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1.
We analyze the effective diffusivity of a passive scalar in a two-dimensional, steady, incompressible random flow that has
mean zero and a stationary stream function. We show that in the limit of small diffusivity or large Peclet number, with convection
dominating, there is substantial enhancement of the effective diffusivity. Our analysis is based on some new variational principles
for convection diffusion problems and on some facts from continuum percolation theory, some of which are widely believed to
be correct but have not been proved yet. We show in detail how the variational principles convert information about the geometry
of the level lines of the random stream function into properties of the effective diffusivity and substantiate the result
of Isichenko and Kalda that the effective diffusivity behaves likeɛ
3/13 when the molecular diffusivityɛ is small, assuming some percolation-theoretic facts. We also analyze the effective diffusivity for a special class of convective
flows, random cellular flows, where the facts from percolation theory are well established and their use in the variational
principles is more direct than for general random flows. 相似文献
2.
Albert C. Fannjiang 《Comptes Rendus Mathematique》2006,342(2):109-114
A systematic derivations of self-averaging scaling limits of parabolic waves in terms of the Wigner distribution function is presented. The convergence of the Wigner distribution to one of the six deterministic radiative transfer equations is established. One of the main contributions of this Note is a unified framework for space–time scaling limits that lead to radiative transfer. To cite this article: A.C. Fannjiang, C. R. Acad. Sci. Paris, Ser. I 342 (2006). 相似文献
3.
Bal Guillaume Fannjiang Albert Papanicolaou George Ryzhik Leonid 《Journal of statistical physics》1999,95(1-2):479-494
We derive radiative transport equations for solutions of a Schrödinger equation in a periodic structure with small random inhomogeneities. We use systematically the Wigner transform and the Bloch wave expansion. The streaming part of the radiative transport equations is determined entirely by the Bloch spectrum, and the scattering part by the random fluctuations. 相似文献
4.
We consider a reaction-diffusion equation in a cellular flow. We prove that in the strong flow regime there are two possible
scenarios for the initial data that is compactly supported and the size of the support is large enough. If the flow cells
are large compared to the reaction length scale, propagating fronts will always form. For small cell size, any finitely supported
initial data will be quenched by a sufficiently strong flow. We estimate that the flow amplitude required to quench the initial
data of support L0 is
The essence of the problem is the question about the decay of the L∞-norm of a solution to the advection-diffusion equation, and the relation between this rate of decay and the properties of
the Hamiltonian system generated by the two-dimensional incompressible fluid flow.
Received: September 2004 Revision: September 2005 Accepted: September 2005 相似文献
5.
Summary We prove long time diffusive behavior (homogenization) for convection-diffusion in a turbulent flow that it incompressible and has a stationary and square integrable stream matrix. Simple shear flow examples show that this result is sharp for flows that have stationary stream matrices. 相似文献
6.
We study transport in random undirectional wave-like velocity fields with nonlinear dispersion relations. For this simple model, we have several interesting findings: (1) In the absence of molecular diffusion the entire family of fractional Brownian motions (FBMs), persistent or anti-persistent, can arise in the scaling limit. (2) The infrared cutoff may alter the scaling limit depending on whether the cutoff exceeds certain critical value or not. (3) Small, but nonzero, molecular diffusion can drastically change the scaling limit. As a result, some regimes stay intact; some (persistent) FBM regimes become non-Gaussian and some other FBM regimes become Brownian motions with enhanced diffusion coefficients. Moreover, in the particular regime where the scaling limit is a Brownian motion in the absence of molecular diffusion, the vanishing molecular diffusion limit of the enhanced diffusion coefficient is strictly larger than the diffusion coefficient with zero molecular diffusion. This is the first such example that we are aware of to demonstrate rigorously a nonperturbative effect of vanishing molecular diffusion on turbulent diffusion coefficient. 相似文献
7.
The Fourier-domain Douglas–Rachford (FDR) algorithm is analyzed for phase retrieval with a single random mask. Since the uniqueness of phase retrieval solution requires more than a single oversampled coded diffraction pattern, the extra information is imposed in either of the following forms: 1) the sector condition on the object; 2) another oversampled diffraction pattern, coded or uncoded.For both settings, the uniqueness of projected fixed point is proved and for setting 2) the local, geometric convergence is derived with a rate given by a spectral gap condition. Numerical experiments demonstrate global, power-law convergence of FDR from arbitrary initialization for both settings as well as for 3 or more coded diffraction patterns without oversampling. In practice, the geometric convergence can be recovered from the power-law regime by a simple projection trick, resulting in highly accurate reconstruction from generic initialization. 相似文献
8.
We introduce the notion of relaxation time for noisy quantum maps on the 2d-dimensional torus – generalization of previously studied dissipation time. We show that the relaxation time is sensitive to the chaotic behavior of the corresponding classical system if one simultaneously
considers the semiclassical limit
together with the limit of small noise strength (ε → 0).
Focusing on quantized smooth Anosov maps, we exhibit a semiclassical régime
(where E > 1) in which classical and quantum relaxation times share the same asymptotics: in this régime, a quantized Anosov map relaxes
to equilibrium fast, as the classical map does. As an intermediate result, we obtain rigorous estimates of the quantum-classical
correspondence for noisy maps on the torus, up to times logarithmic in
On the other hand, we show that in the “quantum régime”
quantum and classical relaxation times behave very differently. In the special case of ergodic toral symplectomorphisms (generalized
“Arnold’s cat” maps), we obtain the exact asymptotics of the quantum relaxation time and precise the régime of correspondence
between quantum and classical relaxations.
Communicated by Jens Marklof
submitted 4/01/05, accepted 2/02/05 相似文献
9.
Albert Fannjiang 《Journal of Differential Equations》2002,179(2):433-455
We study the long time transport property of conservative systems perturbed by a small white noise. We introduce the dissipation and martingale times and show how they are related to the diffusion time on which a limit theorem is valid. The limit theorem is a probabilistic version of homogenization with vanishing molecular diffusion. Examples of nontrivial time scales are given. 相似文献
10.
Albert C. Fannjiang 《Journal of statistical physics》2004,114(1-2):115-135
We prove that the passive scalar field in the Ornstein–Uhlenbeck velocity field with wave-number dependent correlation times converges, in the white-noise limit, to that of Kraichnan's model with higher spatial regularity. 相似文献