Probability Distribution Function of Passive Scalars in Shell Models

A shell-model version of passive scalar problem is introduced, which is inspired by the model of K. Ohkitani and M. Yakhot [K. Ohkitani and M. Yakhot, Phys. Rev. Lett. 60 (1988) 983; K. Ohkitani and M. Yakhot, Prog. Theor. Phys. 81 (1988) 329]. As in the original problem, the prescribed random velocity field is Gaussian and δ correlated in time. Deterministic differential equations are regarded as nonlinear Langevin equation. Then, the Fokker-Planck equations of PDF for passive scalars are obtained and solved numerically. In energy input range (n<5, n is the shell number.), the probability distribution function (PDF) of passive scalars is near the Gaussian distribution. In inertial range (5≤n≤16) and dissipation range (n≥17), the probability distribution function (PDF) of passive scalars has obvious intermittence. And the scaling power of passive scalar is anomalous. The results of numerical simulations are compared with experimental measurements.     

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.     

Closure Approximations for Passive Scalar Turbulence: A Comparative Study on an Exactly Solvable Model with Complex Features

Some standard closure approximations used in turbulence theory are analyzed by examining systematically the predictions these approximations produce for a passive scalar advection model consisting of a shear flow with a fluctuating cross sweep. This model has a general geometric structure of a jet flow with transverse disturbances, which occur in a number of contexts, and it encompasses a wide variety of possible spatio-temporal statistical structures for the velocity field, including strong long-range correlations. Even though the Eulerian and Lagrangian velocity statistics are not equal and the passive scalar statistics exhibit broader-than-Gaussian intermittency, this model is nevertheless simple enough so that many passive scalar statistics can be computed exactly and compared systematically with the predictions of the closure approximations. Our comparative study illustrates the strength and weaknesses of the closure approximations and points out the physical phenomena that these approximations are able or not able to describe properly. In particular it is shown that the direct interaction approximation (DIA), one of the most sophisticated closure approximations available, fails to reproduce adequately the statistical features of the scalar and may even lead to absdurd predictions, even though the equations it produces are rather complicated and difficult to analyze. Two alternative closure approximations, the Modified DIA (MDIA) and the Renormalized Lagrangian Approximation (RLA), with different levels of sophistication, both are simpler to use than the DIA and perform better. In particular, it is shown that both closure approximations always reproduce exactly the second order statistics for the scalar and that the MDIA is even able to capture intermittency effects.     

A minimal flow unit for the study of turbulence with passive scalars

The concept of a minimal flow unit (MFU) for the study of the basic physics of turbulent flows is introduced. The MFU is an initial vorticity configuration that consists of a few simple well-defined large-scale vortex structures. The form and position of these structures are chosen so that their interaction produces turbulence capturing many of the essential characteristics of isotropic homogeneous turbulence produced from random-phase initial conditions or that produced by continual random-phase forcing. The advantage of using the MFU is that the evolution of the vortex structures can be followed more clearly and the relationship between the evolving vortex structures and the various ranges in the energy spectrum can be more clearly defined. The addition of passive scalar fields to the MFU permits an investigation of passive scalar mixing that is relevant to the study of combustion. With a particular choice of the MFU, one that produces a trend to a finite-time singularity in the vorticity field, it is demonstrated that passive scalar distributed in the original large-scale vortices will develop intense gradients in the region where the vorticity is tending toward a singularity. In viscous flow, the evolution of the MFU clearly shows how the volume of the regions where originally well-separated passive scalars come into contact increases with increasing Reynolds number.     

Reassessment of the classical closures for scalar turbulence

In deducing the consequences of the Direct Interaction Approximation, Kraichnan was sometimes led to consider the properties of special classes of nonlinear interactions in degenerate triads in which one wavevector is very small. Such interactions can be described by simplified models closely related to elementary closures for homogeneous isotropic turbulence such as the Heisenberg and Leith models. These connections can be exploited to derive considerably improved versions of the Heisenberg and Leith models that are only slightly more complicated analytically. This paper applies this approach to derive some new simplified closure models for passive scalar advection and investigates the consistency of these models with fundamental properties of scalar turbulence. Whereas some properties, such as the existence of the Kolmogorov–Obukhov range and the existence of thermal equilibrium ensembles, follow the velocity case closely, phenomena special to the scalar case arise when the diffusive and viscous effects become important at different scales of motion. These include the Batchelor and Batchelor–Howells–Townsend ranges pertaining, respectively, to high and low molecular Schmidt number. We also consider the spectrum in the diffusive range that follows the Batchelor range. We conclude that improved elementary models can be made consistent with many nontrivial properties of scalar turbulence, but that such models have unavoidable limitations.     

Mixing of a passive scalar across a thin shearless layer: concentration of intermittency on the sides of the turbulent interface

The advection of a passive scalar through an initial flat interface separating two different isotropic decaying turbulent fields is investigated in two and three dimensions. Simulations have been performed for a range of Taylor’s microscale Reynolds numbers from 45 to 250 and for a Schmidt number equal to 1. Different to the case where the transport involves the momentum and kinetic energy only and one intermittency layer is formed in the low-turbulent energy side of the system, in the passive scalar concentration field two intermittent layers are observed to develop at the sides of the interface. The layers move normally to the interface in opposite directions. The dimensionality produces different time scaling of the passive scalar diffusion, which is much faster in the two-dimensional case. In two dimensions, the propagation of the intermittent layers exhibits a significant asymmetry with respect to the initial position of the interface and is deeper for the layer which moves towards the high kinetic energy side of the system. In three dimensions, the two intermittent layers propagate nearly symmetrically with respect the centre of the mixing region. During the temporal decay, inside the mixing, which is both inhomogeneous and anisotropic but devoid of a mean velocity shear, the passive scalar spectra are computed. In three dimensions, the exponent in the scaling range gets in time a value close to that of the kinetic energy spectrum of isotropic turbulence (?5/3). In two dimensions, instead the exponent settles down to a value that is about one-half of the corresponding isotropic case. By means of an analysis based on simple wavy perturbations of the interface we show that the formation of the double layer of intermittency is a dynamic general feature not specific to the turbulent transport. These results of our numerical study are discussed in the context of experimental results and numerical simulations.     

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.     

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.     

Scaling,Renormalization and Statistical Conservation Laws in the Kraichnan Model of Turbulent Advection

We present a systematic way to compute the scaling exponents of the structure functions of the Kraichnan model of turbulent advection in a series of powers of ξ, adimensional coupling constant measuring the degree of roughness of the advecting velocity field. We also investigate the relation between standard and renormalization group improved perturbation theory. The aim is to shed light on the relation between renormalization group methods and the statistical conservation laws of the Kraichnan model, also known as zero modes.     

微粒形体特征概率分布函数的光散射频谱分析法

针对微粒(包括生物微粒)光散射测量中微粒形体对测量结果的影响以及形体分布不能实时测量的问题,本文根据光散射测量原理,从微粒群的统计性出发,对散射脉冲的脉宽、脉峰分布及其与微粒大小和形体分布物理动态特征的相互关系进行了系统的分析,得到了微粒形体、微粒大小分布与其散射谱分布的特征关系,提出了等值概率面积确定法则,并据此建立了一种实时由散射谱一体分析获取微粒形体特征分布的方法,得到了较好的实验验证,由此可有效地修正因微粒形体变化而对测量精度的影响,从而为粒度分析仪以及以光散射原理为测量理论的流式细胞仪技术的提升提供了有力的基础。     

The Finite-Size Scaling Relation for the Order-Parameter Probability Distribution of the Six-Dimensional Ising Model

The six dimensional Ising model with nearest-neighbor pair interactions has been simulated and verified numerically on the Creutz Cellular Automaton by using five bit demons near the infinite-lattice critical temperature with the linear dimensions L=4,6,8,10. The order parameter probability distribution for six dimensional Ising model has been calculated at the critical temperature. The constants of the analytical function have been estimated by fitting to probability function obtained numerically at the finite size critical point.     

Universal Probability Distribution for the Wave Function of a Quantum System Entangled with its Environment

A quantum system (with Hilbert space \({\mathcal {H}_{1}}\)) entangled with its environment (with Hilbert space \({\mathcal {H}_{2}}\)) is usually not attributed to a wave function but only to a reduced density matrix \({\rho_{1}}\). Nevertheless, there is a precise way of attributing to it a random wave function \({\psi_{1}}\), called its conditional wave function, whose probability distribution \({\mu_{1}}\) depends on the entangled wave function \({\psi \in \mathcal {H}_{1} \otimes \mathcal {H}_{2}}\) in the Hilbert space of system and environment together. It also depends on a choice of orthonormal basis of \({\mathcal {H}_{2}}\) but in relevant cases, as we show, not very much. We prove several universality (or typicality) results about \({\mu_{1}}\), e.g., that if the environment is sufficiently large then for every orthonormal basis of \({\mathcal {H}_{2}}\), most entangled states \({\psi}\) with given reduced density matrix \({\rho_{1}}\) are such that \({\mu_{1}}\) is close to one of the so-called GAP (Gaussian adjusted projected) measures, \({GAP(\rho_{1})}\). We also show that, for most entangled states \({\psi}\) from a microcanonical subspace (spanned by the eigenvectors of the Hamiltonian with energies in a narrow interval \({[E, E+ \delta E]}\)) and most orthonormal bases of \({\mathcal {H}_{2}}\), \({\mu_{1}}\) is close to \({GAP(\rm {tr}_{2} \rho_{mc})}\) with \({\rho_{mc}}\) the normalized projection to the microcanonical subspace. In particular, if the coupling between the system and the environment is weak, then \({\mu_{1}}\) is close to \({GAP(\rho_\beta)}\) with \({\rho_\beta}\) the canonical density matrix on \({\mathcal {H}_{1}}\) at inverse temperature \({\beta=\beta(E)}\). This provides the mathematical justification of our claim in Goldstein et al. (J Stat Phys 125: 1193–1221, 2006) that GAP measures describe the thermal equilibrium distribution of the wave function.     

Stationary Probability Distribution of Colored-noise Saturation Model for a Single-mode Dye Laser

It is applied the interpolation procedure to calculate the stationary probability distribution of colored-gain-noise model of a single-mode dye laser which operates above threshold with correlation time τ covering a very wide rang. By use of Stochastic Runge-Kutta Algorithm, it also has carried out numerical simulations of the steady-state properties. Comparing the results of the interpolation procedure and the unified colored-noise approximation with simulation results, the agreement between the results of the interpolation procedure and simulation results is much better than that of the unified colored-noise approximation when correlation time τ covers range from moderate to large.     

Formation of Delta Standing Wave for a Scalar Conservation Law with a Linear Flux Function Involving Discontinuous Coefficients

Abstract

The aim of this paper is to study the formation of delta standing wave for a scalar conservation law with a linear flux function involving discontinuous coefficients. In order to deal with it, we approximate the discontinuous coefficients by piecewise affine ones and then apply the method of characteristics to construct a global solution to the original equation by approximation. Finally, it is proved rigorously that the delta standing wave can be obtained in the limit of the approximate solution as the perturbation parameter tends to zero. In contrast to the classical method of vanishing viscosity, it can be seen clearly here how the delta standing wave forms naturally along the characteristics.     

线性离散成像系统调制传递函数的随机条纹测试法

离散光电成像系统因其不再具有平移不变特性,它的调制传递函数测试具有相当的复杂性和不稳定性。为提高光学系统调制传递函数测试的灵活性和可重复性,在基于统计光学和随机信号相关性理论的基础上,分析了离散型空间随机信号经线性光学系统成像后的物像相关性,提出了应用随机条纹靶来测定调制传递函数的方法。并给出了计算数据结果。该方法从统计意义上考虑了离散采样成像器件的采样场景相位的影响,大大提高了现场测试的灵活性和自动化程度,具有很好的应用前景。此方法同样适用于一般光学成像系统的调制传递函数测量。     

任意两个相干态的叠加态的相位分布和Wigner函数(英文)

从相位分布和Wigner函数两个方面研究了任意两个相干态|β〉and |mβeiδ〉的叠加态的量子统计性质.结果表明这种叠加态的非经典特性与β2,振幅系数m,相干态间的位相差δ以及叠加系数间的位相差都有关.当参量选择合适,这种叠加态存在着量子效应.计算了两个相干态等几率混合系综的相位分布和Wigner函数,经过与前者比较,结果表明由于相干项的存在,使得叠加态具有很好的量子力学行为.     

The Homotopic Probability Distribution and the Partition Function for the Entangled System Around a Ribbon Segment Chain

Using the Feynman's path integral with topological constraints arising from the presence of one singular line, we find the homotopic probability distribution P_Lⁿ for the winding number n and the partition function P_L of the entangled system around a ribbon segment chain. We find that when the width of the ribbon segment chain 2a increases,the partition function exponentially decreases, whereas the free energy increases an amount, which is proportional to the square of the width. When the width tends to zero we obtain the same results as those of a single chain with one singular point.     

Statistical Estimates for the Navier–Stokes Equations and the Kraichnan Theory of 2-D Fully Developed Turbulence

A mathematical formulation of the Kraichnan theory for 2-D fully developed turbulence is given in terms of ensemble averages of solutions to the Navier–Stokes equations. A simple condition is given for the enstrophy cascade to hold for wavenumbers just beyond the highest wavenumber of the force up to a fixed fraction of the dissipation wavenumber, up to a logarithmic correction. This is followed by partial rigorous support for Kraichnan's eddy breakup mechanism. A rigorous estimate for the total energy is found to be consistent with Kraichnan's theory. Finally, it is shown that under our conditions for fully developed turbulence the fractal dimension of the attractor obeys a sharper upper bound than in the general case.     

Stationary Probability Distribution of Colored-noise Saturation Model for a Single-mode Dye Laser

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任意两个相干态的叠加态的相位分布和Wigner函数

从相位分布和Wigner函数两个方面研究了任意两个相干态|β〉 and |mβeiδ〉的叠加态的量子统计性质.结果表明这种叠加态的非经典特性与β2,振幅系数m,相干态间的位相差δ以及叠加系数间的位相差都有关.当参量选择合适,这种叠加态存在着量子效应.计算了两个相干态等几率混合系综的相位分布和Wigner函数,经过与前者比较,结果表明由于相干项的存在,使得叠加态具有很好的量子力学行为.