Gaussian Concentration Research Articles

Incorporating meander to account for the impact of low winds in area source modeling; AERMOD as a case study

ABSTRACT A variety of sources of pollutant emissions can be represented as area sources. These include manure lagoons, landfills, wastewater treatment ponds, and highways. A group of point sources can also be treated as an area source. The impact of an area source is usually computed by representing the area source as a set of line sources perpendicular to the wind direction. As for point sources, the Gaussian horizontal concentration distribution used to compute the contributions of the line sources is likely to overestimate ground-level concentrations when the wind speed is comparable to the standard deviation of the horizontal velocity fluctuations. A variety of methods are used to mitigate this overestimation under these conditions, referred to as meander. As an example of one these approaches, we examine that of AERMOD, EPA’s regulatory model. AERMOD includes meander in modeling the impact of point and volume sources, but has not yet incorporated it into AERMOD’s area source algorithm. This paper describes an approach to include meander in AERMOD’s area source algorithm and demonstrates its impact on concentrations associated with area sources. Implications: Inclusion of wind direction meander in modeling dispersion when the wind speed is low is important in ensuring that AERMOD does not overestimate concentrations under these conditions. In view of the importance of area sources of pollution, the results presented in this paper represent a potential enhancement of AERMOD’s ability to estimate the upper end of the concentration distribution, which forms the basis of the regulatory acceptance of the model.

Fluctuations of the Occupation Density for a Parking Process

Consider the following simple parking process on Λn:={-n,…,n}d,d≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda _n:= \\{-n, \\ldots , n\\}^d,d\\ge 1$$\\end{document}: at each step, a site i is chosen at random in Λn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda _n$$\\end{document} and if i and all its nearest neighbor sites are empty, i is occupied. Once occupied, a site remains so forever. The process continues until all sites in Λn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda _n$$\\end{document} are either occupied or have at least one of their nearest neighbors occupied. The final configuration (occupancy) of Λn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda _n$$\\end{document} is called the jamming limit and is denoted by XΛn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X_{\\Lambda _n}$$\\end{document}. Ritchie (J Stat Phys 122:381–398, 2006) constructed a stationary random field on Zd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Z}^d$$\\end{document} obtained as a (thermodynamic) limit of the XΛn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X_{\\Lambda _n}$$\\end{document}’s as n tends to infinity. As a consequence of his construction, he proved a strong law of large numbers for the proportion of occupied sites in the box Λn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda _n$$\\end{document} for the random field X. Here we prove the central limit theorem, the law of iterated logarithm, and a gaussian concentration inequality for the same statistics. A particular attention will be given to the case d=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=1$$\\end{document}, in which we also obtain new asymptotic properties for the sequence XΛn,n≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X_{\\Lambda _n},n\\ge 1$$\\end{document}.

Studying the Effect of the Instantaneous and Continuous Pollutants Sources on the Advection-diffusion Equation and Its Applications

This study investigates instantaneous and continuous sources as point, line, and area sources. Gaussian concentration in the case of the puff model with an instantaneous point source inhomogeneous longitudinal diffusion is investigated. The concentration is calculated using different dispersion parameters to get the proposed normalized concentration of the puff model at ground level around the centerline, which is compared with observed data by the Copenhagen experiment and previous work [1]. Also, the continuous point source is used to get the Gaussian plume model in three dimensions using dispersion parameters to compare with the observed concentration data measured by the Egyptian Atomic Energy Authority for Iodine-135 (I135) in an unstable condition.

On Gilles Pisier’s approach to Gaussian concentration, isoperimetry, and Poincaré-type inequalities

On Gilles Pisier’s approach to Gaussian concentration, isoperimetry, and Poincaré-type inequalities

Gaussian concentration bounds for stochastic chains of unbounded memory

Gaussian concentration bounds for stochastic chains of unbounded memory

Application of Resuspension Data of Respirable Particles for Early Phase Inhalation Following Deposition Contamination in Radiological Emergencies.

Abstract In model experiments, wind resuspension rates, Rr, of respirable particles (<10 μm, aerodynamic diameter) and their time dependence after deposition on representatively contaminated urban surfaces were measured under real-life conditions, taking into account different weather conditions (wind velocities, precipitation) as well as countermeasures (fixation). For contamination by dry deposition of particles, short-term resuspension rates averaged over the first half hour, after exposure to wind with a friction velocity, u*, of 0.8 m s−1 ranged between 0.1% h−1 and 2% h−1 depending on the surface material, whereas after wet deposition, resuspension rates were two orders of magnitude lower. The time pattern of the resuspension rates followed a power law decrease, Rr ∝ t −μ , with values of μ close to 1. The measurements revealed a strong dependence on friction velocity, . For a larger area of 200 × 800 m2 contaminated with 106 Bq m−2 of particles in the respirable size range, applying a simple Gaussian dispersion model airborne activity concentration (Bq m−3) was calculated. On this basis, for an initial time of high windspeed, a cautious resuspension factor of Rf = 5 10−6 m−1 is derived for the first day, which diminishes inversely with time. The exposure of persons to external radiation and to inhaled airborne radioactivity is compared with examples for three radionuclides representing strong beta-, gamma- and alpha-radiation.

The Kraichnan Model and Non-equilibrium Statistical Physics of Diffusive Mixing

We discuss application of methods from the Kraichnan model of turbulent advection to the study of non-equilibrium concentration fluctuations arising during diffusion in liquid mixtures at high Schmidt numbers. This approach treats nonlinear advection of concentration fluctuations exactly, without linearization. Remarkably, we find that static and dynamic structure functions obtained by this method reproduce precisely the predictions of linearized fluctuating hydrodynamics. It is argued that this agreement is an analogue of anomaly non-renormalization which does not, however, protect higher-order multi-point correlations. The latter should thus yield non-vanishing cumulants, unlike those for the Gaussian concentration fluctuations predicted by linearized theory.

Feasibility of sparse large Lotka-Volterra ecosystems.

Consider a large ecosystem (foodweb) with n species, where the abundances follow a Lotka-Volterra system of coupled differential equations. We assume that each species interacts with [Formula: see text] other species and that their interaction coefficients are independent random variables. This parameter d reflects the connectance of the foodweb and the sparsity of its interactions especially if d is much smaller that n. We address the question of feasibility of the foodweb, that is the existence of an equilibrium solution of the Lotka-Volterra system with no vanishing species. We establish that for a given range of d, namely [Formula: see text] or [Formula: see text] with an extra condition on the sparsity structure, there exists an explicit threshold depending on n and d and reflecting the strength of the interactions, which guarantees the existence of a positive equilibrium as the number of species n gets large. From a mathematical point of view, the study of feasibility is equivalent to the existence of a positive solution [Formula: see text] (component-wise) to the equilibrium linear equation: [Formula: see text]where [Formula: see text] is the [Formula: see text] vector with components 1 and [Formula: see text] is a large sparse random matrix, accounting for the interactions between species. The analysis of such positive solutions essentially relies on large random matrix theory for sparse matrices and Gaussian concentration of measure. The stability of the equilibrium is established. The results in this article extend to a sparse setting the results obtained by Bizeul and Najim in Bizeul and Najim (2021).

Relative Entropy, Gaussian Concentration and Uniqueness of Equilibrium States.

For a general class of lattice spin systems, we prove that an abstract Gaussian concentration bound implies positivity of the lower relative entropy density. As a consequence, we obtain uniqueness of translation-invariant Gibbs measures from the Gaussian concentration bound in this general setting. This extends earlier results with a different and very short proof.

Variations and extensions of the Gaussian concentration inequality, Part I

The classical Gaussian concentration inequality for Lipschitz functions is adapted to a setting where the classical assumptions (i.e. Lipschitz and Gaussian) are not met. The theory is more direct than much of the existing theory designed to handle related generalizations. An application is presented to linear combinations of heavy tailed random variables.

Quantum Concentration Inequalities

We establish Transportation Cost Inequalities (TCIs) with respect to the quantum Wasserstein distance by introducing quantum extensions of well-known classical methods: First, we generalize the Dobrushin uniqueness condition to prove that Gibbs states of 1D commuting Hamiltonians satisfy a TCI at any positive temperature and provide conditions under which this first result can be extended to non-commuting Hamiltonians. Next, using a non-commutative version of Ollivier’s coarse Ricci curvature, we prove that high temperature Gibbs states of commuting Hamiltonians on arbitrary hypergraphs H=(V,E) satisfy a TCI with constant scaling as O(|V|). Third, we argue that the temperature range for which the TCI holds can be enlarged by relating it to recently established modified logarithmic Sobolev inequalities. Fourth, we prove that the inequality still holds for fixed points of arbitrary reversible local quantum Markov semigroups on regular lattices, albeit with slightly worsened constants, under a seemingly weaker condition of local indistinguishability of the fixed points. Finally, we use our framework to prove Gaussian concentration bounds for the distribution of eigenvalues of quasi-local observables and argue the usefulness of the TCI in proving the equivalence of the canonical and microcanonical ensembles and an exponential improvement over the weak Eigenstate Thermalization Hypothesis.

Effect of sampling duration on the estimate of pollutant concentration behind a heavy-duty vehicle: A large-eddy simulation

Plume chasing is cost-effective, measuring individual, on-road vehicular emissions. Whereas, wake-flow-generated turbulence results in intermittent, rapid pollutant dilution and substantial fluctuating concentrations right behind the vehicle being chased. The sampling duration is therefore one of the important factors for acquiring representative (average) concentrations, which, however, has been seldom addressed. This paper, which is based on the detailed spatio-temporal dispersion data after a heavy-duty truck calculated by large-eddy simulation (LES), examines how sampling duration affects the uncertainty of the measured concentrations in plume chasing. The tailpipe dispersion is largely driven by the jet-like flows through the vehicle underbody with approximate Gaussian concentration distribution for x ≤ 0.6h, where x is the distance after the vehicle and h the characteristic vehicle size. Thereafter for x ≥ 0.6h, the major recirculation plays an important role in near-wake pollutant transport whose concentrations are highly fluctuating and positively shewed. Plume chasing for a longer sampling duration is more favourable but is logistically impractical in busy traffic. Sampling duration, also known as averaging time in the statistical analysis, thus has a crucial role in sampling accuracy. With a longer sampling (averaging) duration, the sample mean concentration converges to the population mean, improving the sample reliability. However, this effect is less pronounced in long sampling duration. The sampling accuracy is also influenced by the locations of sampling points. For the region x > 0.6h, the sampling accuracy is degraded to a large extent. As a result, acceptable sample mean is hardly achievable. Finally, frequency analysis unveils the mechanism leading to the variance in concentration measurements which is attributed to sampling duration. Those data with frequency higher than the sampling frequency are filtered out by moving average in the statistical analyses.

Evolution of Concentration Under Lattice Spin-Flip Dynamics

We consider spin-flip dynamics of Ising lattice spin systems and study the time evolution of concentration inequalities. For “weakly interacting” dynamics we show that the Gaussian concentration bound is conserved in the course of time and it is satisfied by the unique stationary Gibbs measure. Next we show that, for a general class of translation-invariant spin-flip dynamics, it is impossible to evolve in finite time from a low-temperature Gibbs state towards a measure satisfying the Gaussian concentration bound. Finally, we consider the time evolution of the weaker uniform variance bound, and show that this bound is conserved under a general class of spin-flip dynamics.

The Quantum Wasserstein Distance of Order 1

We propose a generalization of the Wasserstein distance of order 1 to the quantum states of $n$ qudits. The proposal recovers the Hamming distance for the vectors of the canonical basis, and more generally the classical Wasserstein distance for quantum states diagonal in the canonical basis. The proposed distance is invariant with respect to permutations of the qudits and unitary operations acting on one qudit and is additive with respect to the tensor product. Our main result is a continuity bound for the von Neumann entropy with respect to the proposed distance, which significantly strengthens the best continuity bound with respect to the trace distance. We also propose a generalization of the Lipschitz constant to quantum observables. The notion of quantum Lipschitz constant allows us to compute the proposed distance with a semidefinite program. We prove a quantum version of Marton's transportation inequality and a quantum Gaussian concentration inequality for the spectrum of quantum Lipschitz observables. Moreover, we derive bounds on the contraction coefficients of shallow quantum circuits and of the tensor product of one-qudit quantum channels with respect to the proposed distance. We discuss other possible applications in quantum machine learning, quantum Shannon theory, and quantum many-body systems.

Deepwater gas concentration feature extraction based on fluid mechanics

In order to solve the complex production process, high technical content, bad working environment, frequent blowout accidents, and other problems. A method based on the concentration distribution characteristics of explosion flue gas emission in deepwater area is proposed. Based on the computational fluid dynamics theory, the basic governing equations of blowout and explosion of natural gas in deepwater are obtained. Based on the basic governing equations of deepwater natural gas blowout and explosion, the diffusion of natural gas in deepwater natural gas blowout and explosion is studied by using the K-turbulence model. The diffusion process of gas explosion in deepwater is obtained. Two Gaussian smoke model parameters, atmospheric diffusion parameter and average wind speed, were realized by combining the segmenting smoke flow with Gaussian smoke mass concentration distribution. The blowout and explosion process of natural gas in deepwater is simulated by MATLAB software, and the distribution characteristics of flue gas concentration are analyzed. The results showed that the smoke concentration and the change of smoke concentration were not affected by the change.

Simulation implementation of air pollution traceability algorithm based on unmanned aerial vehicle

Air pollution has serious harm to the ecological environment and human health. However, current ground monitoring methods and mobile robot traceability methods are difficult to accurately and quickly trace air pollution sources after pollution events. To solve this problem, this paper proposes an air pollution traceability algorithm based on unmanned aerial vehicle (UAV), which combines the mobile and flexible UAV with the hill climb traceability algorithm to realize the monitoring and tracking of air pollution source in a large area. Gaussian concentration field and turbulent concentration field are built by MATLAB, and the simulation experiment is carried out in these two concentration fields. Experimental results show that the algorithm can trace air pollution sources quickly and accurately.

Gaussian Concentration and Uniqueness of Equilibrium States in Lattice Systems

We consider equilibrium states (that is, shift-invariant Gibbs measures) on the configuration space $S^{\mathbb{Z}^d}$ where $d\geq 1$ and $S$ is a finite set. We prove that if an equilibrium state for a shift-invariant uniformly summable potential satisfies a Gaussian concentration bound, then it is unique. Equivalently, if there exist several equilibrium states for a potential, none of them can satisfy such a bound.

Gaussian concentration bound and Ensemble equivalence in generic quantum many-body systems including long-range interactions

This work explores fundamental statistical and thermodynamic properties of short-and long-range-interacting systems. The purpose of this study is twofold. Firstly, we rigorously prove that the probability distribution of arbitrary few-body observables is restricted by a Gaussian concentration bound (or Chernoff–Hoeffding inequality) above some threshold temperature. This bound is then derived for arbitrary Gibbs states of systems that include long-range interactions Secondly, we establish a quantitative relationship between the concentration bound of the Gibbs state and the equivalence of canonical and micro-canonical ensembles. We then evaluate the difference in the averages of thermodynamic properties between the canonical and the micro-canonical ensembles. Under the assumption of the Gaussian concentration bound on the canonical ensemble, the difference between the ensemble descriptions is upper-bounded by n−1log(n3∕2Δ−1)1∕2 with n being the system size and Δ being the width of the energy shell of the micro-canonical ensemble This limit gives a non-trivial upper bound exponentially small energy width with respect to the system size. By combining these two results, we prove the ensemble equivalence as well as the weak eigenstate thermalization in arbitrary long-range-interacting systems above a threshold temperature.

Evaluation of the van Deemter equation in terms of open-ended flow to chromatography.

Plate theory and adsorption theory are the main tools available for understanding chromatographic experiments. Both theories predict a Gaussian distribution of solute molecules within the tubular system. However, Gaussian concentration distributions are observed predominantly at slow linear flow rates, while asymmetric concentration distributions are observed at the linear flow rates most used in chromatography. Allegedly, this asymmetry originates from an inhomogeneous distribution of grain sizes in the column and column overload. However, it is an experimental fact that the distribution of chemicals within an injected volume of solute changes as a function of time, while the response is measured simultaneously. Accordingly, the obtained signal cannot be compared to the theory before some type of time-based deconvolution of the data has been performed. Adjustments to high-performance liquid chromatography data were thus proposed through empirical equations that describe the relevant time values, peak height, peak area, and parameters of the van Deemter equation. It was proposed that the transfer of solute from the front to the rear part of the pulse during laminar open-ended flow occurs at rate that depends on the linear flow rate, and to a lesser extent, on properties of the response function.

Stability of the Bakry-Émery theorem on [formula omitted

We establish quantitative stability estimates for the Bakry-Émery bound on logarithmic Sobolev and Poincaré constants of uniformly log-concave measures. More specifically, we show that if a 1-uniformly log-concave measure has almost the same logarithmic Sobolev or Poincaré constant as the standard Gaussian measure, then it almost splits off a Gaussian factor. Our results are dimension-free, leading to dimension-free stability estimates for Gaussian concentration of Lipschitz functions. The proofs are based on Stein's method, optimal transport, and an approximate integration by parts identity relating measures and near-extremals in the associated functional inequality.