Evolution of the Probability Measure for the Majda Model: New Invariant Measures and Breathing PDFs

In 1993, Majda proposed a simple, random shear model from which scalar intermittency was rigorously predicted for the invariant probability measure of passive tracers. In this work, we present an integral formulation for the tracer measure, which leads to a new, comprehensive study on its temporal evolution based on Monte Carlo simulation and direct numerical integration. An interesting, non-monotonic “breathing” phenomenon is discovered from these results and carefully defined, with a solid example for special initial data to predict such phenomenon. The signature of this phenomenon may persist at long time, characterized by the approach of the PDF core to its infinite time, invariant value. We find that this approach may be strongly dependent on the non-dimensional Péclet number, of which the invariant measure itself is independent. Further, the “breathing” PDF is recovered as a new invariant measure in a distinguished time scale in the diffusionless limit. Rigorous asymptotic analysis is also performed to identify the Gaussian core of the invariant measures, and the critical rate at which the heavy, stretched exponential regime propagates towards the tail as a function of time is calculated.     

New Challenges for Classical and Quantum Probability

The discovery that any classical random variable with all moments gives rise to a full quantum theory (that in the Gaussian and Poisson cases coincides with the usual one) implies that a quantum–type formalism will enter into practically all applications of classical probability and statistics. The new challenge consists in finding the classical interpretation, for different types of classical contexts, of typical quantum notions such as entanglement, normal order, equilibrium states, etc. As an example, every classical symmetric random variable has a canonically associated conjugate momentum. In usual quantum mechanics (associated with Gaussian or Poisson classical random variables), the interpretation of the momentum operator was already clear to Heisenberg. How should we interpret the conjugate momentum operator associated with classical random variables outside the Gauss–Poisson class? The Introduction is intended to place in historical perspective the recent developments that are the main object of the present exposition.     

Real Roots of Random Polynomials and Zero Crossing Properties of Diffusion Equation

We study various statistical properties of real roots of three different classes of random polynomials which recently attracted a vivid interest in the context of probability theory and quantum chaos. We first focus on gap probabilities on the real axis, i.e. the probability that these polynomials have no real root in a given interval. For generalized Kac polynomials, indexed by an integer d, of large degree n, one finds that the probability of no real root in the interval [0,1] decays as a power law n ^−θ(d) where θ(d)>0 is the persistence exponent of the diffusion equation with random initial conditions in spatial dimension d. For n≫1 even, the probability that they have no real root on the full real axis decays like n ^{−2(θ(2)+θ(d))}. For Weyl polynomials and Binomial polynomials, this probability decays respectively like and where θ _∞ is such that in large dimension d. We also show that the probability that such polynomials have exactly k roots on a given interval [a,b] has a scaling form given by where N _ab is the mean number of real roots in [a,b] and a universal scaling function. We develop a simple Mean Field (MF) theory reproducing qualitatively these scaling behaviors, and improve systematically this MF approach using the method of persistence with partial survival, which in some cases yields exact results. Finally, we show that the probability density function of the largest absolute value of the real roots has a universal algebraic tail with exponent −2. These analytical results are confirmed by detailed numerical computations. Some of these results were announced in a recent letter (Schehr and Majumdar in Phys. Rev. Lett. 99:060603, 2007).     

Dissipation Statistics of a Passive Scalar in a Multidimensional Smooth Flow

We compute analytically the probability distribution function () of the dissipation field =()² of a passive scalar advected by a d-dimensional random flow, in the limit of large Peclet and Prandtl numbers (Batchelor–Kraichnan regime). The tail of the distribution is a stretched exponential: for , ln ()–(d ² )^1/3.     

Inertial-Range Behavior of a Passive Scalar Field in a Random Shear Flow: Renormalization Group Analysis of a Simple Model

Infrared asymptotic behavior of a scalar field, passively advected by a random shear flow, is studied by means of the field theoretic renormalization group and the operator product expansion. The advecting velocity is Gaussian, white in time, with correlation function of the form μ d(t-t￠) / k_^^d-1+x\propto\delta(t-t') / k_{\bot}^{d-1+\xi}, where k _⊥=|k _⊥| and k _⊥ is the component of the wave vector, perpendicular to the distinguished direction (‘direction of the flow’)—the d-dimensional generalization of the ensemble introduced by Avellaneda and Majda (Commun. Math. Phys. 131:381, 1990). The structure functions of the scalar field in the infrared range exhibit scaling behavior with exactly known critical dimensions. It is strongly anisotropic in the sense that the dimensions related to the directions parallel and perpendicular to the flow are essentially different. In contrast to the isotropic Kraichnan’s rapid-change model, the structure functions show no anomalous (multi)scaling and have finite limits when the integral turbulence scale tends to infinity. On the contrary, the dependence of the internal scale (or diffusivity coefficient) persists in the infrared range. Generalization to the velocity field with a finite correlation time is also obtained. Depending on the relation between the exponents in the energy spectrum E μ k_^^1-e\mathcal{E} \propto k_{\bot}^{1-\varepsilon} and in the dispersion law w μ k_^^2-h\omega\propto k_{\bot}^{2-\eta}, the infrared behavior of the model is given by the limits of vanishing or infinite correlation time, with the crossover at the ray η=0, ε>0 in the ε–η plane. The physical (Kolmogorov) point ε=8/3, η=4/3 lies inside the domain of stability of the rapid-change regime; there is no crossover line going through this point.     

Convergence in High Probability of the Quantum Diffusion in a Random Band Matrix Model

We consider Hermitian random band matrices H in \(d \geqslant 1 \) dimensions. The matrix elements \(H_{xy},\) indexed by \(x, y \in \varLambda \subset \mathbb {Z}^d,\) are independent, uniformly distributed random variable if \(|x-y| \) is less than the band width W,  and zero otherwise. We update the previous results of the converge of quantum diffusion in a random band matrix model from convergence of the expectation to convergence in high probability. The result is uniformly in the size \(|\varLambda | \) of the matrix.     

Explicit inertial range renormalization theory in a model for turbulent diffusion

The inertial range for a statistical turbulent velocity field consists of those scales that are larger than the dissipation scale but smaller than the integral scale. Here the complete scale-invariant explicit inertial range renormalization theory for all the higher-order statistics of a diffusing passive scalar is developed in a model which, despite its simplicity, involves turbulent diffusion by statistical velocity fields with arbitrarily many scales, infrared divergence, long-range spatial correlations, and rapid fluctuations in time-such velocity fields retain several characteristic features of those in fully developed turbulence. The main tool in the development of this explicit renormalization theory for the model is an exact quantum mechanical analogy which relates higher-order statistics of the diffusing scalar to the properties of solutions of a family ofN- body parabolic quantum problems. The canonical inertial range renormalized statistical fixed point is developed explicitly here as a function of the velocity spectral parameter, which measures the strength of the infrared divergence: for<2, mean-field behavior in the inertial range occurs with Gaussian statistical behavior for the scalar and standard diffusive scaling laws; for>2 a phase transition occurs to a fixed point with anomalous inertial range scaling laws and a non-Gaussian renormalized statistical fixed point. Several explicit connections between the renormalization theory in the model and intermediate asymptotics are developed explicitly as well as links between anomalous turbulent decay and explicit spectral properties of Schrödinger operators. The differences between this inertial range renormalization theory and the earlier theories for large-scale eddy diffusivity developed by Avellaneda and the author in such models are also discussed here.     

Self-Similar Decay in the Kraichnan Model of a Passive Scalar

We study the two-point correlation function of a freely decaying scalar in Kraichnan's model of advection by a Gaussian random velocity field that is stationary and white noise in time, but fractional Brownian in space with roughness exponent 0<<2, appropriate to the inertial-convective range of the scalar. We find all self-similar solutions by transforming the scaling equation to Kummer's equation. It is shown that only those scaling solutions with scalar energy decay exponent a(d/)+1 are statistically realizable, where d is space dimension and =2–. An infinite sequence of invariants J _p, p=0, 1, 2,..., is pointed out, where J ₀ is Corrsin's integral invariant but the higher invariants appear to be new. We show that at least one of the invariants J ₀ or J ₁ must be nonzero (possibly infinite) for realizable initial data. Initial datum with a finite, nonzero invariant—the first being J _p—converges at long times to a scaling solution _p with a=(d/)+p, p=0, 1. The latter belongs to an exceptional series of self-similar solutions with stretched-exponential decay in space. However, the domain of attraction includes many initial data with power-law decay. When the initial datum has all invariants zero or infinite and also it exhibits power-law decay, then the solution converges at long times to a nonexceptional scaling solution with the same power-law decay. These results support a picture of a two-scale decay with breakdown of self-similarity for a range of exponents (d+)/<a<(d+2)/, analogous to what has recently been found in the decay of Burgers turbulence.     

A Family of Balance Relations for the Two-Dimensional Navier--Stokes Equations with Random Forcing

For the 2D Navier--Stokes equation perturbed by a random force of a suitable kind we show that, if g(F) is an arbitrary real continuous function with (at most) polynomial growth, then the stationary in time vorticity field (t,x) satisfies where M_1 is a number, independent of g, which measures the strength of the random forcing. Another way of stating this result is that, in the unique stationary measure of this system, the random variables g((t,x) and |(t,x)|² are uncorrelated for each t and each x.This revised version was published online in March 2005 with corrections to the page numbers.     

Exact Results for Conditional Means of a Passive Scalar in Certain Statistically Homogeneous Flows

For a passive scalar T(r, t) randomly advected by a statistically homogeneous flow, the probability density function (pdf) of its fluctuation can in general be expressed in terms of two conditional means: 〈?² T|T〉 and 〈|?T|²|T〉. We find that in some special cases, either one of the two conditional means can be obtained explicitly from the equation of motion. In the cases when there is no external source and that the normalized fluctuation reaches a steady state or when a steady state results from a negative damping, 〈?² T|T〉=?(〈|?T|²〉/〈T ²〉)T. The linearity of the conditional mean in these cases follows directly from the fact that the advection equation of a passive scalar is linear. On the other hand, when the scalar is supported by a homogeneous white-in-time external source, 〈|?T|²|T〉=〈|?T|²〉.     

Computer Simulations for Some One-Dimensional Models of Random Walks in Fluctuating Random Environment

We report some results of computer simulations for two models of random walks in random environment (rwre) on the one-dimensional lattice for fixed space–time configuration of the environment (“quenched rwre”): a “Markov model” with Markov dependence in time, and a “quasi stationary” model with long range space–time correlations. We compare with the corresponding results for a model with i.i.d. (in space time) environment. In the range of times available to us the quenched distributions of the random walk displacement are far from gaussian, but as the behavior is similar for all three models one cannot exclude asymptotic gaussianity, which is proved for the model with i.i.d. environment. We also report results on the random drift and on some time correlations which show a clear power decay     

Vicious random walkers in the limit of a large number of walkers

The vicious random walker problem on a line is studied in the limit of a large number of walkers. The multidimensional integral representing the probability that thep walkers will survive a timet (denotedP _t ^(p) ) is shown to be analogous to the partition function of a particular one-component Coulomb gas. By assuming the existence of the thermodynamic limit for the Coulomb gas, one can deduce asymptotic formulas forP _t ^(p) in the large-p, large-t limit. A straightforward analysis gives rigorous asymptotic formulas for the probability that after a timet the walkers are in their initial configuration (this event is termed a reunion). Consequently, asymptotic formulas for the conditional probability of a reunion, given that all walkers survive, are derived. Also, an asymptotic formula for the conditional probability density that any walker will arrive at a particular point in timet, given that allp walkers survive, is calculated in the limittp.     

A Note on Transience Versus Recurrence for a Branching Random Walk in Random Environment

We consider a branching random walk in random environment on ^d where particles perform independent simple random walks and branch, according to a given offspring distribution, at a random subset of sites whose density tends to zero at infinity. Given that initially one particle starts at the origin, we identify the critical rate of decay of the density of the branching sites separating transience from recurrence, i.e., the progeny hits the origin with probability <1 resp. =1. We show that for d3 there is a dichotomy in the critical rate of decay, depending on whether the mean offspring at a branching site is above or below a certain value related to the return probability of the simple random walk. The dichotomy marks a transition from local to global behavior in the progeny that hits the origin. We also consider the situation where the branching sites occur in two or more types, with different offspring distributions, and show that the classification is more subtle due to a possible interplay between the types. This note is part of a series of papers by the second author and various co-authors investigating the problem of transience versus recurrence for random motions in random media.     

A Diffusion Limit for a Test Particle in a Random Distribution of Scatterers

We consider a point particle moving in a random distribution of obstacles described by a potential barrier. We show that, in a weak-coupling regime, under a diffusion limit suggested by the potential itself, the probability distribution of the particle converges to the solution of the heat equation. The diffusion coefficient is given by the Green–Kubo formula associated to the generator of the diffusion process dictated by the linear Landau equation.     

Probability Density and Statistical Properties for a Three-State Markovian Noise and Escape of Particles for a System Driven by This Noise

A three-state Markovian noise is investigated. Its probability density and statistical properties are obtained. Escape of particles for a system with potential barrier only driven by this noise is investigated. It is shown that, in some circumstances, this noise can make the particles escape over the potential barrier; but in other circumstances, it cannot. Resonant activation phenomenon appears for the system considered by us.     

实际大气中激光闪烁的概率分布

根据湍流大气激光对数强度的最低几阶中心矩建立了一种最大似然概率分布模型,该模型可以方便和准确地描述实际概率分布,根据实验结果分析了激光地数强度的概率分布的偏斜度和陆峭度的特征。发现在弱起伏条件下,对数强度的概率分布一般接近于正态分布,当偏离正态分布时,概率密度分布的偏斜度总为负,陆峭度总是为正。     

圆柱热尾流中温度的概率密度函数

本文通过实验研究圆柱热尾流中温度的概率密度函数在不同雷诺数下沿中心线的演变以及其与湍流混合程度的关系.实验中雷诺数的取值范围是1200-8600,温度是由直径为0.63 μm的冷线探针测量的.实验结果表明,温度概率密度函数在尾流中场区随空间位置变化显著.雷诺数的增加加快了这个变化过程,特别加速了尾流中心线温度的概率密度函数从非高斯向近似高斯分布的演变.