/ξ}$. For these quasi-equilibrium states, the usual Boltzmann-Gibbs recipe does not work, since the partition function is divergent due to the flatness of the potential at long distances. However, we show that the standard Boltzmann-Gibbs (BG) statistical framework and thermodynamic relations can still be applied through proper regularization. This can be a valuable tool for the analysis of metastability in the non-confining potential fields that characterize a vast number of systems.

cond-mat.stat-mech↗
arXiv2020

Infinite ergodic theory meets Boltzmann statistics

We investigate the overdamped stochastic dynamics of a particle in an asymptotically flat external potential field, in contact with a thermal bath. For an infinite system size, the particles may escape the force field and diffuse freely at large length scales. The partition function diverges and hence the standard canonical ensemble fails. This is replaced with tools stemming from infinite ergodic theory. Boltzmann-Gibbs statistics, even though not normalized, still describes integrable observables, like energy and occupation times. The Boltzmann infinite density is derived heuristically using an entropy maximization principle, as well as via a first-principles calculation using an eigenfunction expansion in the continuum of low-energy states. A generalized virial theorem is derived, showing how the virial coefficient describes the delay in the diffusive spreading of the particles, found at large distances. When the process is non-recurrent, e.g. diffusion in three dimensions with a Coulomb-like potential, we use weighted time averages to restore basic canonical relations between time and ensemble averages.

cond-mat.stat-mech↗
arXiv2020

Rare events in generalized Lévy Walks and the Big Jump principle

The prediction and control of rare events is an important task in disciplines that range from physics and biology, to economics and social science. The Big Jump principle deals with a peculiar aspect of the mechanism that drives rare events. According to the principle, in heavy-tailed processes a rare huge fluctuation is caused by a single event and not by the usual coherent accumulation of small deviations. We consider generalized Lévy walks, a class of stochastic processes with power law distributed step durations, which model complex microscopic dynamics in the single stretch. We derive the bulk of the probability distribution and using the big jump principle, the exact form of the tails that describes rare events. We show that the tails of the distribution present non-universal and non-analytic behaviors, which depend crucially on the dynamics of the single step. The big jump estimate also provides a physical explanation of the processes driving the rare events, opening new possibilities for their correct prediction.

cond-mat.stat-mech↗
arXiv2020

Packets of Diffusing Particles Exhibit Universal Exponential Tails

Brownian motion is a Gaussian process described by the central limit theorem. However, exponential decays of the positional probability density function $P(X,t)$ of packets of spreading random walkers, were observed in numerous situations that include glasses, live cells and bacteria suspensions. We show that such exponential behavior is generally valid in a large class of problems of transport in random media. By extending the Large Deviations approach for a continuous time random walk we uncover a general universal behavior for the decay of the density. It is found that fluctuations in the number of steps of the random walker, performed at finite time, lead to exponential decay (with logarithmic corrections) of $P(X,t)$. This universal behavior holds also for short times, a fact that makes experimental observations readily achievable.

cond-mat.stat-mech↗
arXiv2019

Transport in disordered systems: the single big jump approach

In a growing number of strongly disordered and dense systems, the dynamics of a particle pulled by an external force field exhibits super-diffusion. In the context of glass forming systems, super cooled glasses and contamination spreading in porous medium it was suggested to model this behavior with a biased continuous time random walk. Here we analyze the plume of particles far lagging behind the mean, with the single big jump principle. Revealing the mechanism of the anomaly, we show how a single trapping time, the largest one, is responsible for the rare fluctuations in the system. These non typical fluctuations still control the behavior of the mean square displacement, which is the most basic quantifier of the dynamics in many experimental setups. We show how the initial conditions, describing either stationary state or non-equilibrium case, persist for ever in the sense that the rare fluctuations are sensitive to the initial preparation. To describe the fluctuations of the largest trapping time, we modify Fréchet's law from extreme value statistics, taking into consideration the fact that the large fluctuations are very different from those observed for independent and identically distributed random variables.

cond-mat.stat-mech↗
arXiv2019

Uncertainty and symmetry bounds for the quantum total detection probability

We investigate a generic discrete quantum system prepared in state $|ψ_\text{in}\rangle$, under repeated detection attempts aimed to find the particle in state $|d\rangle$, for example a quantum walker on a finite graph searching for a node. For the corresponding classical random walk, the total detection probability $P_\text{det}$ is unity. Due to destructive interference, one may find initial states $|ψ_\text{in}\rangle$ with $P_\text{det}<1$. We first obtain an uncertainty relation which yields insight on this deviation from classical behavior, showing the relation between $P_\text{det}$ and energy fluctuations: $ ΔP \,\mathrm{Var}[\hat{H}]_d \ge | \langle d| [\hat{H}, \hat{D}] | ψ_\text{in} \rangle |^2$ where $ΔP = P_\text{det} - |\langleψ_\text{in}|d\rangle |^2$, and $\hat{D} = |d\rangle\langle d|$ is the measurement projector. Secondly, exploiting symmetry we show that $P_\text{det}\le 1/ν$ where the integer $ν$ is the number of states equivalent to the initial state. These bounds are compared with the exact solution for small systems, obtained from an analysis of the dark and bright subspaces, showing the usefulness of the approach. The upper bounds works well even in large systems, and we show how to tighten the lower bound in this case.

quant-ph↗
arXiv2019

Infinite horizon billiards: Transport at the border between Gauss and Lévy universality classes

We consider transport in two billiard models, the infinite horizon Lorentz gas and the stadium channel, presenting analytical results for the spreading packet of particles. We first obtain the cumulative distribution function of traveling times between collisions, which exhibits non-analytical behavior. Using a renewal assumption and the Lévy walk model, we obtain the particles' probability density. For the Lorentz gas, it shows a distinguished difference when compared with the known Gaussian propagator, as the latter is valid only for extremely long times. In particular, we show plumes of particles spreading along the infinite corridors, creating power-law tails of the density. We demonstrate the slow convergence rate via summation of independent identically distributed random variables on the border between Lévy and Gauss laws. The renewal assumption works well for the Lorentz gas with intermediately sized scattering centers, but fails for the stadium channel due to strong temporal correlations. Our analytical results are supported with numerical samplings.

cond-mat.stat-mech↗
arXiv2019

Large fluctuations of the first detected quantum return time

How long does it take a quantum particle to return to its origin? As shown previously under repeated projective measurements aimed to detect the return, the closed cycle yields a geometrical phase which shows that the average first detected return time is quantized. For critical sampling times or when parameters of the Hamiltonian are tuned this winding number is modified. These discontinuous transitions exhibit gigantic fluctuations of the return time. While the general formalism of this problem was studied at length, the magnitude of the fluctuations, which is quantitatively essential, remains poorly characterized. Here, we derive explicit expressions for the variance of the return time, for quantum walks in finite Hilbert space. A classification scheme of the diverging variance is presented, for four different physical effects: the Zeno regime, when the overlap of an energy eigenstate and the detected state is small and when two or three phases of the problem merge. These scenarios present distinct physical effects which can be analyzed with the fluctuations of return times investigated here, leading to a topology-dependent time-energy uncertainty principle.

cond-mat.stat-mech↗
arXiv2019

Quantum total detection probability from repeated measurements II. Exploiting symmetry

A quantum walker on a graph, prepared in the state $| ψ_{\rm in} \rangle$, e.g. initially localized at node $r_{\rm in}$, is repeatedly probed, with fixed frequency $1/τ$, to test its presence at some target node $r_{\rm d}$ until the first successful detection. This is a quantum version of the first-passage problem. We investigate the total detection probability $P_{\rm det}$, i.e. the probability to eventually detect the particle after an arbitrary number of detection attempts. It is demonstrated that this total detection probability is less than unity in symmetric systems, where it is possible to find initial states which are shielded from the detector by destructive interference, so-called dark states. The identification of physically equivalent initial states yields an upper bound for $P_{\rm det}$ in terms of the reciprocal of the number $ν$ of physically equivalent states. The relevant subgroup of the system's symmetry operations is found to be the stabilizer of the detection state. Using this, we prove that all bright, i.e. surely detectable, states are symmetric with respect to the stabilizer. This implies that $P_{\rm det}$ can be obtained from a diagonalization of the "symmetrized" Hamiltonian, instead of having to find all eigenstates of the Hamiltonian.

quant-ph↗
arXiv2019

The Single Big Jump Principle in Physical Modelling

The big jump principle is a well established mathematical result for sums of independent and identically distributed random variables extracted from a fat tailed distribution. It states that the tail of the distribution of the sum is the same as the distribution of the largest summand. In practice, it means that when in a stochastic process the relevant quantity is a sum of variables, the mechanism leading to rare events is peculiar: instead of being caused by a set of many small deviations all in the same direction, one jump, the biggest of the lot, provides the main contribution to the rare large fluctuation. We reformulate and elevate the big jump principle beyond its current status to allow it to deal with correlations, finite cutoffs, continuous paths, memory and quenched disorder. Doing so we are able to predict rare events using the extended big jump principle in Lévy walks, in a model of laser cooling, in a scattering process on a heterogeneous structure and in a class of Lévy walks with memory. We argue that the generalized big jump principle can serve as an excellent guideline for reliable estimates of risk and probabilities of rare events in many complex processes featuring heavy tailed distributions, ranging from contamination spreading to active transport in the cell.

cond-mat.stat-mech↗
arXiv2019

Running Measurement Protocol for the Quantum First-detection problem

The problem of the detection statistics of a quantum walker has received increasing interest, connected as it is to the problem of quantum search. We investigate the effect of employing a moving detector, using a projective measurement approach with fixed sampling time $τ$, with the detector moving right before every detection attempt. For a tight-binding quantum walk on the line, the moving detector allows one to target a specific range of group velocities of the walker, qualitatively modifying the behavior of the quantum first-detection probabilities. We map the problem to that of a stationary detector with a modified unitary evolution operator and use established methods for the solution of that problem to study the first-detection statistics for a moving detector on a finite ring and on an infinite 1D lattice. On the line, the system exhibits a dynamical phase transition at a critical value of $τ$, from a state where detection decreases exponentially in time and the total detection is very small, to a state with power-law decay and a significantly higher probability to detect the particle. The exponent describing the power-law decay of the detection probability at this critical $τ$ is 10/3, as opposed to 3 for every larger $τ$. In addition, the moving detector strongly modifies the Zeno effect.

cond-mat.stat-mech↗
arXiv2019

Non-self-averaging behaviors and ergodicity in quenched trap model with finite system size

Tracking tracer particles in heterogeneous environments plays an important role in unraveling the material properties. These heterogeneous structures are often static and depend on the sample realizations. Sample-to-sample fluctuations of such disorder realizations sometimes become considerably large. When we investigate the sample-to-sample fluctuations, fundamental averaging procedures are a thermal average for a single disorder realization and the disorder average for different disorder realizations. Here, we report on non-self-averaging phenomena in quenched trap models with finite system sizes, where we consider the periodic and the reflecting boundary conditions. Sample-to-sample fluctuations of diffusivity greatly exceeds trajectory-to-trajectory fluctuations of diffusivity in the corresponding annealed model. For a single disorder realization, the time-averaged mean square displacement and position-dependent observables converge to constants with the aid of the existence of the equilibrium distribution. This is a manifestation of ergodicity. As a result, the time-averaged quantities do not depend on the initial condition nor on the thermal histories but depend crucially on the disorder realization.

cond-mat.stat-mech↗
arXiv2018

From Non-normalizable Boltzmann-Gibbs statistics to infinite-ergodic theory

We study a particle immersed in a heat bath, in the presence of an external force which decays at least as rapidly as $1/x$, for example a particle interacting with a surface through a Lennard-Jones or a logarithmic potential. As time increases, our system approaches a non-normalizable Boltzmann state. We study observables, such as the energy, which are integrable with respect to this asymptotic thermal state, calculating both time and ensemble averages. We derive a useful canonical-like ensemble which is defined out of equilibrium, using a maximum entropy principle, where the constraints are: normalization, finite averaged energy and a mean-squared displacement which increases linearly with time. Our work merges infinite-ergodic theory with Boltzmann-Gibbs statistics, thus extending the scope of the latter while shedding new light on the concept of ergodicity.

cond-mat.stat-mech↗
arXiv2018

Renewal theory with fat tailed distributed sojourn times: typical versus rare

Renewal processes with heavy-tailed power law distributed sojourn times are commonly encountered in physical modelling and so typical fluctuations of observables of interest have been investigated in detail. To describe rare events the rate function approach from large deviation theory does not hold and new tools must be considered. Here we investigate the large deviations of the number of renewals, the forward and backward recurrence time, the occupation time, and the time interval straddling the observation time. We show how non-normalized densities describe these rare fluctuations, and how moments of certain observables are obtained from these limiting laws. Numerical simulations illustrate our results showing the deviations from arcsine, Dynkin, Darling-Kac, L{é}vy and Lamperti laws.

cond-mat.stat-mech↗
arXiv2018

The spectral dimension controls the decay of the quantum first detection probability

We consider a quantum system that is initially localized at $x_{in}$ and that is repeatedly projectively probed with a fixed period $τ$ at position $x_d$. We ask for the probability that the system is detected in $x_d$ for the very first time, $F_n$, where $n$ is the number of detection attempts. We relate the asymptotic decay and oscillations of $F_n$ with the system's energy spectrum, which is assumed to be absolutely continuous. In particular $F_n$ is determined by the Hamiltonian's measurement spectral density of states (MSDOS) $f(E)$ that is closely related to the density of energy states (DOS). We find that $F_n$ decays like a power law whose exponent is determined by the power law exponent $d_S$ of $f(E)$ around its singularities $E^*$. Our findings are analogous to the classical first passage theory of random walks. In contrast to the classical case, the decay of $F_n$ is accompanied by oscillations with frequencies that are determined by the singularities $E^*$. This gives rise to critical detection periods $τ_c$ at which the oscillations disappear. In the ordinary case $d_S$ can be identified with the spectral dimension found in the DOS. Furthermore, the singularities $E^*$ are the van Hove singularities of the DOS in this case. We find that the asymptotic statistics of $F_n$ depend crucially on the initial and detection state and can be wildly different for out-of-the-ordinary states, which is in sharp contrast to the classical theory. The properties of the first detection probabilities can alternatively be derived from the transition amplitudes. All our results are confirmed by numerical simulations of the tight-binding model, and of a free particle in continuous space both with a normal and with an anomalous dispersion relation. We provide explicit asymptotic formulae for the first detection probability in these models.

cond-mat.stat-mech↗
arXiv2017

First detected arrival of a quantum walker on an infinite line

The first detection of a quantum particle on a graph has been shown to depend sensitively on the sampling time τ . Here we use the recently introduced quantum renewal equation to investigate the statistics of first detection on an infinite line, using a tight-binding lattice Hamiltonian with nearest- neighbor hops. Universal features of the first detection probability are uncovered and simple limiting cases are analyzed. These include the small τ limit and the power law decay with attempt number of the detection probability over which quantum oscillations are superimposed. When the sampling time is equal to the inverse of the energy band width, non-analytical behaviors arise, accompanied by a transition in the statistics. The maximum total detection probability is found to occur for τ close to this transition point. When the initial location of the particle is far from the detection node we find that the total detection probability attains a finite value which is distance independent.

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