Discrete Sampling of Extreme Events Modifies Their Statistics.

Extreme value (EV) statistics of correlated systems are widely investigated in many fields, spanning the spectrum from weather forecasting to earthquake prediction. Does the unavoidable discrete sampling of a continuous correlated stochastic process change its EV distribution? We explore this question for correlated random variables modeled via Langevin dynamics for a particle in a potential field. For potentials growing at infinity faster than linearly and for long measurement times, we find that the EV distribution of the discretely sampled process diverges from that of the full continuous dataset and converges to that of independent and identically distributed random variables drawn from the process's equilibrium measure. However, for processes with sublinear potentials, the long-time limit is the EV statistics of the continuously sampled data. We treat processes whose equilibrium measures belong to the three EV attractors: Gumbel, Fréchet, and Weibull. Our Letter shows that the EV statistics can be extremely sensitive to the sampling rate of the data.

Gas of sub-recoiled laser cooled atoms described by infinite ergodic theory.

The velocity distribution of a classical gas of atoms in thermal equilibrium is the normal Maxwell distribution. It is well known that for sub-recoiled laser cooled atoms, Lévy statistics and deviations from usual ergodic behavior come into play. In a recent letter, we showed how tools from infinite ergodic theory describe the cool gas. Here, using the master equation, we derive the scaling function and the infinite invariant density of a stochastic model for the momentum of laser cooled atoms, recapitulating results obtained by Bertin and Bardou [Am. J. Phys. 76, 630 (2008)] using life-time statistics. We focus on the case where the laser trapping is strong, namely, the rate of escape from the velocity trap is R(v) ∝ |v|α for v → 0 and α > 1. We construct a machinery to investigate time averages of physical observables and their relation to ensemble averages. The time averages are given in terms of functionals of the individual stochastic paths, and here we use a generalization of Lévy walks to investigate the ergodic properties of the system. Exploring the energy of the system, we show that when α = 3, it exhibits a transition between phases where it is either an integrable or a non-integrable observable with respect to the infinite invariant measure. This transition corresponds to very different properties of the mean energy and to a discontinuous behavior of fluctuations. While the integrable phase is described by universal statistics and the Darling-Kac law, the more challenging case is the exploration of statistical properties of non-integrable observables. Since previous experimental work showed that both α = 2 and α = 4 are attainable, we believe that both phases could also be explored experimentally.

Transitions in the Ergodicity of Subrecoil-Laser-Cooled Gases.

With subrecoil-laser-cooled atoms, one may reach nanokelvin temperatures while the ergodic properties of these systems do not follow usual statistical laws. Instead, due to an ingenious trapping mechanism in momentum space, power-law-distributed sojourn times are found for the cooled particles. Here, we show how this gives rise to a statistical-mechanical framework based on infinite ergodic theory, which replaces ordinary ergodic statistical physics of a thermal gas of atoms. In particular, the energy of the system exhibits a sharp discontinuous transition in its ergodic properties. Physically, this is controlled by the fluorescence rate, but, more profoundly, it is a manifestation of a transition for any observable, from being an integrable to becoming a nonintegrable observable, with respect to the infinite (non-normalized) invariant density.

Uncertainty Relation between Detection Probability and Energy Fluctuations.

A classical random walker starting on a node of a finite graph will always reach any other node since the search is ergodic, namely it fully explores space, hence the arrival probability is unity. For quantum walks, destructive interference may induce effectively non-ergodic features in such search processes. Under repeated projective local measurements, made on a target state, the final detection of the system is not guaranteed since the Hilbert space is split into a bright subspace and an orthogonal dark one. Using this we find an uncertainty relation for the deviations of the detection probability from its classical counterpart, in terms of the energy fluctuations.

Non-Normalizable Quasi-Equilibrium Solution of the Fokker-Planck Equation for Nonconfining Fields.

We investigate the overdamped Langevin motion for particles in a potential well that is asymptotically flat. When the potential well is deep as compared to the temperature, physical observables, like the mean square displacement, are essentially time-independent over a long time interval, the stagnation epoch. However, the standard Boltzmann-Gibbs (BG) distribution is non-normalizable, given that the usual partition function is divergent. For this regime, we have previously shown that a regularization of BG statistics allows for the prediction of the values of dynamical and thermodynamical observables in the non-normalizable quasi-equilibrium state. In this work, based on the eigenfunction expansion of the time-dependent solution of the associated Fokker-Planck equation with free boundary conditions, we obtain an approximate time-independent solution of the BG form, being valid for times that are long, but still short as compared to the exponentially large escape time. The escaped particles follow a general free-particle statistics, where the solution is an error function, which is shifted due to the initial struggle to overcome the potential well. With the eigenfunction solution of the Fokker-Planck equation in hand, we show the validity of the regularized BG statistics and how it perfectly describes the time-independent regime though the quasi-stationary state is non-normalizable.

Fractional Advection-Diffusion-Asymmetry Equation.

Fractional kinetic equations employ noninteger calculus to model anomalous relaxation and diffusion in many systems. While this approach is well explored, it so far failed to describe an important class of transport in disordered systems. Motivated by work on contaminant spreading in geological formations, we propose and investigate a fractional advection-diffusion equation describing the biased spreading packet. While usual transport is described by diffusion and drift, we find a third term describing symmetry breaking which is omnipresent for transport in disordered systems. Our work is based on continuous time random walks with a finite mean waiting time and a diverging variance, a case that on the one hand is very common and on the other was missing in the kaleidoscope literature of fractional equations. The fractional space derivatives stem from long trapping times, while previously they were interpreted as a consequence of spatial Lévy flights.

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 also holds for short times, a fact that makes experimental observations readily achievable.

Large Deviations for Continuous Time Random Walks.

Recently observation of random walks in complex environments like the cell and other glassy systems revealed that the spreading of particles, at its tails, follows a spatial exponential decay instead of the canonical Gaussian. We use the widely applicable continuous time random walk model and obtain the large deviation description of the propagator. Under mild conditions that the microscopic jump lengths distribution is decaying exponentially or faster i.e., Lévy like power law distributed jump lengths are excluded, and that the distribution of the waiting times is analytical for short waiting times, the spreading of particles follows an exponential decay at large distances, with a logarithmic correction. Here we show how anti-bunching of jump events reduces the effect, while bunching and intermittency enhances it. We employ exact solutions of the continuous time random walk model to test the large deviation theory.

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, e.g., 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.

First Detected Arrival of a Quantum Walker on an Infinite Line.

The first detection of a quantum particle on a graph is shown to depend sensitively on the distance ξ between the detector and initial location of the particle, and 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 large ξ limit, the small τ limit, and the power law decay with the attempt number of the detection probability over which quantum oscillations are superimposed. For large ξ the first detection probability assumes a scaling form and when the sampling time is equal to the inverse of the energy band width nonanalytical 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 that is distance independent.

Large Fluctuations for Spatial Diffusion of Cold Atoms.

We use a new approach to study the large fluctuations of a heavy-tailed system, where the standard large-deviations principle does not apply. 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 falloff 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 non-normalized infinite-covariant density approach. The small and large fluctuations give rise to a bifractal nature of the spreading packet. We derive general relations which extend our theory to a class of systems with multifractal moments.

Universal Fluctuations of Single-Particle Diffusivity in a Quenched Environment.

Local diffusion coefficients in disordered materials such as living cells are highly heterogeneous. 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 square displacement depends crucially on the disorder; i.e., the system is ergodic but non-self-averaging. Moreover, we show that the disorder average of the time-averaged mean square displacement decreases with the system size. We find a universal distribution for diffusivity in the sense that the shape of the distribution does not depend on the dimension. 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.

Deviations from Boltzmann-Gibbs Statistics in Confined Optical Lattices.

We investigate the semiclassical phase-space probability distribution P(x,p) of cold atoms in a Sisyphus cooling lattice with an additional harmonic confinement. We pose the question of whether this nonequilibrium steady state satisfies the equivalence of energy and probability. This equivalence is the foundation of Boltzmann-Gibbs and generalized thermostatic statistics, and a prerequisite for the description in terms of a temperature. At large energies, P(x,p) depends only on the Hamiltonian H(x,p) and the answer to the question is yes. In distinction to the Boltzmann-Gibbs state, the large-energy tails are power laws P(x,p)∝H(x,p)(-1/D), where D is related to the depth of the optical lattice. At intermediate energies, however, P(x,p) cannot be expressed as a function of the Hamiltonian and the equivalence between energy and probability breaks down. As a consequence the average potential and kinetic energy differ and no well-defined temperature can be assigned. The Boltzmann-Gibbs state is regained only in the limit of deep optical lattices. For strong confinement relative to the damping, we derive an explicit expression for the stationary phase-space distribution.

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

Strong anomalous diffusion, where ⟨|x(t)|(q)⟩ ∼ tqν(q) with a nonlinear spectrum ν(q) ≠ 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 phenomenon 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 multifractal anomalous diffusion, as it is complementary to the central limit theorem.

Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking.

Modern microscopic techniques following the stochastic motion of labelled tracer particles have uncovered significant deviations from the laws of Brownian motion in a variety of animate and inanimate systems. Such anomalous diffusion can have different physical origins, which can be identified from careful data analysis. In particular, single particle tracking provides the entire trajectory of the traced particle, which allows one to evaluate different observables to quantify the dynamics of the system under observation. We here provide an extensive overview over different popular anomalous diffusion models and their properties. We pay special attention to their ergodic properties, highlighting the fact that in several of these models the long time averaged mean squared displacement shows a distinct disparity to the regular, ensemble averaged mean squared displacement. In these cases, data obtained from time averages cannot be interpreted by the standard theoretical results for the ensemble averages. Here we therefore provide a comparison of the main properties of the time averaged mean squared displacement and its statistical behaviour in terms of the scatter of the amplitudes between the time averages obtained from different trajectories. We especially demonstrate how anomalous dynamics may be identified for systems, which, on first sight, appear to be Brownian. Moreover, we discuss the ergodicity breaking parameters for the different anomalous stochastic processes and showcase the physical origins for the various behaviours. This Perspective is intended as a guidebook for both experimentalists and theorists working on systems, which exhibit anomalous diffusion.

Stretched-exponential relaxation in weakly confined Brownian systems through large deviation theory.

Stretched-exponential relaxation is a widely observed phenomenon found in ordered ferromagnets as well as glassy systems. One modeling approach connects this behavior to a droplet dynamics described by an effective Langevin equation for the droplet radius with an r^{2/3} potential. Here, we study a Brownian particle under the influence of a general confining, albeit weak, potential field that grows with distance as a sublinear power law. We find that for this memoryless model, observables display stretched-exponential relaxation. The probability density function of the system is studied using a rate-function ansatz. We obtain analytically the stretched-exponential exponent along with an anomalous power-law scaling of length with time. The rate function exhibits a point of nonanalyticity, indicating a dynamical phase transition. In particular, the rate function is double valued both to the left and right of this point, leading to four different rate functions, depending on the choice of initial conditions and symmetry.

Fluctuations of 1/f noise and the low-frequency cutoff paradox.

Recent experiments on blinking quantum dots, weak turbulence in liquid crystals, and nanoelectrodes reveal the fundamental connection between 1/f noise and power law intermittency. The nonstationarity of the process implies that the power spectrum is random--a manifestation of weak ergodicity breaking. Here, we obtain the universal distribution of the power spectrum, which can be used to identify intermittency as the source of the noise. We solve in this case an outstanding paradox on the nonintegrability of 1/f noise and the violation of Parseval's theorem. We explain why there is no physical low-frequency cutoff and therefore why it cannot be found in experiments.

Aging effects and population splitting in single-particle trajectory averages.

We study time averages of single particle trajectories in scale-free anomalous diffusion processes, in which the measurement starts at some time t(a)>0 after initiation of the process at t=0. Using aging renewal theory, we show that for such nonstationary processes a large class of observables are affected by a unique aging function, which is independent of boundary conditions or the external forces. Moreover, we discuss the implications of aging induced population splitting: with growing age t(a) of the process, an increasing fraction of particles remains motionless in a measurement of fixed duration. Consequences for single biomolecule tracking in live cells are discussed.

Noisy continuous time random walks.

Experimental studies of the diffusion of biomolecules within biological cells are routinely confronted with multiple sources of stochasticity, whose identification renders the detailed data analysis of single molecule trajectories quite intricate. Here, we consider subdiffusive continuous time random walks that represent a seminal model for the anomalous diffusion of tracer particles in complex environments. This motion is characterized by multiple trapping events with infinite mean sojourn time. In real physical situations, however, instead of the full immobilization predicted by the continuous time random walk model, the motion of the tracer particle shows additional jiggling, for instance, due to thermal agitation of the environment. We here present and analyze in detail an extension of the continuous time random walk model. Superimposing the multiple trapping behavior with additive Gaussian noise of variable strength, we demonstrate that the resulting process exhibits a rich variety of apparent dynamic regimes. In particular, such noisy continuous time random walks may appear ergodic, while the bare continuous time random walk exhibits weak ergodicity breaking. Detailed knowledge of this behavior will be useful for the truthful physical analysis of experimentally observed subdiffusion.

Statistics of long-range force fields in random environments: Beyond Holtsmark.

Since the times of Holtsmark (1911), statistics of fields in random environments have been widely studied, for example in astrophysics, active matter, and line-shape broadening. The power-law decay of the two-body interaction of the form 1/|r|^{δ}, and assuming spatial uniformity of the medium particles exerting the forces, imply that the fields are fat-tailed distributed, and in general are described by stable Lévy distributions. With this widely used framework, the variance of the field diverges, which is nonphysical, due to finite size cutoffs. We find a complementary statistical law to the Lévy-Holtsmark distribution describing the large fields in the problem, which is related to the finite size of the tracer particle. We discover biscaling with a sharp statistical transition of the force moments taking place when the order of the moment is d/δ, where d is the dimension. The high-order moments, including the variance, are described by the framework presented in this paper, which is expected to hold for many systems. The new scaling solution found here is nonnormalized similar to infinite invariant densities found in dynamical systems.