High frequency chaotic waves
Stéphane NONNENMACHER
Laboratoire de Mathématiques d'Orsay, Université Paris-Saclay
I will address the dynamics of linear scalar waves, but in systems where the corresponding classical dynamics is chaotic (this is where the nonlinearity is hidden). A typical example is the scalar wave equation (or the Schrödinger equation) in a box, in which the classical billiard flow is chaotic. Due to the linearity of the wave equation, one may focus on the stationary wave modes, identical with the eigenmodes of quantum particles in the box.
One objective is to understand the properties of these stationary modes, in particular in the high frequency regime, and at various scales. At the wavelength scale (microscopic scale), these modes feature intricate nodal patterns, which can be analyzed by statistical tools. At the macroscopic scale (the scale of the box), most of these modes are equidistributed across the full box, a wave manifestation of the ergodicity of the classical flow (one speaks of Quantum Ergodicity). Does this equidistribution property hold down to some mesoscopic scales? Does it concern all high frequency modes, or can some exceptional modes feature some partial localization? I will give some old and more recent mathematical results on those questions, and sketches of the semiclassical methods used to address them.
IMAGES
Water slider (plus de détails...)
CONFÉRENCES
New Challenges in Turbulence Research VII, École de Physique des Houches, 10 Février 2025
New Challenges in Turbulence Research VII, École de Physique des Houches, 10 Février 2025
New Challenges in Turbulence Research VII, École de Physique des Houches, 10 Février 2025