The stochastic acceleration problem in two dimensions

Komorowski, Tomasz; Ryzhik, Lenya

doi:10.1007/bf02773954

Cited by 12 publications

(30 citation statements)

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“…We restrict our attention in this paper also to the case d ≥ 3 though a generalization using the results of [5] and [10] is possible. If S = 0 then solution of (2.6) is given explicitly in terms of the random characteristics (X δ (t), K δ (t)): define…”

Section: A Review Of the Case In The Absence Of A Mismatchmentioning

confidence: 99%

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Wave Field Correlations in Weakly Mismatched Random Media

Bal

Ryzhik

2006

Stoch. Dyn.

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This paper concerns the derivation of a Fokker-Planck equation for the correlation of two high frequency wave fields propagating in two different random media. The mismatch between the random media need be small, on the order of the wavelength, and their correlation length need be large relative to the wavelength. The loss of correlation caused by the mismatch in the random media is quantified and the limit process for the phase difference is obtained. The derivation is based on a random Liouville equation to model high frequency correlations and on the method of characteristics to characterize mixing in the random Liouville equation. Applications of such correlation loss include the monitoring in time of random media and the analysis of time reversed waves in changing heterogeneous domains.

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Section: A Review Of the Case In The Absence Of A Mismatchmentioning

confidence: 99%

“…This potential return was a major obstacle in the proofs in [2,8,9,10]. The strategy of the proof is as follows.…”

Section: The Proof Of Proposition 41mentioning

confidence: 99%

Wave Field Correlations in Weakly Mismatched Random Media

Bal

Ryzhik

2006

Stoch. Dyn.

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“…In a certain weak-coupling/space-adiabatic limit and under some mixing hypotheses on the random potential, we show that the long-time, large-distance behavior of the soliton is described by momentum diffusion in dimension N ≥ 2: The soliton center of mass undergoes Brownian motion on the energy sphere of constant momentum. This is analogous to the long-time/large-distance behavior of the classical particle in a random potential [29], [30], [15]. Moreover, in dimension N ≥ 3, the long-time limit of a momentum diffusion is spatial diffusion; see [31].…”

Section: Overview Of Earlier Results and Heuristic Discussionmentioning

confidence: 91%

“…The dynamics of a classical particle in a weak, strongly mixing, and homogeneous random potential has been the object of extensive studies in the literature; see [29], [15], [30], and [31].…”

Section: Description Of the Problemmentioning

confidence: 99%

“…As for the fluctuation, we control its H 1 -norm almost surely using an approximate Lyapunov functional. To study the limiting dynamics in the weak-coupling/spaceadiabatic limit, we rely on explicit control of the difference between the effective dynamics of the soliton and a classical particle in the random potential and on the results of [29], [30], and [31] on the stochastic acceleration of a classical particle in a random potential.…”

Section: Description Of the Problemmentioning

confidence: 99%

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Stochastic Acceleration of Solitons for the Nonlinear Schrödinger Equation

Salem¹,

Sulem²

2009

SIAM J. Math. Anal.

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We study the effective dynamics of solitons for the generalized nonlinear Schrödinger equation in a random potential. We show that when the external potential varies slowly in space compared to the size of the soliton, the dynamics of the center of the soliton is almost surely described by Hamilton's equations for a classical particle in the random potential, plus error terms due to radiation damping. Furthermore, we prove a limit theorem for the dynamics of the center of mass of the soliton in the weak-coupling and space-adiabatic limit in two and higher dimensions: Under certain mixing hypotheses for the potential, the momentum of the center of mass of the soliton converges in law to a diffusion process on a sphere of constant momentum. In three and higher dimensions, the dynamics of the center of mass of the soliton converges to spatial diffusion.

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Brownian Motion, “Diverse and Undulating”

Duplantier

2005

Einstein, 1905–2005

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Truly man is a marvelously vain, diverse, and undulating object. It is hard to found any constant and uniform judgment on him. Michel de Montaigne, Les Essais, Book I, Chapter 1: "By diverse means we arrive at the same end"; in The Complete Essays of Montaigne, Donald M. Frame transl., Stanford University Press (1958).Pour distinguer les choses les plus simples de celles qui sont compliquées et pour les chercher avec ordre, il faut, dans chaque série de choses où nous avons déduit directement quelques vérités d'autres vérités, voir quelle est la chose la plus simple, et comment toutes les autres en sont plus, ou moins, ouégalementéloignées. René Descartes, Règles pour la direction de l'esprit, Règle VI.In order to distinguish what is most simple from what is complex, and to deal with things in an orderly way, what we must do, whenever we have a series in which we have directly deduced a number of truths one from another, is to observe which one is most simple, and how far all the others are removed from this-whether more, or less, or equally. René Descartes, Rules for the Direction of the Mind, Rule VI. 1 Expanded and updated version (13 May 2007). 2 Note: From géomance, a way to foretell the future; a form of divination. 202 B. Duplantier Poincaré Seminar 2005 L'antimodernisme, c'est la liberté des modernes. Antoine Compagnon, about his book "Les antimodernes : de Joseph de Maistreà Roland Barthes," Bibliothèque des Idées, Gallimard, March 2005.Antimodernism is the liberty of modern men.Here we briefly describe the history of Brownian motion, as well as the contributions of Einstein, Sutherland, Smoluchowski, Bachelier, Perrin and Langevin to its theory. The always topical importance in physics of the theory of Brownian motion is illustrated by recent biophysical experiments, where it serves, for instance, for the measurement of the pulling force on a single DNA molecule.In the second part, we stress the mathematical importance of the theory of Brownian motion, illustrated by two chosen examples. The by-now classic representation of the Newtonian potential by Brownian motion is explained in an elementary way. We conclude with the description of recent progress seen in the geometry of the planar Brownian curve. At its heart lie the concepts of conformal invariance and multifractality, associated with the potential theory of the Brownian curve itself. A brief history of Brownian motionSeveral classic works give a historical view of Brownian motion. Amongst them, we cite those of Brush, 3 Nelson, 4 Nye, 5 Pais 6 , Stachel 7 and Wax. 8 We also cite a number of essays in mathematics, 9 physics, 10 11 especially those which have appeared very recently for the centenary of Einstein's 1905 articles, 12 and in biology. 13 3 Stephen G.

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The stochastic acceleration problem in two dimensions

Abstract: We consider the motion of a particle in a two-dimensional spatially homogeneous mixing potential and show that its momentum converges to the Brownian motion on a circle. This complements the limit theorem of Kesten and Papanicolaou [4] proved in dimensions d ≥ 3.

Cited by 12 publications

References 8 publications

Wave Field Correlations in Weakly Mismatched Random Media

Wave Field Correlations in Weakly Mismatched Random Media

Stochastic Acceleration of Solitons for the Nonlinear Schrödinger Equation

Brownian Motion, “Diverse and Undulating”

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