Colloquium Abstracts Fall 2021
Abstracts will be posted here for the colloquium talks when they are available.
CNRS and ENS de Lyon, France
September 16, 2021
Path large deviations for kinetic theories: beyond the Boltzmann, the Landau and the Lenard—Balescu kinetic equations
In many physical systems one seeks to describe effectively mesoscopic or macroscopic variables. Kinetic theories and kinetic equations are examples where the average mesoscopic dynamics is obtained through very clear theoretical procedures and can possibly lead to mathematical proofs. A few works go beyond the average evolution and describe, for instance, Gaussian fluctuations. However, for many physical systems, rare event can be of importance, and Gaussian fluctuations are not relevant. This is the case for instance if one wants to understand the irreversibility paradox associated to the kinetic equations, or to understand the dynamics that leads to rare but important events.
The aim of this presentation is to describe recent results where we derived explicitly the functional that describes the path large deviations for the empirical measure of dilute gases, plasma and systems of particles with long range interactions. The associated kinetic equations (the average evolution) are then either the Boltzmann, the Landau or the Balescu--Lenard—Guernsey equations. After making the classic assumptions in theoretical physics textbooks for deriving the kinetic equation, our derivation of the large deviation functional is exact. I will explain how we plan to generalize these results to turbulence problem.
These path large deviation principles give a very nice and transparent new interpretation of the classical irreversibility paradox. This new explanation is fully compatible with the classical one, but it gives a deeper insight.
For the large deviations associated to the Boltzmann equation (dilute gazes), and a general introduction (published in J. Stat. Phys. in 2020): F. Bouchet, 2020, Is the Boltzmann equation reversible? A large deviation perspective on the irreversibility paradox and the Boltzmann equation, Journal of Statistical Physics, 181, 515–550.
For the large deviations associated to the Landau equation (plasma below the Debye length, accepted for publication in J. Stat. Phys. in March 2021): O. Feliachi and F. Bouchet, 2021, Dynamical large deviations for plasma below the Debye length and the Landau equation, Journal of Statistical Physics, 183, 42.
For the large deviations associated to the Balescu—Guernsey--Lenard equation (plasma and systems with long range interactions, submitted to publication in J. Stat. Phys. in March 2021): O. Feliachi and F. Bouchet, 2021, Dynamical large deviations for plasma and other systems with long range interactions associated to the Lenard-Balescu-Guernsey equation, submitted to Journal of Statistical Physics, https://arxiv.org/abs/2105.05644.
Carnegie Mellon University
September 23, 2021
Dissipation Enhancement, Mixing and Blow-up Suppression
Diffusion and mixing are two fundamental phenomena that arise in a wide variety of applications. In this talk we quantitatively study the interaction between diffusion and mixing in the context of problems arising in fluid dynamics. The first question we address is how fast the energy can decay in the advection diffusion equation. Even though this is a simple linear equation, the energy decay rate is intrinsically related to the mixing properties of the advecting velocity field, and there are many unresolved open questions. I will present a few recent results involving both upper and lower bounds, and then consider applications to studying the long time dynamics of a few model non-linear equations.