Section: New Results

Macroscopic behaviors of large interacting particle systems

Stochastic acceleration and approach to equilibrium

S. De Bièvre, Carlos Mejia-Monasterio (Madrid) and Paul E. Parris (Missouri)  [57] studied thermal equilibration in a two-component Lorentz gas, in which the obstacles are modeled by rotating disks. They show that a mechanism of dynamical friction leads to a fluctuation-dissipation relation that is responsible for driving the system to equilibrium.

Stephan De Bièvre, Jeremy Faupin (Metz) and Schuble (Metz)  [59] studied a related model quantum mechanically. Here a quantum particle moves through a field of quantized bose fields, modeling membranes that exchange energy and momentum with the particle. They establish a number of spectral properties of this model, that will be essential to study the time-asymptotic behavior of the system.

S. De Bièvre and collaborators analyse in [20] a multi-particle, kinetic version of a Hamiltonian model describing the interaction of a gas of particles with a vibrating medium. They prove existence results for weak solutions, and identify an asymptotic regime where the model, quite surprisingly, approaches the attractive Vlasov—Poisson system.

Towards the weak KPZ universality conjecture

One may start by considering the microscopic system in equilibrium (its measure is parametrized by the thermodynamical quantities under investigation). By removing the mean to the empirical measure and by scaling it properly, one would like to show that the random process, obtained by this rescaling, converges, as the size of the system is taken to infinity, to another random process which is a solution of some generalized stochastic PDE. Thanks to the remarkable recent result of M. Jara and P. Gonçalves [66], one has now all in hands to establish the latter result for a particular stochastic PDE known as the stochastic Burgers equation, and its companion, the Kardar-Parisi-Zhang (KPZ) equation. Indeed, in the latter paper, the authors introduce a new tool, called the second order Boltzmann-Gibbs principle, which permits to replace certain additive functionals of the dynamics by similar functionals given in terms of the density of the particles.

In [28], M. Simon in collaboration with T. Franco and P. Gonçalves, investigate the case of a microscopic dynamics with local defects, which is much harder. More precisely, the microscopic particle system is locally perturbed, and depending on the type of perturbation, the macroscopic laws can hold different boundary conditions. Since the ideas of [66] do not apply to the model considered there, they propose a new way to estimate the error in the replacement performed in the Boltzmann-Gibbs principle.

In the same spirit, M. Simon in collaboration with O. Blondel and P. Gonçalves investigate in [7] the class of kinetically constrained lattice gases that have been introduced and intensively studied in the literature in the past few years. In these models, particles are subject to restrictive constraints that make both approaches of [66] and [28] not work, so that new mathematical tools are needed. The main technical difficulty is that their model exhibits configurations that do not evolve under the dynamics and are locally non-ergodic. Their proof does not impose any knowledge on the spectral gap for the microscopic models. Instead, it relies on the fact that, under the equilibrium measure, the probability to find a blocked configuration in a finite box is exponentially small in the size of the box.

With these two recent results, M. Simon and coauthors contribute towards the weak KPZ universality conjecture, which states that a large class of one-dimensional weakly asymmetric conservative systems should converge to the KPZ equation.

Diffusion and fractional diffusion of energy

The rigorous derivation of the heat equation from deterministic systems of Newtonian particles is one of the most fundamental questions in mathematical physics. The main issue is that the existence of conservation laws and the high number of degrees of freedom impose very poor ergodic properties to the associated dynamical systems. A possible way out of this lack of ergodicity is to introduce stochastic models, in such a way that in one hand ergodicity issues are solved by the stochastic dynamics and in the other hand the qualitative behaviour of the system is not modified by the randomness. In these models, one starts with a chain of oscillators with a Hamiltonian dynamics, and one adds a stochastic component in such a way that the fundamental conservation laws (energy, momentum and stretch in this case) are maintained, and the corresponding Gibbs measures become ergodic.

It was already proved in [51] that these stochastic chains model correctly the behaviour of the conductivity. In particular, it is prove that Fourier law holds in dimension $d\ge 3$ if energy and momentum are conserved, and in any dimension if only energy is conserved. Once the conductivity has been successfully understood, one investigates the existence of the hydrodynamic limit, which fully describes the macroscopic evolution of the empirical profiles associated to the conserved quantity. In [41], M. Simon in collaboration with T. Komorowski and S. Olla consider the unpinned harmonic chain where the velocities of particles can randomly change sign. The only conserved quantities of the dynamics are the energy and the elongation. Using a diffusive space-time scaling, the profile of elongation evolves independently of the energy and follows a linear diffusive equation. The energy profile evolves following a non-linear diffusive equation involving the elongation. The presence of non-linearity makes the macroscopic limit non-trivial, and its mathematical proof requires very sophisticated arguments.

In [52] and [69] it has been previously shown that in the case of one-dimensional harmonic oscillators with noise that preserves the momentum, the scaling limit of the energy fluctuations is ruled by the fractional heat equation

This equation does not only predict the superdiffusivity of energy in momentum-conserving models, but it also predicts the speed at which it diverges. This result opens a way to a myriad of open problems. The main goal is to observe anomalous fractional superdiffusion type limit in the context of low dimensional asymmetric systems with several conserved quantities. In two recent papers by M. Simon in collaboration with C. Bernardin, P. Gonçalves, M. Jara, M. Sasada [53] & [32], they confirmed rigorously recent Spohn's predictions on the Lévy form of the energy fluctuations for a harmonic chain perturbed by an energy-volume conservative noise. In [32] they also showed the existence of a crossover between a normal diffusion regime and a fractional superdiffusion regime by tuning a parameter of a supplementary stochastic noise conserving the energy but not the volume.