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102
Why stock market crash
, 2003
"... The young science of complexity, which studies systems as diverse as the human body, the earth and the universe, offers novel insights on the question raised in the title. The science of complexity explains largescale collective behavior, such as wellfunctioning capitalistic markets, and also pre ..."
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Cited by 106 (13 self)
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The young science of complexity, which studies systems as diverse as the human body, the earth and the universe, offers novel insights on the question raised in the title. The science of complexity explains largescale collective behavior, such as wellfunctioning capitalistic markets, and also predicts that financial crashes and depressions are intrinsic properties resulting from the repeated nonlinear interactions between investors. Applying concepts and methods from complex theory and statistical physics, we have developed mathematical measures to successfully predict the emergence and development of speculative bubbles as well as depressions. This essay attempts to capture and extend the essence of the book with the same title published in January 2003 by Princeton University Press. Recent novelties and live predictions are available at
Kicked Burgers turbulence
 J. Fluid Mech
"... J. Fluid Mech.; in press Burgers turbulence subject to a force f(x,t) = ∑ j fj(x)δ(t − tj), where the tj’s are “kicking times ” and the “impulses ” fj(x) have arbitrary space dependence, combines features of the purely decaying and the continuously forced cases. With largescale forcing this “kicke ..."
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Cited by 26 (6 self)
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J. Fluid Mech.; in press Burgers turbulence subject to a force f(x,t) = ∑ j fj(x)δ(t − tj), where the tj’s are “kicking times ” and the “impulses ” fj(x) have arbitrary space dependence, combines features of the purely decaying and the continuously forced cases. With largescale forcing this “kicked ” Burgers turbulence presents many of the regimes proposed by E, Khanin, Mazel and Sinai (1997) for the case of random whiteintime forcing. It is also amenable to efficient numerical simulations in the inviscid limit, using a modification of the Fast Legendre Transform method developed for decaying Burgers turbulence by Noullez and Vergassola (1994). For the kicked case, concepts such as “minimizers ” and “main shock”, which play crucial roles in recent developments for forced Burgers turbulence, become elementary since everything can be constructed from simple twodimensional areapreserving Euler– Lagrange maps.
Hyperbolic lines and the stratospheric polar vortex
 Chaos
"... The necessary and sufficient conditions for Lagrangian hyperbolicity recently derived in the literature are reviewed in the light of older concepts of effective local rotation in strain coordinates. In particular, we introduce the simple interpretation of the necessary condition as a constraint on ..."
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Cited by 24 (0 self)
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The necessary and sufficient conditions for Lagrangian hyperbolicity recently derived in the literature are reviewed in the light of older concepts of effective local rotation in strain coordinates. In particular, we introduce the simple interpretation of the necessary condition as a constraint on the local angular displacement in strain coordinates. These mathematically rigorous conditions are applied to the winter stratospheric circulation of the southern hemisphere, using analyzed wind data from the European Center for MediumRange Weather Forecasts. Our results demonstrate that the sufficient condition is too strong and the necessary condition is too weak, so that both conditions fail to identify hyperbolic lines in the stratosphere. However a phenomenological, nonrigorous, criterion based on the necessary condition reveals the hyperbolic structure of the flow. Another ͑still nonrigorous͒ alternative is the finitesize Lyapunov exponent ͑FSLE͒ which is shown to produce good candidates for hyperbolic lines. In addition, we also tested the sufficient condition for Lagrangian ellipticity and found that it is too weak to detect elliptic coherent structures ͑ECS͒ in the stratosphere, of which the polar vortex is an obvious candidate. Yet, the FSLE method reveals a clear ECSlike barrier to mixing along the polar vortex edge. Further theoretical advancement is needed to explain the apparent success of nonrigorous methods, such as the FSLE approach, so as to achieve a sound kinematic understanding of chaotic mixing in the winter stratosphere and other geophysical flows. © 2002 American Institute of Physics. ͓DOI: 10.1063/1.1480442͔ Transport plays an important role in the distribution of chemicals in the stratosphere "the layer of atmosphere between 12 and 55 km in altitude…. This fact is clearly illustrated by, for instance, the formation of the Antarctic ozone hole every austral winter. In the extratropical stratosphere, chemical transport proceeds in quasihorizontal layers, where air parcels practically conserve entropy for up to about 3 weeks. Transport, stirring, and mixing in these isentropic layers is governed by the Lagrangian chaos generated by organized largescale circulations "of several hundred kilometers and larger…. The spatial organization of chaotic stirring is described by the main hyperbolic lines "i.e., the material lines that are locally the most attracting or repelling… forming at any time a skeleton of paths and lobes through the flow. Gradients of longlived tracers tend to orient normal to and intensify along strongly attracting lines, thereby enhancing the mixing process by smallscale vertical circulations. At the same time, a strong vortical circulation exists in the winter polar region. The polar vortex exemplifies an elliptic coherent structure: its edge forms a partial barrier to mixing. Rigorous mathematical criteria were derived recently to characterize hyperbolic lines and elliptic coherent structures. In this paper, we review and test these criteria in a case study using stratospheric winds from the European Center for MediumRange Weather Forecasts. Our work shows that these criteria fail to pick out hyperbolic lines and elliptic coherent structures in the stratosphere, which are, however, readily identified with other less rigorous methods.
Whitenoise and geometrical optical limits of WignerMoyal equation for wave beams in turbulent media
, 2004
"... Starting with the Wigner distribution formulation for beam wave propagation in Hölder continuous nonGaussian random refractive index fields we show that the wave beam regime naturally leads to the whitenoise scaling limit and converges to a Gaussian whitenoise model which is characterized by the ..."
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Cited by 22 (12 self)
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Starting with the Wigner distribution formulation for beam wave propagation in Hölder continuous nonGaussian random refractive index fields we show that the wave beam regime naturally leads to the whitenoise scaling limit and converges to a Gaussian whitenoise model which is characterized by the martingale problem associated to a stochastic differentialintegral equation of the Itô type. In the simultaneous geometrical optics the convergence to the Gaussian whitenoise model for the Liouville equation is also established if the ultraviolet cutoff or the Fresnel number vanishes sufficiently slowly. The advantage of the Gaussian whitenoise model is that its npoint correlation functions are governed by closed form equations.
Fluctuation relations for diffusion process
 Commun. Math. Phys
"... The paper presents a unified approach to different fluctuation relations for classical nonequilibrium dynamics described by diffusion processes. Such relations compare the statistics of fluctuations of the entropy production or work in the original process to the similar statistics in the timerever ..."
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Cited by 21 (6 self)
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The paper presents a unified approach to different fluctuation relations for classical nonequilibrium dynamics described by diffusion processes. Such relations compare the statistics of fluctuations of the entropy production or work in the original process to the similar statistics in the timereversed process. The origin of a variety of fluctuation relations is traced to the use of different time reversals. It is also shown how the application of the presented approach to the tangent process describing the joint evolution of infinitesimally close trajectories of the original process leads to a multiplicative extension of the fluctuation relations. 1
A public turbulence database cluster and applications to study lagrangian evolution of velocity increments in turbulence,” arXiv.org
, 2008
"... in turbulence ..."
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Fluctuation relations in simple examples of nonequilibrium steady states
, 2008
"... We discuss fluctuation relations in simple cases of nonequilibrium Langevin dynamics. In particular, we show that close to nonequilibrium steady states with nonvanishing probability currents some of these relations reduce to a modified version of the fluctuationdissipation theorem. The latter ma ..."
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Cited by 17 (3 self)
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We discuss fluctuation relations in simple cases of nonequilibrium Langevin dynamics. In particular, we show that close to nonequilibrium steady states with nonvanishing probability currents some of these relations reduce to a modified version of the fluctuationdissipation theorem. The latter may be interpreted as the equilibriumlike relation in the reference frame moving with the mean local velocity determined by the probability current.
On the double cascades of energy and enstrophy in two dimensional turbulence
 B
"... Abstract. This paper is concerned with three interrelated issues on our proposal of double cascades intended to serve as a more realistic theory of twodimensional turbulence. We begin by examining the approach to the KLB limit. We present improved proofs of the result by Fjortoft. We also explain wh ..."
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Cited by 15 (9 self)
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Abstract. This paper is concerned with three interrelated issues on our proposal of double cascades intended to serve as a more realistic theory of twodimensional turbulence. We begin by examining the approach to the KLB limit. We present improved proofs of the result by Fjortoft. We also explain why in that limit the subleading downscale energy cascade and upscale enstrophy cascade are hidden in the energy spectrum. Then we review the experimental evidence from numerical simulations concerning the realizability of the energy and enstrophy cascade. The inverse energy cascade is found to be affected by the presense of a particular solution, and the downscale enstrophy cascade forms only under certain configurations of the dissipative terms. Finally, we amplify the hypothesis that the energy spectrum of the atmosphere reflects a combined downscale cascade of energy and enstrophy. The possibility of the downscale helicity cascade is also considered.
Turbulent pair dispersion of inertial particles
 UNDER CONSIDERATION FOR PUBLICATION IN J. FLUID MECH.
, 2009
"... The relative dispersion of pairs of inertial particles in incompressible, homogeneous, and isotropic turbulence is studied by means of direct numerical simulations at two values of the Taylorscale Reynolds number Reλ ∼ 200 and Reλ ∼ 400, corresponding to resolutions of 512 3 and 2048 3 grid points, ..."
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Cited by 15 (9 self)
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The relative dispersion of pairs of inertial particles in incompressible, homogeneous, and isotropic turbulence is studied by means of direct numerical simulations at two values of the Taylorscale Reynolds number Reλ ∼ 200 and Reλ ∼ 400, corresponding to resolutions of 512 3 and 2048 3 grid points, respectively. The evolution of both heavy and light particle pairs is analysed at varying the particle Stokes number and the fluidtoparticle density ratio. For particles much heavier than the fluid, the range of available Stokes numbers is St ∈ [0.1: 70], while for light particles the Stokes numbers span the range St ∈ [0.1:3] and the density ratio is varied up to the limit of vanishing particle density. For heavy particles, it is found that turbulent dispersion is schematically governed by two temporal regimes. The first is dominated by the presence, at large Stokes numbers, of smallscale caustics in the particle velocity statistics, and it lasts until heavy particle velocities have relaxed towards the underlying flow velocities. At such large scales, a second regime starts where heavy particles separate as tracers particles would do. As a consequence, at increasing inertia, a larger transient stage is observed, and the Richardson diffusion of simple tracers is recovered only at large times and large scales. These
Consistent families of Brownian motions and stochastic flows of kernels. ArXiv: math.PR/0611292
"... Consider the following mechanism for the random evolution of a distribution of mass on the integer lattice Z. At unit rate, independently for each site, the mass at the site is split into two parts by choosing a random proportion distributed according to some specified probability measure on [0,1] a ..."
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Cited by 13 (1 self)
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Consider the following mechanism for the random evolution of a distribution of mass on the integer lattice Z. At unit rate, independently for each site, the mass at the site is split into two parts by choosing a random proportion distributed according to some specified probability measure on [0,1] and dividing the mass in that proportion. One part then moves to each of the two adjacent sites. This paper considers a continuous analogue of this evolution, which may be described by means of a stochastic flow of kernels, the theory of which was developed by Le Jan and Raimond. One of their results is that such a flow is characterized by specifying its N point motions, which form a consistent family of Brownian motions. This means for each dimension N we have a diffusion in R N, whose N coordinates are all Brownian motions. Any M coordinates taken from the Ndimensional process are distributed as the Mdimensional process in the family. Moreover, in this setting, the only interactions between coordinates are local: when coordinates differ in value they evolve independently of each other. In this paper we explain how such multidimensional diffusions may be constructed and characterized via martingale problems.