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Start Over Topic mathematical physics Publication Year Range Last 50 years
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1. Rigorous study of the equilibria of collision kernels appearing in the theory of weak turbulence

Author

Maxime Breden, Laurent Desvillettes, Technische Universität Munchen - Université Technique de Munich [Munich, Allemagne] (TUM), Université Paris Diderot, Sorbonne Paris Cité, Paris, France, Université Paris Diderot - Paris 7 (UPD7), and The research leading to this paper was partly funded by Université SorbonneParis Cité, in the framework of the 'Investissements d'Avenir', convention ANR-11-IDEX-0005. MB also acknowledges partial support from a Lichtenberg Professorship grant ofthe VolkswagenStiftung awarded to C. Kuehn.

Subjects
Keys words: weak turbulence, Complex system, 82C40, FOS: Physical sciences, Type (model theory), [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA], 01 natural sciences, symbols.namesake, equilibria Mathematical Subject Classification: 76F99, Mathematics - Analysis of PDEs, Mathematics (miscellaneous), [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Statistical physics, 0101 mathematics, Mathematical Physics, Physics, 76P05, Turbulence, Mechanical Engineering, 010102 general mathematics, Mathematical Subject Classification: 76F99, 76P05, 82C40, Turbulence theory, Mathematical Physics (math-ph), Collision, equilibria, Functional Analysis (math.FA), Mathematics - Functional Analysis, 010101 applied mathematics, collision kernels, Boltzmann constant, Boltzmann's H-theorem, symbols, Particle, weak turbulence, Analysis, Analysis of PDEs (math.AP)
Abstract

In this paper, we rigorously obtain all the equilibria of collision kernels of type ``two particles give two particles'' appearing in weak turbulence theory under very general assumptions, thus completing the ``equality case'' in Boltzmann's H-theorem for those models. We also provide some rigorous results for collision kernels of type ``two particles give one particle'', under assumptions which include some of the most classical kernels of this type. The method of proof is inspired by the quantitative estimates obtained for the Landau equation by Desvillettes in [J. Funct. Anal. 269:1359-1403, 2015].

Published
2018

2. Extreme Value Laws for non stationary processes generated by sequential and random dynamical systems

Author

Ana C. Freitas, Sandro Vaienti, Jorge Milhazes Freitas, Centro de Matemática - Universidade do Porto (CMUP), Universidade do Porto = University of Porto, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), CPT - Ex E7 Systèmes dynamiques et théorie ergodique, Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), ACMF was partially supported by FCT (Portugal) grant SFRH/BPD/66174/2009. JMF was partially supported by FCT grant SFRH/BPD/66040/2009. All these grants are financially supported by the program POPH/FSE. ACMF and JMF were partially supported by FCT project FAPESP/19805/2014 and by CMUP (UID/MAT/00144/2013), which is funded by FCT with national (MEC) and European structural funds through the programs ,FEDER, under the partnership agreement PT2020. SV was supported by the ANRProject Perturbations and by the project Atracción de Capital Humano Avanzado del Extranjero MEC 80130047, CONICYT, at the CIMFAV, University of Valparaiso. All authors were partially supported by FCT project PTDC/MAT/12034 /2010, which is funded by national and European structural funds through the programs FEDER and COMPETE. SV is grateful to N. Haydn, M. Nicol and A. Török for several and fruitful discussions on sequential systems, especially in the framework of indifferent maps. JMF is grateful to M. Todd for fruitful discussions and careful reading of a preliminary version of this paper. All authors acknowledge the Isaac Newton Institute for Mathematical Sciences, where this work was initiated during the program Mathematics for the Fluid Earth., ANR-10-BLAN-0106,PERTURBATIONS,Perturbations aléatoires de systèmes dynamiques: applications non-uniformément dilatantes, isométries, billards et systèmes de fonctions itérées. Grandes déviations et valeurs extrêmes.(2010), Faculdade de Economia, Faculdade de Ciências, and Universidade do Porto

Subjects
Hitting Times, Statistics and Probability, Pure mathematics, Random dynamical systems, 60G70, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], FOS: Physical sciences, Dynamical Systems (math.DS), 01 natural sciences, 010104 statistics & probability, Sequential dynamical systems, FOS: Mathematics, 37A50, Mathematics - Dynamical Systems, 0101 mathematics, Extreme value theory, Mathematical Physics, Mathematics, 37A25, 37B20, Non-stationarity, Probability (math.PR), 010102 general mathematics, Mathematical Physics (math-ph), Non stationarity, Statistics, Probability and Uncertainty, Mathematics - Probability
Abstract

International audience; We develop and generalize the theory of extreme value for non-stationary stochastic processes, mostly by weakening the uniform mixing condition that was previously used in this setting. We apply our results to non-autonomous dynamical systems, in particular to sequential dynamical systems, both given by uniformly expanding maps and by maps with a neutral fixed point, and to a few classes of random dynamical systems. Some examples are presented and worked out in detail.

Published
2017