Statistics of three-dimensional lagrangian turbulence

Spatial heterogeneity of air pollution statistics in Europe

Air pollution is one of the leading causes of death globally, and continues to have a detrimental effect on our health. In light of these impacts, an extensive range of statistical modelling approaches has been devised in order to better understand air pollution statistics. However, the time-varying statistics of different types of air pollutants are far from being fully understood. The observed probability density functions (PDFs) of concentrations depend very much on the spatial location and on the pollutant substance. In this paper, we analyse a large variety of data from 3544 different European monitoring sites and show that the PDFs of nitric oxide (NO), nitrogen dioxide ([Formula: see text]) and particulate matter ([Formula: see text] and [Formula: see text]) concentrations generically exhibit heavy tails and are asymptotically well approximated by q-exponential distributions with a given width parameter [Formula: see text]. We observe that the power-law parameter q and the width parameter [Formula: see text] vary widely for the different spatial locations. For each substance, we find different patterns of parameter clouds in the [Formula: see text] plane. These depend on the type of pollutants and on the environmental characteristics (urban/suburban/rural/traffic/industrial/background). This means the effective statistical physics description of air pollution exhibits a strong degree of spatial heterogeneity.

Scaling of Acceleration Statistics in High Reynolds Number Turbulence

The scaling of acceleration statistics in turbulence is examined by combining data from the literature with new data from well-resolved direct numerical simulations of isotropic turbulence, significantly extending the Reynolds number range. The acceleration variance at higher Reynolds numbers departs from previous predictions based on multifractal models, which characterize Lagrangian intermittency as an extension of Eulerian intermittency. The disagreement is even more prominent for higher-order moments of the acceleration. Instead, starting from a known exact relation, we relate the scaling of acceleration variance to that of Eulerian fourth-order velocity gradient and velocity increment statistics. This prediction is in excellent agreement with the variance data. Our Letter highlights the need for models that consider Lagrangian intermittency independent of the Eulerian counterpart.

Application of superstatistical analysis on fluctuant surface shear in particle-laden turbulence boundary layer

We report on an application of superstatistics to particle-laden turbulent flow. Four flush-mounted hot-film wall shear sensors were used to record the fluctuations of the wall shear stress in sand-laden flow. By comparing the scaling exponent in sand-free with that in sand-laden flows, we found that the sand-laden flow is more intermittent. By applying the superstatistics analysis to the friction velocity, we found that the large time scale is smaller when the flow is sand-laden. The probability density of a fluctuating energy dissipation rate measured in sand-laden flow follows a log-normal distribution with higher variances than for sand-free flow. The variance of this dissipation rate is a power law of the corresponding time scale. The prediction based on the superstatistics model is consistent with our structure function exponents [Formula: see text] for sand-free flow. Nevertheless, it overestimates [Formula: see text] for sand-laden flow, especially at higher Reynolds numbers.

Fluctuations of water quality time series in rivers follow superstatistics

Superstatistics is a general method from nonequilibrium statistical physics which has been applied to a variety of complex systems, ranging from hydrodynamic turbulence to traffic delays and air pollution dynamics. Here, we investigate water quality time series (such as dissolved oxygen concentrations and electrical conductivity) as measured in rivers and provide evidence that they exhibit superstatistical behavior. Our main example is time series as recorded in the River Chess in South East England. Specifically, we use seasonal detrending and empirical mode decomposition to separate trends from fluctuations for the measured data. With either detrending method, we observe heavy-tailed fluctuation distributions, which are well described by log-normal superstatistics for dissolved oxygen. Contrarily, we find a double peaked non-standard superstatistics for the electrical conductivity data, which we model using two combinedχ 2 -distributions.

Aerodynamic wall shear fluctuation in sand-laden flow in a turbulent boundary layer

To improve our understanding on the aerodynamic wall shear fluctuation in particle-laden flow, we perform statistical and spectral analyses of aerodynamic wall shear fluctuation in sand-free and sand-laden flows in a turbulent boundary layer based on the results of a wind tunnel experiment. An increase in the skewness of the probability density function of aerodynamic wall shear fluctuation is found in the sand-laden flow. The turbulent intensity of aerodynamic wall shear stress increases rapidly with the sand mass flux. The decreased convective velocity indicates blocking effects of sand particles in the near-wall region. The power of aerodynamic wall shear fluctuation in the sand-laden flow increases at low frequencies corresponding to the duration of sand streamers. The results of a superstatistical analysis of skin friction velocity show that spatio-temporal fluctuation in the local energy dissipation rate is enhanced in the sand-laden flow. Finally, the local and spatio-drifting force acting on the stochastic system of wall shear fluctuation in sand-laden flow are different, providing a proof for the significant variation in flow condition caused by sand streamers near the wall.

Superstatistical two-temperature Ising model

The previous approach of the nonequilibrium Ising model was based on the local temperature in which each site or part of the system has its own specific temperature. We introduce an approach of the two-temperature Ising model as a prototype of the superstatistic critical phenomena. The model is described by two temperatures (T_{1},T_{2}) in a zero magnetic field. To predict the phase diagram and numerically estimate the exponents, we develop the Metropolis and Swendsen-Wang Monte Carlo method. We observe that there is a nontrivial critical line, separating ordered and disordered phases. We propose an analytic equation for the critical line in the phase diagram. Our numerical estimation of the critical exponents illustrates that all points on the critical line belong to the ordinary Ising universality class.

Maximum entropy approach to H-theory: Statistical mechanics of hierarchical systems

A formalism, called H-theory, is applied to the problem of statistical equilibrium of a hierarchical complex system with multiple time and length scales. In this approach, the system is formally treated as being composed of a small subsystem-representing the region where the measurements are made-in contact with a set of "nested heat reservoirs" corresponding to the hierarchical structure of the system, where the temperatures of the reservoirs are allowed to fluctuate owing to the complex interactions between degrees of freedom at different scales. The probability distribution function (pdf) of the temperature of the reservoir at a given scale, conditioned on the temperature of the reservoir at the next largest scale in the hierarchy, is determined from a maximum entropy principle subject to appropriate constraints that describe the thermal equilibrium properties of the system. The marginal temperature distribution of the innermost reservoir is obtained by integrating over the conditional distributions of all larger scales, and the resulting pdf is written in analytical form in terms of certain special transcendental functions, known as the Fox H functions. The distribution of states of the small subsystem is then computed by averaging the quasiequilibrium Boltzmann distribution over the temperature of the innermost reservoir. This distribution can also be written in terms of H functions. The general family of distributions reported here recovers, as particular cases, the stationary distributions recently obtained by Macêdo et al. [Phys. Rev. E 95, 032315 (2017)10.1103/PhysRevE.95.032315] from a stochastic dynamical approach to the problem.

First-passage time for superstatistical Fokker-Planck models

The first-passage-time (FPT) problem is studied for superstatistical models assuming that the mesoscopic system dynamics is described by a Fokker-Planck equation. We show that all moments of the random intensive parameter associated to the superstatistical approach can be put in one-to-one correspondence with the moments of the FPT. For systems subjected to an additional uncorrelated external force, the same statistical information is obtained from the dependence of the FPT moments on the external force. These results provide an alternative technique for checking the validity of superstatistical models. As an example, we characterize the mean FPT for a forced Brownian particle.

Universality classes of fluctuation dynamics in hierarchical complex systems

A unified approach is proposed to describe the statistics of the short-time dynamics of multiscale complex systems. The probability density function of the relevant time series (signal) is represented as a statistical superposition of a large time-scale distribution weighted by the distribution of certain internal variables that characterize the slowly changing background. The dynamics of the background is formulated as a hierarchical stochastic model whose form is derived from simple physical constraints, which in turn restrict the dynamics to only two possible classes. The probability distributions of both the signal and the background have simple representations in terms of Meijer G functions. The two universality classes for the background dynamics manifest themselves in the signal distribution as two types of tails: power law and stretched exponential, respectively. A detailed analysis of empirical data from classical turbulence and financial markets shows excellent agreement with the theory.

Flow-induced separation in wall turbulence

One of the defining characteristics of turbulence is its ability to promote mixing. We present here a case where the opposite happens-simulation results indicate that particles can separate near the wall of a turbulent channel flow, when they have sufficiently different Schmidt numbers without use of any other means. The physical mechanism of the separation is understood when the interplay between convection and diffusion, as expressed by their characteristic time scales, is considered, leading to the determination of the necessary conditions for a successful separation between particles. Practical applications of these results can be found when very small particles need to be separated or removed from a fluid.

Acceleration of passive tracers in compressible turbulent flow

In compressible turbulence at high Reynolds and Mach numbers, shocklets emerge as a new type of flow structure in addition to intense vortices as in incompressible turbulence. Using numerical simulation of compressible homogeneous isotropic turbulence, we conduct a Lagrangian study to explore the effects of shocklets on the dynamics of passive tracers. We show that shocklets cause very strong intermittency and short correlation time of tracer acceleration. The probability density function of acceleration magnitude exhibits a -2.5 power-law scaling in the high compression region. Through a heuristic model, we demonstrate that this scaling is directly related to the statistical behavior of strong negative velocity divergence, i.e., the local compression. Tracers experience intense acceleration near shocklets, and most of them are decelerated, usually with large curvatures in their trajectories.

Multicanonical distribution: statistical equilibrium of multiscale systems

A multicanonical formalism is introduced to describe the statistical equilibrium of complex systems exhibiting a hierarchy of time and length scales, where the hierarchical structure is described as a set of nested "internal heat reservoirs" with fluctuating "temperatures." The probability distribution of states at small scales is written as an appropriate averaging of the large-scale distribution (the Boltzmann-Gibbs distribution) over these effective internal degrees of freedom. For a large class of systems the multicanonical distribution is given explicitly in terms of generalized hypergeometric functions. As a concrete example, it is shown that generalized hypergeometric distributions describe remarkably well the statistics of acceleration measurements in Lagrangian turbulence.

Generalization of the Beck-Cohen superstatistics

Generalized superstatistics, i.e., a "statistics of superstatistics," is proposed. A generalized superstatistical system comprises a set of superstatistical subsystems and represents a generalized hyperensemble. There exists a random control parameter that determines both the density of energy states and the distribution of the intensive parameter for each superstatistical subsystem, thereby forming the third, upper level of dynamics. Generalized superstatistics can be used for nonstationary nonequilibrium systems. The system in which a supercritical multitype age-dependent branching process takes place is an example of a nonstationary generalized superstatistical system. The theory is applied to pair production in a neutron star magnetosphere.

Tsallis distributions and 1/f noise from nonlinear stochastic differential equations

Probability distributions that emerge from the formalism of nonextensive statistical mechanics have been applied to a variety of problems. In this article we unite modeling of such distributions with the model of widespread 1/f noise. We propose a class of nonlinear stochastic differential equations giving both the q-exponential or q-Gaussian distributions of signal intensity, revealing long-range correlations and 1/f(β) behavior of the power spectral density. The superstatistical framework to get 1/f(β) noise with q-exponential and q-Gaussian distributions of the signal intensity is proposed, as well.

Nonparametric segmentation of nonstationary time series

The nonstationary evolution of observable quantities in complex systems can frequently be described as a juxtaposition of quasistationary spells. Given that standard theoretical and data analysis approaches usually rely on the assumption of stationarity, it is important to detect in real time series intervals holding that property. With that aim, we introduce a segmentation algorithm based on a fully nonparametric approach. We illustrate its applicability through the analysis of real time series presenting diverse degrees of nonstationarity, thus showing that this segmentation procedure generalizes and allows one to uncover features unresolved by previous proposals based on the discrepancy of low order statistical moments only.

Minding impacting events in a model of stochastic variance

We introduce a generalization of the well-known ARCH process, widely used for generating uncorrelated stochastic time series with long-term non-Gaussian distributions and long-lasting correlations in the (instantaneous) standard deviation exhibiting a clustering profile. Specifically, inspired by the fact that in a variety of systems impacting events are hardly forgot, we split the process into two different regimes: a first one for regular periods where the average volatility of the fluctuations within a certain period of time is below a certain threshold, , and another one when the local standard deviation outnumbers . In the former situation we use standard rules for heteroscedastic processes whereas in the latter case the system starts recalling past values that surpassed the threshold. Our results show that for appropriate parameter values the model is able to provide fat tailed probability density functions and strong persistence of the instantaneous variance characterized by large values of the Hurst exponent (H>0.8), which are ubiquitous features in complex systems.

Anomalous scaling for Lagrangian velocity structure functions in fully developed turbulence

A hierarchical structure model is developed for anomalous scaling of the Lagrangian velocity structure functions in fully developed turbulence. This model is an extension of the Eulerian hierarchical structure model of She and Leveque [Phys. Rev. Lett. 72, 366 (1994)] to the Lagrangian velocity structure functions, where the straining and sweeping hypotheses are used to build up the relationship between the singular scalings of Lagrangian and Eulerian intermittent structures. The Lagrangian scaling exponents obtained from the straining hypothesis are in good agreement with the experimental results of the Bodenschatz group.

Generalized statistical mechanics for superstatistical systems

Mesoscopic systems in a slowly fluctuating environment are often well described by superstatistical models. We develop a generalized statistical mechanics formalism for superstatistical systems, by mapping the superstatistical complex system onto a system of ordinary statistical mechanics with modified energy levels. We also briefly review recent examples of applications of the superstatistics concept for three very different subject areas, namely train delay statistics, turbulent tracer dynamics and cancer survival statistics.

Fluctuations of entropy and log-normal superstatistics

Nonequilibrium complex systems are often effectively described by the mixture of different dynamics on different time scales. Superstatistics, which is "statistics of statistics" with two largely separated time scales, offers a consistent theoretical framework for such a description. Here, a theory is developed for log-normal superstatistics based on the fluctuation theorem for entropy changes as well as the maximum entropy method. This gives novel physical insight into log-normal statistics, other than the traditional multiplicative random processes. A comment is made on a possible application of the theory to the fluctuating energy dissipation rate in turbulence.

Generalized Fisher information matrix in nonextensive systems with spatial correlation

By using the q -Gaussian distribution derived by the maximum entropy method for spatially correlated N -unit nonextensive systems, we have calculated the generalized Fisher information matrix of gthetanthetam for (theta1,theta2,theta3)=(muq,sigmaq2,s) , where muq, sigmaq2, and s denote the mean, variance, and degree of spatial correlation, respectively, for a given entropic index q . It has been shown from the Cramér-Rao theorem that (1) an accuracy of an unbiased estimate of muq is improved (degraded) by a negative (positive) correlation s, (2) that of sigmaq2 is worsen with increasing s, and (3) that of s is much improved for s approximately= -1/(N-1) or s approximately 1.0 though it is worst at s=(N-2)/2(N-1). Our calculation provides a clear insight to the long-standing controversy whether the spatial correlation is beneficial or detrimental to decoding in neuronal ensembles. We discuss also a calculation of the q-Gaussian distribution applying the superstatistics to the Langevin model subjected to spatially correlated inputs.

Superstatistical fluctuations in time series: applications to share-price dynamics and turbulence

We report a general technique to study a given experimental time series with superstatistics. Crucial for the applicability of the superstatistics concept is the existence of a parameter beta that fluctuates on a large time scale as compared to the other time scales of the complex system under consideration. The proposed method extracts the main superstatistical parameters out of a given data set and examines the validity of the superstatistical model assumptions. We test the method thoroughly with surrogate data sets. Then the applicability of the superstatistical approach is illustrated using real experimental data. We study two examples, velocity time series measured in turbulent Taylor-Couette flows and time series of log returns of the closing prices of some stock market indices.

Bose-Einstein and Fermi-Dirac distributions in nonextensive quantum statistics: exact and interpolation approaches

Generalized Bose-Einstein and Fermi-Dirac distributions in nonextensive quantum statistics have been discussed by the maximum-entropy method (MEM) with the optimum Lagrange multiplier based on the exact integral representation [A. K. Rajagopal, R. S. Mendes, and E. K. Lenzi, Phys. Rev. Lett. 80, 3907 (1998)]. It has been shown that the (q-1) expansion in the exact approach agrees with the result obtained by the asymptotic approach valid for O(q-1). Model calculations have been made with a uniform density of states for electrons and with the Debye model for phonons. Based on the result of the exact approach, we have proposed the interpolation approximation to the generalized distributions, which yields results in agreement with the exact approach within O(q-1) and in high- and low-temperature limits. By using the four methods of the exact, interpolation, factorization, and superstatistical approaches, we have calculated coefficients in the generalized Sommerfeld expansion and electronic and phonon specific heats at low temperatures. A comparison among the four methods has shown that the interpolation approximation is potentially useful in the nonextensive quantum statistics. Supplementary discussions have been made on the (q-1) expansion of the generalized distributions based on the exact approach with the use of the un-normalized MEM, whose results also agree with those of the asymptotic approach.

Superstatistical distributions from a maximum entropy principle

We deal with a generalized statistical description of nonequilibrium complex systems based on least biased distributions given some prior information. A maximum entropy principle is introduced that allows for the determination of the distribution of the fluctuating intensive parameter beta of a superstatistical system, given certain constraints on the complex system under consideration. We apply the theory to three examples: the superstatistical quantum-mechanical harmonic oscillator, the superstatistical classical ideal gas, and velocity time series as measured in a turbulent Taylor-Couette flow.

Lagrangian statistical model for transport in highly heterogeneous velocity fields

We define an effective Lagrangian statistical model in phase space (x, t, v) for describing transport in highly heterogeneous velocity fields with complex spatial organizations. The spatial Markovian nature (and temporal non-Markovian nature) of Lagrangian velocities leads to an effective transport description that turns out to be a correlated continuous time random walk. This model correctly captures the Lagrangian velocity correlation properties and is demonstrated to represent a forward model for predicting transport in highly heterogeneous porous media for different types of velocity organizations.

Lagrangian statistical theory of fully developed hydrodynamical turbulence

The Lagrangian velocity structure functions in the inertial range of fully developed fluid turbulence are for the first time derived based on the Navier-Stokes equation. For time tau much smaller than the correlation time, the structure functions are shown to obey the scaling relations K_{n}(tau) proportional, varianttau;{zeta_{n}}. The scaling exponents zeta_{n} are calculated analytically without any fitting parameters. The obtained values are in amazing agreement with the experimental results of the Bodenschatz group. A new relation connecting the Lagrangian structure functions of different orders analogously to the extended self-similarity ansatz is found.

Superstatistics, thermodynamics, and fluctuations

A thermodynamiclike formalism is developed for superstatistical systems based on conditional entropies. This theory takes into account large-scale variations of intensive variables of systems in nonequilibrium stationary states. Ordinary thermodynamics is recovered as a special case of the present theory, and corrections to it can systematically be evaluated. A generalization of Einstein's relation for fluctuations is presented using a maximum entropy condition.