Spatiotemporal Correlation Function Research Articles

Effective Spatial Degrees of Freedom of Natural Temperature Variability as a Function of Frequency

Abstract A fundamental statistic of climate variability is its spatiotemporal correlation function. Its complex structure can be concisely summarized by a frequency-dependent measure of the effective spatial degrees of freedom (ESDOF). Here we present, for the first time, frequency-dependent ESDOF estimates of global natural surface temperature variability from purely instrumental measurements, using the HadCRUT4 dataset (1850–2014). The approach is based on a newly developed method for estimating the frequency-dependent spatial correlation function from gappy data fields. Results reveal a multicomponent structure of the spatial correlation function, including a large-amplitude short-distance component (with weak time scale dependence) and a small-amplitude long-distance component (with increasing relative amplitude toward the longer time scales). Two frequency-dependent ESDOF measures are applied, each responding mainly to either of the two components. Both measures exhibit a significant ESDOF reduction from monthly to multidecadal time scales, implying an increase of the effective spatial scale of natural surface temperature fluctuations. Moreover, it is found that a good approximation to the global number of equally spaced samples needed to estimate the variance of global mean temperature is given, at any frequency, by the greater one of the two ESDOF measures, decreasing from ∼130 at monthly to ∼30 at multidecadal time scales. Finally, the multicomponent structure of the correlation function together with the detected ESDOF scaling properties indicate that the ESDOF reduction toward the longer time scales cannot be explained simply by diffusion acting on stochastically driven anomalies, as it might be suggested from simple stochastic-diffusive energy balance models.

Circuits of space and time quantum channels

Exact solutions in interacting many-body systems are scarce but extremely valuable since they provide insights into the dynamics. Dual-unitary models are examples in one spatial dimension where this is possible. These brick-wall quantum circuits consist of local gates, which remain unitary not only in time, but also when interpreted as evolutions along the spatial directions. However, this setting of unitary dynamics does not directly apply to real-world systems due to their imperfect isolation, and it is thus imperative to consider the impact of noise to dual-unitary dynamics and its exact solvability. In this work we generalise the ideas of dual-unitarity to obtain exact solutions in noisy quantum circuits, where each unitary gate is substituted by a local quantum channel. Exact solutions are obtained by demanding that the noisy gates yield a valid quantum channel not only in time, but also when interpreted as evolutions along one or both of the spatial directions and possibly backwards in time. This gives rise to new families of models that satisfy different combinations of unitality constraints along the space and time directions. We provide exact solutions for the spatio-temporal correlation functions, spatial correlations after a quantum quench, and the structure of steady states for these families of models. We show that noise unbiased around the dual-unitary family leads to exactly solvable models, even if dual-unitarity is strongly violated. We prove that any channel unital in both space and time directions can be written as an affine combination of a particular class of dual-unitary gates. Finally, we extend the definition of solvable initial states to matrix-product density operators. We completely classify them when their tensor admits a local purification.

Liquid Structures and Diffusion Dynamics of Alkyl-Pyrene Liquids Studied by Molecular Dynamics Simulations.

Functional molecular liquids (FMLs) based on alkylated π-conjugated molecules have attracted attention as solvent-free and nonvolatile liquid materials with prominent optoelectronic features. Recently, novel FML compounds containing pyrene as the functional core were synthesized, and their rheological and photochemical properties were investigated. Although the molecules differ only in the number of alkyl chain substituents and their substitution positions, their viscosity coefficients are largely different beyond the Stokes-Einstein relation on the assumption of identical microscopic friction, indicating that local microscopic molecular interactions are crucial for the macroscopic rheological properties. Here, we report a theoretical study on the rheological properties of the alkyl-pyrene liquids by means of atomistic molecular dynamics (MD) simulations. We performed long-time MD simulations for tens of microseconds to obtain ample statistical samples of the alkyl-pyrene liquids and analyzed their liquid structures and diffusion dynamics based on spatiotemporal correlation functions. We found the formation of characteristic local liquid structures of π-π stacking of the pyrene moieties and locally anisotropic and anomalous diffusion dynamics, which remarkably vary depending on the alkyl substituent patterns. The present results provide an atomistic insight into the macroscopic rheological properties of alkyl-π FMLs and molecular design strategy for them.

Turbulence video simulation using the spatiotemporal cross correlation of distortion fields.

Previous simulations of atmospheric turbulence in videos are computationally complex. The purpose of this study is to develop an efficient algorithm for simulating spatiotemporal videos affected by atmospheric turbulence, given a static image. We extend an existing method for the simulation of atmospheric turbulence in a single image by incorporating turbulence properties in the time domain and the blurring effect. We accomplish this through analysis of the correlation between turbulence image distortions in time and in space. The significance of this method is the ease with which it will be possible to produce a simulation, given properties of the turbulence (including turbulence strength, object distance, and height). We apply the simulation to low and high frame rate videos, and we show that the spatiotemporal cross correlation of the distortion fields in the simulated video matches the physical spatiotemporal cross correlation function. Such a simulation can be useful when developing algorithms that apply to videos degraded by atmospheric turbulence and require a large amount of imaging data for training.

Large‐Domain Multisite Precipitation Generation: Operational Blueprint and Demonstration for 1,000 Sites

AbstractStochastic simulations of spatiotemporal patterns of hydroclimatic processes, such as precipitation, are needed to build alternative but equally plausible inputs for water‐related design and management, and to estimate uncertainty and assess risks. However, while existing stochastic simulation methods are mature enough to deal with relatively small domains and coarse spatiotemporal scales, additional work is required to develop simulation tools for large‐domain analyses, which are more and more common in an increasingly interconnected world. This study proposes a methodological advancement in the CoSMoS framework, which is a flexible simulation framework preserving arbitrary marginal distributions and correlations, to dramatically decrease the computational burden and make the algorithm fast enough to perform large‐domain simulations in short time. The proposed approach focuses on correlated processes with mixed (zero‐inflated) Uniform marginal distributions. These correlated processes act as intermediates between the target process to simulate (precipitation) and parent Gaussian processes that are the core of the simulation algorithm. Working in the mixed‐Uniform space enables a substantial simplification of the so‐called correlation transformation functions, which represent a computational bottle neck in the original CoSMoS formulation. As a proof of concept, we simulate 40 years of daily precipitation records from 1,000 gauging stations in the Mississippi River basin. Moreover, we extend CoSMoS incorporating parent non‐Gaussian processes with different degrees of tail dependence and suggest potential improvements including the separate simulation of occurrence and intensity processes, and the use of advection, anisotropy, and nonstationary spatiotemporal correlation functions.

Spatiotemporal characteristics of dynamic speckle from a 3D target in atmospheric turbulence

Spatiotemporal correlation function is the basic characteristic of dynamic laser speckle and the basis of various applications. The correlation function of the speckle intensity from a 3D target in turbulent atmosphere is derived based on the model of random phase screen and Fresnel Kirchhoff Diffraction Formula, and a fast algorithm based on FFT is developed. The particularity of dynamic speckle in turbulence is numerically analyzed and discussed. The results show that the speckle intensity fluctuates at two independent scales both in space and time domain, which are affected by target size and atmospheric turbulence respectively. In particular, the time scale caused by turbulence is also affected by the target translating velocity and the wind direction. The theory and algorithm developed in this paper can play important roles in applications of laser speckle such as remote detection in atmospheric environment.

Dual unitary circuits in random geometries

Recently introduced dual unitary brickwork circuits have been recognised as paradigmatic exactly solvable quantum chaotic many-body systems with tunable degree of ergodicity and mixing. Here we show that regularity of the circuit lattice is not crucial for exact solvability. We consider a circuit where random 2-qubit dual unitary gates sit at intersections of random arrangements of straight lines in two dimensions (mikado) and analytically compute the variance of the spatio-temporal correlation function of local operators. Note that the average correlator vanishes due to local Haar randomness of the gates. The result can be physically motivated for two random mikado settings. The first corresponds to the thermal state of free particles carrying internal qubit degrees of freedom which experience interaction at kinematic crossings, while the second represents rotationally symmetric (random euclidean) space-time.

Spatiotemporal dynamics of classical and quantum density profiles in low-dimensional spin systems

We provide a detailed comparison between the dynamics of high-temperature spatiotemporal correlation functions in quantum and classical spin models. In the quantum case, our large-scale numerics are based on the concept of quantum typicality, which exploits the fact that random pure quantum states can faithfully approximate ensemble averages, allowing the simulation of spin-$1/2$ systems with up to $40$ lattice sites. Due to the exponentially growing Hilbert space, we find that for such system sizes even a single random state is sufficient to yield results with extremely low noise that is negligible for most practical purposes. In contrast, a classical analog of typicality is missing. In particular, we demonstrate that, in order to obtain data with a similar level of noise in the classical case, extensive averaging over classical trajectories is required, no matter how large the system size. Focusing on (quasi-)one-dimensional spin chains and ladders, we find a remarkably good agreement between quantum and classical dynamics. This applies not only to cases where both the quantum and classical model are nonintegrable, but also to cases where the quantum spin-$1/2$ model is integrable and the corresponding classical $s\to\infty$ model is not. Our analysis is based on the comparison of space-time profiles of the spin and energy correlation functions, where the agreement is found to hold not only in the bulk but also in the tails of the resulting density distribution. The mean-squared displacement of the density profiles reflects the nature of emerging hydrodynamics and is found to exhibit similar scaling for quantum and classical models.

Determining Ionospheric Drift and Anisotropy of Irregularities from LOFAR Core Measurements: Testing Hypotheses behind Estimation

We try to assess the validity of assumptions taken when deriving drift velocity. We give simple formulas for characteristics of the spatiotemporal correlation function of the observed diffraction pattern for the frozen flow and the more general Briggs model. Using Low-Frequency Array (LOFAR) Cassiopeia intensity observation, we compare the experimental velocity scaling factor with a theoretical one to show that both models do not follow observations. We also give a qualitative comparison of our drift velocity estimates with SuperDARN convection maps. The article is essentially an extended version of the conference paper: “Determining ionospheric drift and anisotropy of irregularities from LOFAR core measurements”, Signal Processing Symposium 2021 (SPSympo 2021).

Shock-Induced Mesoparticles and Turbulence Occurrence

The development of a new approach to describe turbulent motions in condensed matter on the basis of nonlocal modeling of highly non-equilibrium processes in open systems is performed in parallel with an experiment studying the mesostructure of dynamically deformed solids. The shock-induced mesostructure formation inside the propagating waveform registered in real time allows the transient stages of non-equilibrium processes to be qualitatively and quantitatively revealed. A new nonlocal approach, developed on the basis of the nonlocal and retarded transport equations obtained within the non-equilibrium statistical physics, is used to describe the occurrence of turbulence. Within the approach, the reason for the transition to turbulence is that the non-equilibrium spatiotemporal correlation function generates the dynamic structures in the form of finite-size clusters on the mesoscale, with almost identical values of macroscopic densities moving as almost solid particles that can interact and rotate. The fragmentation of spatiotemporal correlations upon impact forms the mesoparticles that move at different speeds and transfer mass, momentum and energy-like wave packets. The movements recorded simultaneously at two scale levels indicate the energy exchange between them. Its description required a redefinition of the concept of energy far from local thermodynamic equilibrium. The experimental results show that the irreversible part of the dynamic mesostructure remains frozen into material as a new defect.

On two reversible cellular automata with two particle species

We introduce a pair of time-reversible models defined on the discrete space–time lattice with three states per site, specifically, a vacancy and a particle of two flavours (species). The local update rules reproduce the rule 54 reversible cellular automaton when only a single species of particles is present, and satisfy the requirements of flavour exchange (C), space-reversal (P), and time-reversal (T) symmetries. We find closed-form expressions for three local conserved charges and provide an explicit matrix product form of the grand canonical Gibbs states, which are identical for both models. For one of the models this family of Gibbs states seems to be a complete characterisation of equilibrium (i.e. space and time translation invariant) states, while for the other model we empirically find a sequence of local conserved charges, one for each support size larger than 2, hinting to its algebraic integrability. Finally, we numerically investigate the behaviour of spatio-temporal correlation functions of charge densities, and test the hydrodynamic prediction for the model with exactly three local charges. Surprisingly, the numerically observed ‘sound velocity’ does not match the hydrodynamic value. The deviations are either significant, or they decay extremely slowly with the simulation time, which leaves us with an open question for the mechanism of such a glassy behaviour in a deterministic locally interacting system.

Spatio-Temporal Correlation-Based False Data Injection Attack Detection Using Deep Convolutional Neural Network

There are lots of cyber-attack, especially false data injection attacks, in modern power systems. This attack can circumvent traditional residual-based detection methods, and destroy the integrity of control information, thus hindering the stability of the power system. In this paper, a novel Spatio-temporal detection mechanism is proposed to evaluate and locate false data injection attacks. In the proposed method, temporal correlation and spatial correlation are analyzed by cubature Kalman filter and Gaussian process regression, respectively, to capture the dynamic features of state vectors. Then, a deep convolutional neural network is trained to depict the functional relationship between Spatio-temporal correlation functions and the output, which is set as the detection indicator to access whether the power system under attack or not. Furthermore, the performance of the proposed mechanism is evaluated with comprehensive numerical simulation on IEEE 39-bus test system. The results of the case studies showed that the proposed method can achieve 99.84%-100% accuracy.

Wind noise performance analysis of a prototype compact infrasound array

Previously, presentations have been given at ASA meetings that demonstrated the potential for compact arrays of infrasound sensors achieving high gain against wind noise under appropriate environmental conditions without the use of large windscreens. These presentations focused on using specific models for the spatiotemporal correlation function of the wind noise as part of the processing methodology, and they did not directly address the features of the compact array design that influence performance. This presentation will study the effects of several compact array features on performance using a recently published cross-spectral density model for wall-bounded turbulent pressure fluctuations. Additionally, data from several experiments will be presented including data from a prototype compact infrasound array.

Simulating Hydrodynamics on Noisy Intermediate-Scale Quantum Devices with Random Circuits.

In a recent milestone experiment, Google's processor Sycamore heralded the era of "quantum supremacy" by sampling from the output of (pseudo-)random circuits. We show that such random circuits provide tailor-made building blocks for simulating quantum many-body systems on noisy intermediate-scale quantum (NISQ) devices. Specifically, we propose an algorithm consisting of a random circuit followed by a trotterized Hamiltonian time evolution to study hydrodynamics and to extract transport coefficients in the linear response regime. We numerically demonstrate the algorithm by simulating the buildup of spatiotemporal correlation functions in one- and two-dimensional quantum spin systems, where we particularly scrutinize the inevitable impact of errors present in any realistic implementation. Importantly, we find that the hydrodynamic scaling of the correlations is highly robust with respect to the size of the Trotter step, which opens the door to reach nontrivial time scales with a small number of gates. While errors within the random circuit are shown to be irrelevant, we furthermore unveil that meaningful results can be obtained for noisy time evolutions with error rates achievable on near-term hardware. Our work emphasizes the practical relevance of random circuits on NISQ devices beyond the abstract sampling task.

Spatio-temporal correlations in three-dimensional homogeneous and isotropic turbulence

We use direct numerical simulations (DNSs) of the forced Navier–Stokes equation for a three-dimensional incompressible fluid in order to test recent theoretical predictions. We study the two- and three-point spatiotemporal correlation functions of the velocity field in stationary, isotropic, and homogeneous turbulence. We compare our numerical results to the predictions from the Functional Renormalization Group (FRG) which were obtained in the large wavenumber limit. DNSs are performed at various Reynolds numbers and the correlations are analyzed in different time regimes focusing on the large wavenumbers. At small time delays, we find that the two-point correlation function decays as a Gaussian in the variable kt, where k is the wavenumber and t the time delay. We compute a triple correlation from the time-dependent advection-velocity correlations, and find that it also follows a Gaussian decay at small t with the same prefactor as the one of the two-point functions. These behaviors are in precise agreement with the FRG results, and can be simply understood as a consequence of sweeping. At large time delays, the FRG predicts a crossover to an exponential in k2t, which we were not able to resolve in our simulations. However, we analyze the two-point spatiotemporal correlations of the modulus of the velocity and show that they exhibit this crossover from a Gaussian to an exponential decay, although we lack of a theoretical understanding in this case. This intriguing phenomenon calls for further theoretical investigation.

Correlations in Perturbed Dual-Unitary Circuits: Efficient Path-Integral Formula

Interacting many-body systems with explicitly accessible spatio-temporal correlation functions are extremely rare, especially in the absence of integrability. Recently, we identified a remarkable class of such systems and termed them dual-unitary quantum circuits. These are brick-wall type local quantum circuits whose dynamics are unitary in both time and space. For these systems the spatio-temporal correlation functions are non-trivial only at the edge of the causal light cone and can be computed in terms of one-dimensional transfer matrices. Dual-unitarity, however, requires fine-tuning and the degree of generality of the observed dynamical features remained unclear. Here we address this question by introducing arbitrary perturbations of the local gates. Considering fixed perturbations, we prove that for a particular class of unperturbed elementary dual-unitary gates the correlation functions are still expressed in terms of one-dimensional transfer matrices. These matrices, however, are now contracted over generic paths connecting the origin to a fixed endpoint inside the causal light cone. The correlation function is given as a sum over all such paths. Our statement is rigorous in the "dilute limit", where only a small fraction of the gates is perturbed, and in the presence of random longitudinal fields, but we provide theoretical arguments and stringent numerical checks supporting its validity even in the clean case and when all gates are perturbed. As a byproduct, in the case of random longitudinal fields -- which turns out to be equivalent to certain classical Markov chains -- we find four types of non-dual-unitary(and non-integrable) interacting many-body systems where the correlation functions are exactly given by the path-sum formula.

Phase transitions and correlations in fracture processes where disorder and stress compete

We study the effect of the competition between disorder and stress enhancement in fracture processes using the local load sharing fiber bundle model, a model that hovers on the border between analytical tractability and numerical accessibility. We implement a disorder distribution with one adjustable parameter. The model undergoes a localization transition as a function of this parameter. We identify an order parameter for this transition and find that the system is in the localized phase over a finite range of values of the parameter bounded by a transition to the non-localized phase on both sides. The transition is first order at the lower transition and second order at the upper transition. The critical exponents characterizing the second order transition are close to those characterizing the percolation transition. We determine the spatiotemporal correlation function in the localized phase. It is characterized by two power laws as in invasion percolation. We find exponents that are consistent with the values found in that problem.

Time-dependent properties of interacting active matter: Dynamical behavior of one-dimensional systems of self-propelled particles

We study an interacting high-density one-dimensional system of self-propelled particles described by the Active Ornstein-Uhlenbeck particle (AOUP) model where, even in the absence of alignment interactions, velocity and energy domains spontaneously form in analogy with those already observed in two dimensions. Such domains are regions where the individual velocities are spatially correlated as a result of the interplay between self-propulsion and interactions. Their typical size is controlled by a characteristic correlation length. In the present work, we focus on a novel and lesser-known aspect of the model, namely its dynamical behavior. To this purpose, we investigate theoretically and numerically the time-dependent velocity autocorrelation and spatio-temporal velocity correlation functions. The study of these correlations provides a measure of the average life-time and, thus, the stability in time of the velocity domains.

Collective movement analysis reveals coordination tactics of team players in football matches

Abstract Collective behavior is a hallmark of every living system and utilizing methods from statistical physics (such as correlation functions) could aid in our understanding of their underlying rules. We analyzed five football (soccer) matches as this game provides a unique but yet mostly unexplored example to study a system of collective cooperation and competition. The aim of our study was to analyze the collective motion patterns exhibited by football players to unfold the underlying coordination among them in order to understand collective strategies associated with team performance. By analyzing pairwise relationships among all the players using spatio-temporal correlation functions we reveal that there exist identifiable collective dynamics that characterize winning and losing teams. Using our metric we find clear and robust differences between the players, indicating a difference in their behavior and their interactions. And this enables us to assign a unique behavioral pattern - a ‘fingerprint’ - for each individual and for each team. Furthermore, we reveal there exists a relationship between the market value of the players and the metrics introduced here, suggesting that these metrics could potentially serve as valuable performance indicators in the future, with applications ranging from talent identification to player scouting. In a broader context team sports could open up new directions for quantitative analyses of human collective behavior.

Stochastic and Heterogeneous Cancer Cell Migration: Experiment and Theory

Cell migration, an essential process for normal cell development and cancer metastasis, differs from a simple random walk: the mean-square displacement (〈(Δr)2(t)〉) of cells sometimes shows non-Fickian behavior, and the spatiotemporal correlation function (G(r, t)) of cells is often non-Gaussian. We find that this intriguing cell migration should be attributed to heterogeneity in a cell population, even one with a homogeneous genetic background. There are two limiting types of heterogeneity in a cell population: cellular heterogeneity and temporal heterogeneity. Cellular heterogeneity accounts for the cell-to-cell variation in migration capacity, while temporal heterogeneity arises from the temporal noise in the migration capacity of single cells. We illustrate that both cellular and temporal heterogeneity need to be taken into account simultaneously to elucidate cell migration. We investigate the two-dimensional migration of A549 lung cancer cells using time-lapse microscopy and find that the migration of A549 cells is Fickian but has a non-Gaussian spatiotemporal correlation. We find that when a theoretical model considers both cellular and temporal heterogeneity, the model reproduces all of the anomalous behaviors of cancer cell migration.