arXiv subjects

E. Barkai

Publications and source records attributed to E. Barkai.

At least 19 recordsLinked to original sources

arXiv2024

Telomeres in Lamin-A Depleted Cells Exhibit Directed Motion and Dynamic Coherence

Investigating the dynamics of chromatin loci and the factors that influence them provides valuable insights into the organization and functionality of the genome within the cell nucleus. We control the expression of Lamin-A, an important organizer of chromatin and nuclear structure. By simultaneously tracking hundreds of telomeres in Lamin-A knocked-out (KO) and wild-type (WT) nuclei, we find that telomere motion in Lamin-A depleted cells is both faster and more directed on micrometer scales, comparable to the size of chromosome territories. In contrast, telomere trajectories in WT cells exhibit pronounced anti-persistent behavior, consistent with caging by the surrounding chromatin environment. We further observe correlated motion between distinct telomeres in both WT and KO cells, with significantly stronger correlations in the KO case, indicating enhanced collective behavior. These correlations reflect cross-correlations among different loci rather than temporal correlations along individual trajectories. Together, these findings highlight the central role of Lamin-A in regulating both local confinement and collective telomere dynamics.

physics.bio-ph↗

arXiv2021

Measurement induced quantum walks

We investigate a tight binding quantum walk on a graph. Repeated stroboscopic measurements of the position of the particle yield a measured "trajectory", and a combination of classical and quantum mechanical properties for the walk are observed. We explore the effects of the measurements on the spreading of the packet on a one dimensional line, showing that except for the Zeno limit, the system converges to Gaussian statistics similarly to a classical random walk. A large deviation analysis and an Edgeworth expansion yield quantum corrections to this normal behavior. We then explore the first passage time to a target state using a generating function method, yielding properties like the quantization of the mean first return time. In particular, we study the effects of certain sampling rates which cause remarkable change in the behavior in the system, like divergence of the mean detection time in finite systems and a decomposition of the phase space into mutually exclusive regions, an effect that mimics ergodicity breaking, whose origin here is the destructive interference in quantum mechanics. For a quantum walk on a line we show that in our system the first detection probability decays classically like $(\text{time})^{-3/2}$, this is dramatically different compared to local measurements which yield a decay rate of $(\text{time})^{-3}$, indicating that the exponents of the first passage time depends on the type of measurements used.

cond-mat.stat-mech↗

arXiv2021

Randomly repeated measurements on quantum systems: Correlations and topological invariants of the quantum evolution

Randomly repeated measurements during the evolution of a closed quantum system create a sequence of probabilities for the first detection of a certain quantum state. The related discrete monitored evolution for the return of the quantum system to its initial state is investigated. We found that the mean number of measurements until the first detection is an integer, namely the dimensionality of the accessible Hilbert space. Moreover, the mean first detected return time is equal to the average time step between successive measurements times the mean number of measurements. Thus, the mean first detected return time scales linearly with the dimensionality of the accessible Hilbert space. The main goal of this work is to explain the quantization of the mean return time in terms of a quantized Berry phase.

quant-ph↗

arXiv2021

Cusp of non-Gaussian density of particles for a diffusing diffusivity model

We study a two state ``jumping diffusivity'' model for a Brownian process alternating between two different diffusion constants, $D_{+}>D_{-}$, with random waiting times in both states whose distribution is rather general. In the limit of long measurement times Gaussian behavior with an effective diffusion coefficient is recovered. We show that for equilibrium initial conditions and when the limit of the diffusion coefficient $D_-\to0$ is taken, the short time behavior leads to a cusp, namely a non - analytical behavior, in the distribution of the displacements $P(x,t)$ for $x\longrightarrow 0$. Visually this cusp, or tent-like shape, resembles similar behavior found in many experiments of diffusing particles in disordered environments, such as glassy systems and intracellular media. This general result depends only on the existence of finite mean values of the waiting times at the different states of the model. Gaussian statistics in the long time limit is achieved due to ergodicity and convergence of the distribution of the temporal occupation fraction in state $D_{+}$ to a $δ$-function. The short time behavior of the same quantity converges to a uniform distribution, which leads to the non - analyticity in $P(x,t)$. We demonstrate how super - statistical framework is a zeroth order short time expansion of $P(x,t)$, in the number of transitions, that does not yield the cusp like shape. The latter, considered as the key feature of experiments in the field, is found with the first correction in perturbation theory.

cond-mat.stat-mech↗

arXiv2020

The Hitchhiker model for Laplace diffusion processes in the cell environment

Aggregation and fragmentation of single molecules in the cell environment lead to a spectrum of diffusivities and to statistical laws of movement very different from typical Brownian motion. Current models of intracellular transport do not explain at a microscopical level the emergence of theses deviations. Employing a many body approach, which we call the Hitchhiker model, we elucidate how the widely observed exponential tails in the particle spreading, i.e. the Laplace distribution and the modulations of the diffusivities, are controlled by size fluctuations of single molecules. By means of numerical simulations Laplace distributions are obtained whether we track one molecule or many molecules in parallel. However, we show that the diffusivity varies significantly depending on which tracking protocol is applied. Using a renewal process in the space of sizes, we quantify to what extent the average diffusivity in the single molecule technique is decreased compared with the ensemble average.

cond-mat.stat-mech↗

arXiv2020

Quantum walks: the first detected transition time

We consider the quantum first detection problem for a particle evolving on a graph under repeated projective measurements with fixed rate $1/τ$. A general formula for the mean first detected transition time is obtained for a quantum walk in a finite-dimensional Hilbert space where the initial state $|ψ_{\rm in}\rangle$ of the walker is orthogonal to the detected state $|ψ_{\rm d}\rangle$. We focus on diverging mean transition times, where the total detection probability exhibits a discontinuous drop of its value, by mapping the problem onto a theory of fields of classical charges located on the unit disk. Close to the critical parameter of the model, which exhibits a blow-up of the mean transition time, we get simple expressions for the mean transition time. Using previous results on the fluctuations of the return time, corresponding to $|ψ_{\rm in}\rangle = |ψ_{\rm d}\rangle$, we find close to these critical parameters that the mean transition time is proportional to the fluctuations of the return time, an expression reminiscent of the Einstein relation.

cond-mat.stat-mech↗

arXiv2018

Infinite Ergodic Theory for Heterogeneous Diffusion Processes

We show the relation between processes which are modeled by a Langevin equation with multiplicative noise and infinite ergodic theory. We concentrate on a spatially dependent diffusion coefficient that behaves as ${D(x)}\sim |x-\tilde{x}|^{2-2/α}$ in the vicinity of a point $\tilde{x}$, where $α$ can be either positive or negative. We find that a nonnormalized state, also called an infinite density, describes statistical properties of the system. For processes under investigation, the time averages of a wide class of observables, are obtained using an ensemble average with respect to the nonnormalized density. A Langevin equation which involves multiplicative noise may take different interpretation; Itô, Stratonovich, or Hänggi-Klimontovich, so the existence of an infinite density, and the density's shape, are both related to the considered interpretation and the structure of $D(x)$.

cond-mat.stat-mech↗

arXiv2018

Dispersion of particles in an infinite-horizon Lorentz gas

We consider a two-dimensional Lorentz gas with infinite horizon. This paradigmatic model consists of pointlike particles undergoing elastic collisions with fixed scatterers arranged on a periodic lattice. It was rigorously shown that when $t\to\infty$, the distribution of particles is Gaussian. However, the convergence to this limit is ultraslow, hence it is practically unattainable. Here we obtain an analytical solution for the Lorentz gas' kinetics on physically relevant timescales, and find that the density in its far tails decays as a universal power law of exponent $-3$. We also show that the arrangement of scatterers is imprinted in the shape of the distribution.

cond-mat.stat-mech↗

arXiv2017

Conditional $1/f^α$ noise: from single molecules to macroscopic measurements

We demonstrate that the measurement of $1/f^α$ noise at the single molecule or nano-object limit is remarkably distinct from the macroscopic measurement over a large sample. The single particle measurements yield a conditional time-dependent spectrum. However, the number of units fluctuating on the time scale of the experiment is increasing in such a way that the macroscopic measurements appear perfectly stationary. The single particle power spectrum is a conditional spectrum, in the sense that we must make a distinction between idler and non-idler units on the time scale of the experiment. We demonstrate our results based on stochastic and deterministic models, in particular the well known superposition of Lorentzians approach, the blinking quantum dot model, and deterministic dynamics generated by non-linear mapping. Our results show that the $1/f^α$ spectrum is inherently nonstationary even if the macroscopic measurement completely obscures the underlying time dependence of the phenomena.

cond-mat.stat-mech↗

arXiv2017

Superdiffusive dispersals impart the geometry of underlying random walks

It is recognised now that a variety of real-life phenomena ranging from diffuson of cold atoms to motion of humans exhibit dispersal faster than normal diffusion. Lévy walks is a model that excelled in describing such superdiffusive behaviors albeit in one dimension. Here we show that, in contrast to standard random walks, the microscopic geometry of planar superdiffusive Lévy walks is imprinted in the asymptotic distribution of the walkers. The geometry of the underlying walk can be inferred from trajectories of the walkers by calculating the analogue of the Pearson coefficient.

cond-mat.stat-mech↗

arXiv2016

Aging Wiener-Khinchin Theorem and Critical Exponents of $1/f$ Noise

The power spectrum of a stationary process may be calculated in terms of the autocorrelation function using the Wiener-Khinchin theorem. We here generalize the Wiener-Khinchin theorem for nonstationary processes and introduce a time-dependent power spectrum $\left\langle S_{t_m}(ω)\right\rangle$ where $t_m$ is the measurement time. For processes with an aging correlation function of the form $\left\langle I(t)I(t+τ)\right\rangle=t^Υϕ_{\rm EA}(τ/t)$, where $ϕ_{\rm EA}(x)$ is a nonanalytic function when $x$ is small, we find aging $1/f$ noise. Aging $1/f$ noise is characterized by five critical exponents. We derive the relations between the scaled correlation function and these exponents. We show that our definition of the time-dependent spectrum retains its interpretation as a density of Fourier modes and discuss the relation to the apparent infrared divergence of $1/f$ noise. We illustrate our results for blinking quantum dot models, single-file diffusion and Brownian motion in logarithmic potential.

cond-mat.stat-mech↗

arXiv2016

Quantum Renewal Equation for the first detection time of a quantum walk

We investigate the statistics of the first detected passage time of a quantum walk. The postulates of quantum theory, in particular the collapse of the wave function upon measurement, reveal an intimate connection between the wave function of a process free of measurements, i.e. the solution of the Schrödinger equation, and the statistics of first detection events on a site. For stroboscopic measurements a quantum renewal equation yields basic properties of quantum walks. For example, for a tight binding model on a ring we discover critical sampling times, diverging quantities such as the mean time for first detection, and an optimal detection rate. For a quantum walk on an infinite line the probability of first detection decays like $(\mbox{time})^{-3}$ with a superimposed oscillation, critical behavior for a specific choice of sampling time, and vanishing amplitude when the sampling time approaches zero due to the quantum Zeno effect.

cond-mat.stat-mech↗

arXiv2015

Aging Wiener-Khinchin Theorem

The Wiener-Khinchin theorem shows how the power spectrum of a stationary random signal $I(t)$ is related to its correlation function $\left\langle I(t)I(t+τ)\right\rangle$. We consider non-stationary processes with the widely observed aging correlation function $\langle I(t) I(t+τ) \rangle \sim t^γϕ_{\rm EN}(τ/t)$ and relate it to the sample spectrum. We formulate two aging Wiener-Khinchin theorems relating the power spectrum to the time and ensemble averaged correlation functions, discussing briefly the advantages of each. When the scaling function $ϕ_{\rm EN}(x)$ exhibits a non-analytical behavior in the vicinity of its small argument we obtain aging $1/f$ type of spectrum. We demonstrate our results with three examples: blinking quantum dots, single file diffusion and Brownian motion in a logarithmic potential, showing that our approach is valid for a wide range of physical mechanisms.

cond-mat.stat-mech↗

arXiv2014

Asymptotic densities of ballistic Lévy walks

We propose an analytical method to determine the shape of density profiles in the asymptotic long time limit for a broad class of coupled continuous time random walks which operate in the ballistic regime. In particular, we show that different scenarios of performing a random walk step, via making an instantaneous jump penalized by a proper waiting time or via moving with a constant speed, dramatically effect the corresponding propagators, despite the fact that the end points of the steps are identical. Furthermore, if the speed during each step of the random walk is itself a random variable, its distribution gets clearly reflected in the asymptotic density of random walkers. These features are in contrast with more standard non-ballistic random walks.

cond-mat.stat-mech↗

arXiv2014

From the area under the Bessel excursion to anomalous diffusion of cold atoms

Levy flights are random walks in which the probability distribution of the step sizes is fat-tailed. Levy spatial diffusion has been observed for a collection of ultra-cold Rb atoms and single Mg+ ions in an optical lattice. Using the semiclassical theory of Sisyphus cooling, we treat the problem as a coupled Levy walk, with correlations between the length and duration of the excursions. The problem is related to the area under Bessel excursions, overdamped Langevin motions that start and end at the origin, constrained to remain positive, in the presence of an external logarithmic potential. In the limit of a weak potential, the Airy distribution describing the areal distribution of the Brownian excursion is found. Three distinct phases of the dynamics are studied: normal diffusion, Levy diffusion and, below a certain critical depth of the optical potential, x~ t^{3/2} scaling. The focus of the paper is the analytical calculation of the joint probability density function from a newly developed theory of the area under the Bessel excursion. The latter describes the spatiotemporal correlations in the problem and is the microscopic input needed to characterize the spatial diffusion of the atomic cloud. A modified Montroll-Weiss (MW) equation for the density is obtained, which depends on the statistics of velocity excursions and meanders. The meander, a random walk in velocity space which starts at the origin and does not cross it, describes the last jump event in the sequence. In the anomalous phases, the statistics of meanders and excursions are essential for the calculation of the mean square displacement, showing that our correction to the MW equation is crucial, and points to the sensitivity of the transport on a single jump event. Our work provides relations between the statistics of velocity excursions and meanders and that of the diffusivity.

cond-mat.stat-mech↗

arXiv2014

Non-normalizable densities in strong anomalous diffusion: beyond the central limit theorem

Strong anomalous diffusion, where $\langle |x(t)|^q \rangle \sim t^{q ν(q)}$ with a nonlinear spectrum $ν(q) \neq \mbox{const}$, is wide spread and has been found in various nonlinear dynamical systems and experiments on active transport in living cells. Using a stochastic approach we show how this phenomena is related to infinite covariant densities, i.e., the asymptotic states of these systems are described by non-normalizable distribution functions. Our work shows that the concept of infinite covariant densities plays an important role in the statistical description of open systems exhibiting multi-fractal anomalous diffusion, as it is complementary to the central limit theorem.

cond-mat.stat-mech↗

arXiv2013

Scaling Green-Kubo relation and application to three aging systems

The Green-Kubo formula relates the spatial diffusion coefficient to the stationary velocity autocorrelation function. We derive a generalization of the Green-Kubo formula valid for systems with long-range or nonstationary correlations for which the standard approach is no longer valid. For the systems under consideration, the velocity autocorrelation function $\langle v(t+τ) v(t) \rangle$ asymptotically exhibits a certain scaling behavior and the diffusion is anomalous $\langle x^2(t) \rangle \simeq 2 D_νt^ν$. We show how both the anomalous diffusion coefficient $D_ν$ and exponent $ν$ can be extracted from this scaling form. Our scaling Green-Kubo relation thus extends an important relation between transport properties and correlation functions to generic systems with scale invariant dynamics. This includes stationary systems with slowly decaying power law correlations as well as aging systems, whose properties depend on the the age of the system. Even for systems that are stationary in the long time limit, we find that the long time diffusive behavior can strongly depend on the initial preparation of the system. In these cases, the diffusivity $D_ν$ is not unique and we determine its values for a stationary respectively nonstationary initial state. We discuss three applications of the scaling Green-Kubo relation: Free diffusion with nonlinear friction corresponding to cold atoms diffusing in optical lattices, the fractional Langevin equation with external noise recently suggested to model active transport in cells and the Lévy walk with numerous applications, in particular blinking quantum dots. These examples underline the wide applicability of our approach, which is able to treat very different mechanisms of anomalous diffusion.

cond-mat.stat-mech↗

arXiv2013

A no-go theorem for ergodicity and an Einstein relation

We provide a simple no-go theorem for ergodicity and the generalized Einstein relation for anomalous diffusion processes. The theorem states that either ergodicity in the sense of equal time and ensemble averaged mean squared displacements (MSD) is broken, and/or the generalized Einstein relation for time averaged diffusivity and mobility is invalid, which is in complete contrast to normal diffusion processes. We also give a general relation for the time averages of drift and MSD for \textit{ergodic} (in the MSD sense) anomalous diffusion processes, showing that the ratio of these quantities depends on the measurement time. The Lévy walk model is used to exemplify the no-go theorem.

cond-mat.stat-mech↗