arXiv subjects

Théophile Dolmaire

Publications and source records attributed to Théophile Dolmaire.

9 recordsLinked to original sources

arXiv2026

Four collapsing one-dimensional particles: a dynamical system approach of the spherical billiard reduction

We consider a system of four one-dimensional inelastic hard spheres evolving on the real line $\mathbb{R}$, and colliding according to a scattering law characterized by a fixed restitution coefficient $r$. We study the possible orders of collisions when the inelastic collapse occurs, relying on the so-called $\mathfrak{b}$-to-$\mathfrak{b}$ mapping, a two-dimensional dynamical system associated to the original particle system which encodes all the possible collision orders. We prove that the $\mathfrak{b}$-to-$\mathfrak{b}$ mapping is a piecewise projective transformation, which allows one to perform efficient numerical simulations of its orbits. We recover previously known results concerning the one-dimensional four-particle inelastic hard sphere system and we support the conjectures stated in the literature concerning particular periodic orbits. We discover three new families of periodic orbits that coexist depending on the restitution coefficient, we prove rigorously that there exist stable periodic orbits for the $\mathfrak{b}$-to-$\mathfrak{b}$ mapping for restitution coefficients larger than the upper bounds previously known, and we prove the existence of quasi-periodic orbits for this mapping.

math.DS↗

arXiv2025

Boltzmann-Grad limit for the inelastic Lorentz gas: Part I. Existence, uniqueness, and rigorous derivation via weak convergence

In this paper we provide a rigorous derivation of the inelastic linear Boltzmann equation, in the Boltzmann-Grad limit, from a dissipative, random, Lorentz gas in arbitrary dimensions d $\geq$ 2. Specifically, we consider a microscopic particle system where scatterers are randomly distributed according to a Poisson process, and a tagged light particle undergoes inelastic collisions with the scatterers following a reflection law characterized by a fixed restitution coefficient. We establish the existence and uniqueness of weak solutions to the inelastic linear Boltzmann equation within the class of non-negative Radon measures, assuming that the initial data has a finite exponential moment. We first show that the forward dynamics of the dissipative particle system is globally defined almost surely and then prove the weak$-*$ convergence of the microscopic solution towards the weak solutions of the inelastic linear Boltzmann equation, providing an explicit rate of convergence. Furthermore, under the same initial data assumptions, we prove the existence of strong solutions to the inelastic linear Boltzmann equation, constructed via a series representation of the solutions.

math-ph↗

arXiv2025

The global attractor of the inelastic linear Boltzmann equation in a gravity field for Maxwell molecules

In this article we consider the linear inelastic Boltzmann equation in presence of a uniform and fixed gravity field, in the case of Maxwell molecules. We first obtain a well-posedness result in the space of finite, non-negative Radon measures. In addition, we rigorously prove the existence of a stationary solution under the non-equilibrium condition which is induced by the presence of the external field. We further show that this stationary solution is unique in the class of the finite, non-negative Radon measures with finite first order moment, and that all the solutions in this class converge towards the stationary solution in the weak topology of the measures.

math.AP↗

arXiv2025

An Alexander-like theorem for a particle model with inelastic collisions

We consider a finite system of hard spheres that collide inelastically according to a particular model, losing a fixed amount of kinetic energy at each collision. We develop the theory of the Transport-Collision-Transport (TCT) dynamics, which allows to study precisely the evolution of the Lebesgue measure in the phase space under the action of the flows of particle systems that can interact via instantaneous binary collision. We show that the scattering mapping associated to the inelastic hard sphere system we introduce preserves locally the Lebesgue measure in the velocity space, in spite of the fact that a positive amount of kinetic energy is lost at each inelastic collision. We prove the analog of Alexander's theorem for our model, which allows us to deduce the global well-posedness of the trajectories, for almost every initial datum.

math-ph↗

arXiv2025

On the local Maxwellians solving the Boltzmann equation with boundary condition

We derive the expressions of the local Maxwellians that solve the Boltzmann equation in the interior of an open domain. We determine which of these local Maxwellians satisfy the Boltzmann equation in a regular domain with boundary, without assuming the boundedness of the domain. We investigate separately, on the one hand, the case of the bounce-back boundary condition in any dimension, and on the other hand the case of the specular reflection boundary condition, in dimension $d = 2$ and $d = 3$. In the case of the bounce-back boundary condition, we prove that the only local Maxwellians solving the Boltzmann equation with boundary condition are the global Maxwellians. In the case of the specular reflection, we provide a complete classification of the domains for which only the global Maxwellians solve the Boltzmann equation with boundary condition, and we describe all the local Maxwellians that solve the equation for the domains presenting symmetries.

math.AP↗

arXiv2024

One-dimensional inelastic collapse of four particles: asymmetric collision sequences and spherical billiard reduction

We consider a one-dimensional system of four inelastic hard spheres, colliding with a fixed restitution coefficient $r$, and we study the inelastic collapse phenomenon for such a particle system. We study a periodic, asymmetric collision pattern, proving that it can be realized, despite its instability. We prove that we can associate to the four-particle dynamical system another dynamical system of smaller dimension, acting on $\{1,2,3\} \times \mathbb{S}^2$, and that encodes the collision orders of each trajectory. We provide different representations of this new dynamical system, and study numerically its $ω$-limit sets. In particular, the numerical simulations suggest that the orbits of such a system might be quasi-periodic.

math.DS↗

arXiv2024

Properties of some dynamical systems for three collapsing inelastic particles

In this article we continue the study of the collapse of three inelastic particles in dimension $d \geq 2$, complementing the results we obtained in its companion paper. We focus on the particular case of the nearly-linear inelastic collapse, when the order of collisions becomes eventually the infinite repetition of the period $0-1$, $0-2$, under the assumption that the relative velocities of the particles (with respect to the central particle $0$) do not vanish at the time of collapse. Taking as starting point the full dynamical system that describes two consecutive collisions of the nearly-linear collapse, we derive formally a two-dimensional dynamical system, called the two-collision mapping. This mapping governs the evolution of the variables of the full dynamical system. We show in particular that in the so-called Zhou-Kadanoff regime, the orbits of the two-collision mapping can be described in full detail. We study rigorously the two-collision mapping, proving that the Zhou-Kadanoff regime is stable and locally attracting in a certain region of the phase space of the two-collision mapping. We describe all the fixed points of the two-collision mapping in the case when the norms of the relative velocities tend to the same positive limit. We establish conjectures to characterize the orbits that verify the Zhou-Kadanoff regime, motivated by numerical simulations, and we prove these conjectures for a simplified version of the two-collision mapping.

math-ph↗

arXiv2024

Collapse of inelastic hard spheres in dimension $d \geq 2$

We investigate the collapse of three inelastic particles in dimension $d \geq 2$. We obtain general results of convergence and asymptotics concerning the variables of the dynamical system describing a collapsing system of particles. We prove a complete classification of the singularities when a collapse of three particles takes place, obtaining only two possible orders of collisions between the particles. In the first case we recover that the particles arrange in a nearly-linear chain, already studied by Zhou and Kadanoff, and in the second case we obtain that the particles arrange in a triangle, and we show that, after sufficiently many collisions, the particles collide according to a unique order of collisions, which is periodic. Finally, we construct an initial configuration leading to a nearly-linear collapse, stable under perturbations, and such that the angle between the particles at the time of collapse can be chosen a priori, with an arbitrary precision.

math-ph↗

arXiv2021

About Lanford's theorem in the half-space with specular reflection

The present article proposes a rigorous derivation of the Boltzmann equation in the half-space. We show an analog of the Lanford's theorem in this domain, with specular reflection boundary condition, stating the convergence in the low density limit of the first marginal of the density function of a system of N hard spheres towards the solution of the Boltzmann equation associated to the initial data corresponding to the initial state of the one-particle-density function. The original contributions of this work consist in two main points: the rigorous definition of the collision operator and of the functional space in which the BBGKY hierarchy is solved in a strong sense; and the adaptation to the case of the half-space of the control of the recollisions performed by Gallagher, Saint-Raymond and Texier, which is a crucial step to obtain the Lanford's theorem.

math.AP↗