arXiv subjects

Eli Barkai

Publications and source records attributed to Eli Barkai.

139 records · Page 8Linked to original sources

arXiv2005

Stochastic Ergodicity Breaking: a Random Walk Approach

The continuous time random walk (CTRW) model exhibits a non-ergodic phase when the average waiting time diverges. Using an analytical approach for the non-biased and the uniformly biased CTRWs, and numerical simulations for the CTRW in a potential field, we obtain the non-ergodic properties of the random walk which show strong deviations from Boltzmann--Gibbs theory. We derive the distribution function of occupation times in a bounded region of space which, in the ergodic phase recovers the Boltzmann--Gibbs theory, while in the non-ergodic phase yields a generalized non-ergodic statistical law.

cond-mat.stat-mech↗

arXiv2004

Super and Sub-Poissonian photon statistics for single molecule spectroscopy

We investigate the distribution of the number of photons emitted by a single molecule undergoing a spectral diffusion process and interacting with a continuous wave laser field. The spectral diffusion is modeled based on a stochastic approach, in the spirit of the Anderson-Kubo line shape theory. Using a generating function formalism we solve the generalized optical Bloch equations, and obtain an exact analytical formula for the line shape and Mandel's Q parameter. The line shape exhibits well known behaviors, including motional narrowing when the stochastic modulation is fast, and power broadening. The Mandel parameter, describing the line shape fluctuations, exhibits a transition from a Quantum sub-Poissonian behavior in the fast modulation limit, to a classical super-Poissonian behavior found in the slow modulation limit. Our result is applicable for weak and strong laser field, namely for arbitrary Rabi frequency. We show how to choose the Rabi frequency in such a way that the Quantum sub-Poissonian nature of the emission process becomes strongest. A lower bound on $Q$ is found, and simple limiting behaviors are investigated. A non-trivial behavior is obtained in the intermediate modulation limit, when the time scales for spectral diffusion and the life time of the excited state, become similar. A comparison is made between our results, and previous ones derived based on the semi-classical generalized Wiener--Khintchine theorem.

cond-mat.stat-mech↗

arXiv2004

Non-ergodic Intensity Correlation Functions for Blinking Nano Crystals

We investigate the non-ergodic properties of blinking nano-crystals using a stochastic approach. We calculate the distribution functions of the time averaged intensity correlation function and show that these distributions are not delta peaked on the ensemble average correlation function values; instead they are W or U shaped. Beyond blinking nano-crystals our results describe non-ergodicity in systems stochastically modeled using the Levy walk framework for anomalous diffusion, for example certain types of chaotic dynamics, currents in ion-channel, and single spin dynamics to name a few.

cond-mat.stat-mech↗

arXiv2004

Aging Correlation Functions for Blinking Nano-Crystals, and Other On - Off Stochastic Processes

Following recent experiments on power law blinking behavior of single nano-crystals, we calculate two-time intensity correlation functions for these systems. We use a simple two state (on and off) stochastic model to describe the dynamics. We classify possible behaviors of the correlation function and show that aging, e.g., dependence of the correlation function on age of process t, is obtained for classes of the on time and off time distributions relevant to experimental situation. Analytical asymptotic scaling behaviors of the intensity correlation in the double time t and t' domain are obtained. In the scaling limit --> h(x), where four classes of behaviors are found: (i) finite averaged on and off times x=t' (standard behavior) (ii) on and off times with identical power law behaviors x=t/t' (case relevant for capped nano-crystals). (iii) exponential on times and power law off times x=tt' (case relevant for uncapped nano-crystals). (iv) For defected off time distribution we also find x=t+t' . Origin of aging behavior is explained based on simple diffusion model. We argue that the diffusion controlled reaction A+B <--> AB, when followed on a single particle level exhibits aging behavior.

cond-mat.stat-mech↗

arXiv2003

Exact Solution for the Influence of Spectral Diffusion on Single-Molecule Photon-Statistics

We investigate the distribution of number of photons emitted by a single molecule undergoing a spectral diffusion process and interacting with a continuous wave field. Using a generating function formalism an exact analytical formula for Mandel's $Q$ parameter is obtained. The solution exhibits transitions between: (i) Quantum sub-Poissonian and Classical super-Poissonian behaviors, and (ii) fast to slow modulation limits. Our solution yields the conditions on the magnitude of the spectral diffusion time scales on which these transitions are observed. We show how to choose the Rabi frequency in such a way that the Quantum sub-Poissonian nature of the emission process becomes strongest, we find ${Ω^2 \over Γ} = {Γ_{\rm SD} + Γ\over 2}$ where $Γ_{\rm SD}$ ($Γ$) is the spectral diffusion (radiative) contribution to the width of the line shape.

cond-mat.stat-mech↗

arXiv2003

A Stochastic Theory of Single Molecule Spectroscopy

A theory is formulated for time dependent fluctuations of the spectrum of a single molecule in a dynamic environment. In particular, we investigate the photon counting statistics of a single molecule undergoing a spectral diffusion process. Based on the stochastic optical Bloch equation, fluctuations are characterized by Mandel's $Q$ parameter yielding the variance of number of emitted photons and the second order intensity correlation function, $g^{(2)}(t)$. Using a semi-classical approach and linear response theory, we show that the $Q$ parameter can be described by a three-time dipole correlation function. This approach gene ralizes the Wiener-Khintchine formula that gives the average number of fluorescent photons in terms of a one-time dipole correlation function. We classify the time ordering properties of the three-time dipole correlation function, and show that it can be represented by three different pulse shape functions similar to those used in the context of nonlinear spectroscopy. An exact solution is found for a single molecule whose absorption frequency undergoes a two state random telegraph process (i.e., the Kubo-Anderson sudden jump process.) Simple expressions are obtained from the exact solution in the slow and fast modulation regimes based on appropriate approximations for each case. In the slow modulation regime $Q$ can be large even in the long time limit, while in the fast modulation regime it becomes small.

cond-mat.stat-mech↗

arXiv2003

Stable Equilibrium Based on Lévy Statistics: Stochastic Collision Models Approach

We investigate equilibrium properties of two very different stochastic collision models: (i) the Rayleigh particle and (ii) the driven Maxwell gas. For both models the equilibrium velocity distribution is a Lévy distribution, the Maxwell distribution being a special case. We show how these models are related to fractional kinetic equations. Our work demonstrates that a stable power-law equilibrium, which is independent of details of the underlying models, is a natural generalization of Maxwell's velocity distribution.

cond-mat.stat-mech↗

arXiv2003

Stable Equilibrium Based on Lévy Statistics: A Linear Boltzmann Equation Approach

To obtain further insight on possible power law generalizations of Boltzmann equilibrium concepts, a stochastic collision model is investigated. We consider the dynamics of a tracer particle of mass $M$, undergoing elastic collisions with ideal gas particles of mass $m$, in the Rayleigh limit $m<<M$. The probability density function (PDF) of the gas particle velocity is $f(\tilde{v}_m)$. Assuming a uniform collision rate and molecular chaos, we obtain the equilibrium distribution for the velocity of the tracer particle $W_{eq}(V_M)$. Depending on asymptotic properties of $f(\tilde{v}_m)$ we find that $W_{eq}(V_M)$ is either the Maxwell velocity distribution or a Lévy distribution. In particular our results yield a generalized Maxwell distribution based on Lévy statistics using two approaches. In the first a thermodynamic argument is used, imposing on the dynamics the condition that equilibrium properties of the heavy tracer particle be independent of the coupling $ε=m/M$ to the gas particles, similar to what is found for a Brownian particle in a fluid. This approach leads to a generalized temperature concept. In the second approach it is assumed that bath particles velocity PDF scales with an energy scale, i.e. the (nearly) ordinary temperature, as found in standard statistical mechanics. The two approaches yield different types of Lévy equilibrium which merge into a unique solution only for the Maxwell--Boltzmann case. Thus, relation between thermodynamics and statistical mechanics becomes non-trivial for the power law case. Finally, the relation of the kinetic model to fractional Fokker--Planck equations is discussed.

cond-mat.stat-mech↗

arXiv2002

Aging Continuous Time Random Walks

We investigate aging continuous time random walks (ACTRW), introduced by Monthus and Bouchaud [{\em J. Phys. A} {\bf 29}, 3847 (1996)]. Statistical behaviors of the displacement of the random walker ${\bf r}={\bf r}(t) - {\bf r}(0)$ in the time interval $(0,t)$ are obtained, after aging the random walk in the time interval $(-t_a,0)$. In ACTRW formalism, the Green function $P({\bf r}, t_a, t)$ depends on the age of the random walk $t_a$ and the forward time $t$. We derive a generalized Montroll--Weiss equation, which yields an exact expression for the Fourier double Laplace transform of the ACTRW Green function. Asymptotic long times $t_a$ and $t$ behaviors of the Green function are investigated in detail. In the limit of $t\gg t_a$, we recover the standard non-equilibrium CTRW behaviors, while the important regimes $t\ll t_a$ and $t \simeq t_a$ exhibit interesting aging effects. Convergence of the ACTRW results towards CTRW behavior, becomes extremely slow when the diffusion exponent becomes small. In the context of biased ACTRW, we investigate an aging Einstein relation. We briefly discuss aging in Scher-Montroll type of transport in disordered materials.

cond-mat.stat-mech↗

arXiv2002

Lineshape Theory and Photon Counting Statistics for Spectral Fluctuations in Quantum Dots: a Lévy Walk Process

Recent experimental observations have found two different kinds of ``strange kinetic behaviors" in individual semiconductor nanocrystals (or quantum dots). Fluorescence intermittency observed in the quantum dots shows power--law statistics in both {\it on} and {\it off} times. Spectral diffusion of the quantum dots is also described by power--law statistics in the sojourn times. Motivated by these experimental observations we consider two different but related problems: (a) a stochastic lineshape theory for the Kubo-Anderson oscillator whose frequency modulation follows power-law statistics and (b) photon counting statistics of quantum dots whose intensity fluctuation is characterized by power-law kinetics. In the first problem, we derive an analytical expression for the lineshape formula and find rich type of behaviors when compared with the standard theory. For example, new type of resonances and narrowing behavior have been found. We show that the lineshape is extremely sensitive to the way the system is prepared at time $t=0$ and discuss the problem of stationarity. In the second problem, we use semiclassical photon counting statistics to characterize the fluctuation of the photon counts emitted from quantum dots. We show that the photon counting statistics problem can be mapped onto a Lévy walk process. We find unusually large fluctuations in the photon counts that have not been encountered previously. In particular, we show that $Q$ may increase in time even in the long time limit.

cond-mat.stat-mech↗

arXiv2001

CTRW Pathways to the Fractional Diffusion Equation

The foundations of the fractional diffusion equation are investigated based on coupled and decoupled continuous time random walks (CTRW). For this aim we find an exact solution of the decoupled CTRW, in terms of an infinite sum of stable probability densities. This exact solution is then used to understand the meaning and domain of validity of the fractional diffusion equation. An interesting behavior is discussed for coupled memories (i.e., Lévy walks). The moments of the random walk exhibit strong anomalous diffusion, indicating (in a naive way) the breakdown of simple scaling behavior and hence of the fractional approximation. Still the Green function $P(x,t)$ is described well by the fractional diffusion equation, in the long time limit.

cond-mat.stat-mech↗

arXiv2001

Photon Counting Statistics for Single Molecule Spectroscopy

We present a general relation between Mandel's $Q$ parameter describing variance of number of photons emitted from a single molecule and a three time correlation function describing a spectral diffusion process the molecule is undergoing. For a prototype of such a stochastic process an exact solution is found which exhibits rich types of physical behaviors.

cond-mat.stat-mech↗

arXiv2000

Dispersive Lineshape Theory

Motivated by recent experiments we consider a stochastic lineshape theory for the case when the underlying process obeys power-law statistics, based on a generalized Anderson-Kubo oscillator model. We derive an analytical expression for the lineshape and find rich type of behaviors when compared with the standard theory, for example, new type of resonances and narrowing phenomena. We show that the lineshape is extremely sensitive to the way the system is prepared at time $t=0$ and discuss the problem of stationarity.

cond-mat↗