Electron waiting times in mesoscopic conductors

Electron transport in mesoscopic conductors has traditionally involved investigations of the mean current and the fluctuations of the current. A complementary view on charge transport is provided by the distribution of waiting times between charge carriers, but a proper theoretical framework for coherent electronic systems has so far been lacking. Here we develop a quantum theory of electron waiting times in mesoscopic conductors expressed by a compact determinant formula. We illustrate our methodology by calculating the waiting time distribution for a quantum point contact and find a crossover from Wigner-Dyson statistics at full transmission to Poisson statistics close to pinch-off. Even when the low-frequency transport is noiseless, the electrons are not equally spaced in time due to their inherent wave nature. We discuss the implications for renewal theory in mesoscopic systems and point out several analogies with level spacing statistics and random matrix theory.     

Correlations for the circular Dyson brownian motion model with Poisson initial conditions

The circular Dyson brownian motion model refers to the stochastic dynamics of the log-gas on a circle. It also specifies the eigenvalues of certain parameter-dependent ensembles of unitary random matrices. This model is considered with the initial condition that the particles are non-interacting (Poisson statistics). Jack polynomial theory is used to derive a simple exact expression for the density-density correlation with the position of one particle specified in the initial state, and the position of one particle specified at time τ, valid for all β > 0. The same correlation with two particles specified in the initial state is also derived exactly, and some special cases of the theoretical correlations are illustrated by comparison with the empirical correlations calculated from the eigenvalues of certain parameter-dependent Gaussian random matrices. Application to fluctuation formulas for time-displaced linear statistics in made.     

Statistical Analysis of the First Passage Path Ensemble of Jump Processes

The transition mechanism of jump processes between two different subsets in state space reveals important dynamical information of the processes and therefore has attracted considerable attention in the past years. In this paper, we study the first passage path ensemble of both discrete-time and continuous-time jump processes on a finite state space. The main approach is to divide each first passage path into nonreactive and reactive segments and to study them separately. The analysis can be applied to jump processes which are non-ergodic, as well as continuous-time jump processes where the waiting time distributions are non-exponential. In the particular case that the jump processes are both Markovian and ergodic, our analysis elucidates the relations between the study of the first passage paths and the study of the transition paths in transition path theory. We provide algorithms to numerically compute statistics of the first passage path ensemble. The computational complexity of these algorithms scales with the complexity of solving a linear system, for which efficient methods are available. Several examples demonstrate the wide applicability of the derived results across research areas.     

Waiting-time statistics of self-organized-criticality systems

It is argued that a system governed by self-organized-criticality (SOC) dynamics can lack Poisson waiting-time statistics not only when the experimental resolution lies within the self-similar scale range but also if the system is slowly driven in a correlated way. This result thus suggests that waiting time statistics cannot be used as a necessary test for SOC behavior in real physical systems.     

Kink dynamics in the highly discrete sine-Gordon system

We study kink dynamics in a very discrete sine-Gordon system where the kink width is of the order of the lattice spacing. Numerical simulations exhibit new properties of kinks in this case: they lose the memory of their initial velocity and propagate preferentially at well-defined velocities which correspond to quasi-steady states, while a kink moving at other velocities suffers relatively high rates of radiation of small amplitude oscillations. When a small external driving force is applied to the system, the same velocities appear as plateus in the strongly nonlinear mobility of the kink. The energy radiated by the kink is calculated for a simple model that preserves the discrete character of the system, and the preferential velocities for the kink are obtained to good accuracy. Similar results may be expected to be valid for other discrete systems manifesting topological solitons. The numerical simulations reveal also new stable “multiple-kink” excitations which can propagate almost freely in extremely discrete systems where “ordinary” simple kinks are pinned to the lattice by discreteness. The stability of the “multiple-kinks” is discussed.     

Statistics of secondary electrons from W induced by He ions and atoms at 0° –90° incidence

Secondary electrons from a W target, induced by 2 KeV He⁰ and He⁺ projectiles at 0° -90° angles of incidence have been measured for the first time using the statistics of secondary electron emission. The statistics were determined from the analysis of measured pulse height distribution spectra for 8–32 KeV He⁺ ions. The pulse height response of the detector is found to be sensitive to the applied bias to the detector. Its consequences in the interpretation of data are investigated. It was established that the number of electrons emitted per incident ion satisfy a Poisson distribution. Some new suggestions are made on the possible use of these results, in particular, on the role of the zeroth and the first Poisson coefficients, for surface studies.     

Percolation of the Loss of Tension in an Infinite Triangular Lattice

We introduce a new class of bootstrap percolation models where the local rules are of a geometric nature as opposed to simple counts of standard bootstrap percolation. Our geometric bootstrap percolation comes from rigidity theory and convex geometry. We outline two percolation models: a Poisson model and a lattice model. Our Poisson model describes how defects--holes is one of the possible interpretations of these defects--imposed on a tensed membrane result in a redistribution or loss of tension in this membrane; the lattice model is motivated by applications of Hooke spring networks to problems in material sciences. An analysis of the Poisson model is given by Menshikov et al. ⁽⁴⁾ In the discrete set-up we consider regular and generic triangular lattices on the plane where each bond is removed with probability 1–p. The problem of the existence of tension on such lattice is solved by reducing it to a bootstrap percolation model where the set of local rules follows from the geometry of stresses. We show that both regular and perturbed lattices cannot support tension for any p<1. Moreover, the complete relaxation of tension--as defined in Section 4--occurs in a finite time almost surely. Furthermore, we underline striking similarities in the properties of the Poisson and lattice models.     

First passage time statistics of Brownian motion with purely time dependent drift and diffusion

Systems where resource availability approaches a critical threshold are common to many engineering and scientific applications and often necessitate the estimation of first passage time statistics of a Brownian motion (Bm) driven by time-dependent drift and diffusion coefficients. Modeling such systems requires solving the associated Fokker-Planck equation subject to an absorbing barrier. Transitional probabilities are derived via the method of images, whose applicability to time dependent problems is shown to be limited to state-independent drift and diffusion coefficients that only depend on time and are proportional to each other. First passage time statistics, such as the survival probabilities and first passage time densities are obtained analytically. The analysis includes the study of different functional forms of the time dependent drift and diffusion, including power-law time dependence and different periodic drivers. As a case study of these theoretical results, a stochastic model of water resources availability in snowmelt dominated regions is presented, where both temperature effects and snow-precipitation input are incorporated.     

The distribution of first-passage times and durations in FOREX and future markets

Possible distributions are discussed for intertrade durations and first-passage processes in financial markets. The view-point of renewal theory is assumed. In order to represent market data with relatively long durations, two types of distributions are used, namely a distribution derived from the Mittag-Leffler survival function and the Weibull distribution. For the Mittag-Leffler type distribution, the average waiting time (residual life time) is strongly dependent on the choice of a cut-off parameter t_max, whereas the results based on the Weibull distribution do not depend on such a cut-off. Therefore, a Weibull distribution is more convenient than a Mittag-Leffler type if one wishes to evaluate relevant statistics such as average waiting time in financial markets with long durations. On the other hand, we find that the Gini index is rather independent of the cut-off parameter. Based on the above considerations, we propose a good candidate for describing the distribution of first-passage time in a market: The Weibull distribution with a power-law tail. This distribution compensates the gap between theoretical and empirical results more efficiently than a simple Weibull distribution. It should be stressed that a Weibull distribution with a power-law tail is more flexible than the Mittag-Leffler distribution, which itself can be approximated by a Weibull distribution and a power-law. Indeed, the key point is that in the former case there is freedom of choice for the exponent of the power-law attached to the Weibull distribution, which can exceed 1 in order to reproduce decays faster than possible with a Mittag-Leffler distribution. We also give a useful formula to determine an optimal crossover point minimizing the difference between the empirical average waiting time and the one predicted from renewal theory. Moreover, we discuss the limitation of our distributions by applying our distribution to the analysis of the BTP future and calculating the average waiting time. We find that our distribution is applicable as long as durations follow a Weibull law for short times and do not have too heavy a tail.     

Earthquakes descaled: on waiting time distributions and scaling laws

Recently, several authors have used waiting time distributions for large earthquake data sets to draw conclusions regarding the physics of earthquake processes. We show, theoretically and by simulation, that a characteristic kink in observed waiting time distributions does not have the physical significance of separating correlated and uncorrelated earthquakes. It also follows from our discussion that the Omori law is not trivially related to a proposed scaling law and that caution must be taken before the spatial scaling exponent of the law is interpreted as a fractal dimension of seismicity.     

On Asymptotic Stability of Moving Kink for Relativistic Ginzburg-Landau Equation

We prove the asymptotic stability of the moving kinks for the nonlinear relativistic wave equations in one space dimension with a Ginzburg-Landau potential: starting in a small neighborhood of the kink, the solution, asymptotically in time, is the sum of a uniformly moving kink and dispersive part described by the free Klein-Gordon equation. The remainder decays in a global energy norm. Our recent results on the weighted energy decay for the Klein-Gordon equations play a crucial role in the proofs.     

A continuous model of human dynamics

Ideas and tools from statistical physics have recently been applied to the investigation of human dynamics. The timing of human activities, in particular, has been studied both experimentally and analytically. Empirical data show that, in many different situations, the time interval separating two consecutive tasks executed by an individual follows a heavy-tailed probability distribution rather than Poisson statistics. To account for this data, human behaviour has been viewed as a decision-based queuing system where individuals select and execute tasks belonging to a finite list of items as an increasing function of a task priority parameter. It is then possible to obtain analytically the empirical result P(τ)∼1/τ, where P(τ) is the waiting time probability distribution.Here a continuous model of human dynamics is introduced using instead an infinite queuing list. In contrast with the results obtained by other models in the finite case we find a waiting time distribution explicitly depending on the priority distribution density function ρ. The power-law scaling P(τ)∼1/τ is then recovered when ρ is exponentially distributed.     

Temperature dependence of bound state spectra in field theory

We analyze the behaviour of kinks and semiclassical bound states at finite temperatures by applying quantum statistics to the fluctuations which determine the quantum dynamics of these states. We consider two theories in one space dimension — the ?⁴ theory with a dynamical symmetry breaking and the Gross-Neveu model. For the ?⁴ theory, the one-loop temperature corrections are obtained by using temperature-dependent Green function techniques. We show that the same result can be obtained by applying quantum statistics to the fluctuations around the kink. For the Gross-Neveu model, the temperature dependence of the bound states, which correspond to time-independent field configurations, is obtained. We show that for every bound state there exists a critical temperature at which this state breaks up into its constituents. This critical temperature increases with the number of constituents of the bound state.     

Dynamics of isolated twin boundary in (TMTSF)

Intermittent and irregular motion of isolated twin boundary (kink) in organic crystal (TMTSF)₂PF₆ was studied at room temperature. Both the local velocity and the time of intermission are determined not only by external stress and temperature but also by the time (t w) elapsed after the backward passage and before the following forward one. When the kink moves after longer t w, its velocity becomes smaller and the time of intermission longer. Both tend to saturate for t w longer than 10² s. This result indicates that some disorder is induced in the lattice by the backward motion and it is relaxed during t w. We also found that the effect of the backward motion of one kink on its following motion is equivalent quantitatively to that of the forward motion of the pair-created counterpart. Received: 14 April 1998 / Received in final form and Accepted: 1st September 1998     

Effect of Ergodic and Non-Ergodic Fluctuations on a Charge Diffusing in a Stochastic Magnetic Field

In this paper, we study the basic problem of a charged particle in a stochastic magnetic field. We consider dichotomous fluctuations of the magnetic field where the sojourn time in one of the two states are distributed according to a given waiting-time distribution either with Poisson or non-Poisson statistics, including as well the case of distributions with diverging mean time between changes of the field, corresponding to an ergodicity breaking condition. We provide analytical and numerical results for all cases evaluating the average and the second moment of the position and velocity of the particle. We show that the field fluctuations induce diffusion of the charge with either normal or anomalous properties, depending on the statistics of the fluctuations, with distinct regimes from those observed, e.g., in standard Continuous-Time Random Walk models.     

Random-walk model of precompound decay II: Stochastic uncertainties in the lifetimes and cross-sections

The uncertainties arising from the stochastic nature of precompound decay nuclear reactions are analyzed in the framework of the preequilibrium exciton and random-walk models. It is demonstrated that the standard deviations and the mean values of the exciton-state lifetimes are of the same order of magnitude. Their correlations are weakly positive, except for exciton states near the equilibrium number, where the correlations are significant. The usefulness and the limitations of the never-come-back approximation are discussed. A general proof is presented of the conditions under which the master-equation and random-walk approaches to Markov processes are equivalent. Connections between different preequilibrium models, e.g. the multi-step compound model and the microscopic statistical theory of precompound decay, are pointed out. It is shown that the waiting time between subsequent collisions is governed by a Poisson process, suggesting that the variance associated with the nucleon mean free path in nuclear matter, as estimated from preequilibrium models, is considerable. The stochastic uncertainties in the emission cross-sections correspond to those of a Bernoulli process.     

A model of classical thermodynamics and mesoscopic physics based on the notion of hidden parameter,Earth gravitation,and quasiclassical asymptotics. II

This paper presents a new approach to thermodynamics based on two “first principles”: the theory of partitions of integers and Earth gravitation. The self-correlated equation obtained by the author from Gentile statistics is used to describe the effect of accumulation of energy at the moment of passage from the boson branch of the partition to its fermion branch. The branch point in the passage from bosons to fermions is interpreted as an analog of a jump of the spin. A hidden parameter–the measurement time as time of the G¨odel numbering–is introduced.