Thierry Bodineau

Directeur de recherche CNRS

Laboratoire Alexander Grothendieck
Institut des Hautes Études Scientifiques
35 route de Chartres
91440 Bures-sur-Yvette, France
Bureau : 1S9

bodineau@ihes.fr

As a researcher in the field of probability theory and mathematical physics, I’m particularly interested in kinetic theory, non-equilibrium statistical mechanics, phase coexistence and renormalisation.

Conférence (5-7 juin 2023) : - Mathematics of disordered systems : a tribute to Francis Comets

Teaching

Cours de l'École Polytechnique : Modélisation de phénomènes aléatoires - PDF

L'aléa joue un rôle déterminant dans des contextes variés et il est souvent nécessaire de le prendre en compte dans de multiples aspects des sciences de l’ingénieur, citons notamment la gestion de données, les télécommunications, la reconnaissance de formes. Plus généralement, l'aléa intervient aussi en économie (gestion du risque), en médecine (propagation d'une épidémie), en biologie (évolution d'une population) ou en physique statistique (théorie des transitions de phases). Dans les applications, les données observées au cours du temps sont souvent modélisées par des variables aléatoires corrélées dont on aimerait prédire le comportement. L'objet de ce cours est de formaliser la notion de corrélation en étudiant deux types de processus aléatoires fondamentaux en théorie des probabilités : les chaînes de Markov et les martingales.

Cours du M2 Mathématiques de l'aléatoire : Systèmes de particules en interaction - PDF

Si on se place à l’échelle atomique, un gaz ne présente pas de structure cohérente, pourtant à l’échelle macroscopique, on le caractérise par quelques paramètres simples comme sa densité et sa pression qui peuvent évoluer au cours du temps. Mathématiquement le passage du microscopique au macroscopique se formalise en assimilant les atomes à un ensemble de variables aléatoires en interaction et en étudiant leur comportement limite qui est alors décrit par une EDP. Dans ce cours, nous étudierons plus spécifiquement un gaz de sphères dures dans la limite cinétique.

Publications

Dynamics of dilute gases : a statistical approach

Thierry Bodineau, Isabelle Gallagher, Laure Saint-Raymond, Sergio Simonella

The evolution of a gas can be described by different models depending on the observation scale. A natural question, raised by Hilbert in his sixth problem, is whether these models provide consistent predictions. In particular, for rarefied gases, it is expected that continuum laws of kinetic theory can be obtained directly from molecular dynamics governed by the fundamental principles of mechanics. In the case of hard sphere gases, Lanford showed that the Boltzmann equation emerges as the law of large numbers in the low density limit, at least for very short times. The goal of this survey is to present recent progress in the understanding of this limiting process, providing a complete statistical description.

- 2022-03-05

Long-Time Correlations For A Hard-Sphere Gas At Equilibrium

Thierry Bodineau, Isabelle Gallagher, Laure Saint-Raymond, Sergio Simonella

It has been known since Lanford [22] that the dynamics of a hard sphere gas is described in the low density limit by the Boltzmann equation, at least for short times. The classical strategy of proof fails for longer times, even close to equilibrium. In this paper, we introduce a weak convergence method coupled with a sampling argument to prove that the covariance of the fluctuation field around equilibrium is governed by the linearized Boltzmann equation globally in time (including in diffusive regimes). This method is much more robust and simple than the one devised in [4] which was specific to the 2D case.

Communications on Pure and Applied Mathematics - 2021

Fluctuation theory in the Boltzmann--Grad limit

Thierry Bodineau, Isabelle Gallagher, Laure Saint-Raymond, Sergio Simonella

We develop a rigorous theory of hard-sphere dynamics in the kinetic regime, away from thermal equilibrium. In the low density limit, the empirical density obeys a law of large numbers and the dynamics is governed by the Boltzmann equation. Deviations from this behaviour are described by dynamical correlations, which can be fully characterized for short times. This provides both a fluctuating Boltzmann equation and large deviation asymptotics.

Journal of Statistical Physics - 2020

Statistical dynamics of a hard sphere gas: fluctuating Boltzmann equation and large deviations

Thierry Bodineau, Isabelle Gallagher, Laure Saint-Raymond, Sergio Simonella

- 2022-08-25

Long-time derivation at equilibrium of the fluctuating Boltzmann equation

Thierry Bodineau, Isabelle Gallagher, Laure Saint-Raymond, Sergio Simonella

We study a hard sphere gas at equilibrium, and prove that in the low density limit, the fluctuations converge to a Gaussian process governed by the fluctuating Boltzmann equation. This result holds for arbitrarily long times. The method of proof builds upon the weak convergence method introduced in the companion paper [8] which is improved by considering clusters of pseudo-trajectories as in [7].

- 2022-01-12

Spectral gap critical exponent for Glauber dynamics of hierarchical spin models

Roland Bauerschmidt, Thierry Bodineau

We develop a renormalisation group approach to deriving the asymptotics of the spectral gap of the generator of Glauber type dynamics of spin systems at and near a critical point. In our approach, we derive a spectral gap inequality, or more generally a Brascamp-Lieb inequality, for the measure recursively in terms of spectral gap or Brascamp-Lieb inequalities for a sequence of renormalised measures. We apply our method to hierarchical versions of the $4$-dimensional $n$-component $|\varphi|^4$ model at the critical point and its approach from the high temperature side, and the $2$-dimensional Sine-Gordon and the Discrete Gaussian models in the rough phase (Kosterlitz-Thouless phase). For these models, we show that the spectral gap decays polynomially like the spectral gap of the dynamics of a free field (with a logarithmic correction for the $|\varphi|^4$ model), the scaling limit of these models in equilibrium.

Communications in Mathematical Physics - 2019-08-29

A very simple proof of the LSI for high temperature spin systems

Roland Bauerschmidt, Thierry Bodineau

We present a very simple proof that the $O(n)$ model satisfies a uniform logarithmic Sobolev inequality (LSI) if the positive definite coupling matrix has largest eigenvalue less than $n$. This condition applies in particular to the SK spin glass model at inverse temperature $\beta < 1/4$. It is the first result of rapid relaxation for the SK model and requires significant cancellations between the ferromagnetic and anti-ferromagnetic spin couplings that cannot be obtained by existing methods to prove Log-Sobolev inequalities. The proof also applies to more general bounded and unbounded spin systems. It uses a single step of zero range renormalisation and Bakry--Emery theory for the renormalised measure.

Journal of Functional Analysis - 2019-04-15

Phase fluctuations in the ABC model

Thierry Bodineau, Bernard Derrida

We analyze the fluctuations of the steady state profiles in the modulated phase of the ABC model. For a system of $L$ sites, the steady state profiles move on a microscopic time scale of order $L^3$. The variance of their displacement is computed in terms of the macroscopic steady state profiles by using fluctuating hydrodynamics and large deviations. Our analytical prediction for this variance is confirmed by the results of numerical simulations.

Journal of Statistical Physics - 2011-11-01

A universality property for last-passage percolation paths close to the axis

Thierry Bodineau, James Martin

We consider a last-passage directed percolation model in $Z_+^2$, with i.i.d. weights whose common distribution has a finite $(2+p)$th moment. We study the fluctuations of the passage time from the origin to the point $(n,n^a)$. We show that, for suitable $a$ (depending on $p$), this quantity, appropriately scaled, converges in distribution as $n\to\infty$ to the Tracy-Widom distribution, irrespective of the underlying weight distribution. The argument uses a coupling to a Brownian directed percolation problem and the strong approximation of Komlós, Major and Tusnàdy.

Electronic Communications in Probability - 2005-06-09

Cluster expansion for a dilute hard sphere gas dynamics

Thierry Bodineau, Isabelle Gallagher, Laure Saint-Raymond, Sergio Simonella

In [7], a cluster expansion method has been developed to study the fluctuations of the hard sphere dynamics around the Boltzmann equation. This method provides a precise control on the exponential moments of the empirical measure, from which the fluctuating Boltzmann equation and large deviation estimates have been deduced. The cluster expansion in [7] was implemented at the level of the BBGKY hierarchy, which is a standard tool to investigate the deterministic dynamics [11]. In this paper, we introduce an alternative approach, in which the cluster expansion is applied directly on real trajectories of the particle system. This offers a fresh perspective on the study of the hard sphere dynamics in the low density limit, allowing to recover the results obtained in [7], and also to describe the actual clustering of particle trajectories.

Proceedings of the International Congress on Mathematical Physics, special issue of Journal of Mathematical Physics - 2022-07-01

Activity phase transition for constrained dynamics

Thierry Bodineau, Cristina Toninelli

We consider two cases of kinetically constrained models, namely East and FA-1f models. The object of interest of our work is the activity A(t) defined as the total number of configuration changes in the interval [0,t] for the dynamics on a finite domain. It has been shown in [GJLPDW1,GJLPDW2] that the large deviations of the activity exhibit a non-equilibirum phase transition in the thermodynamic limit and that reducing the activity is more likely than increasing it due to a blocking mechanism induced by the constraints. In this paper, we study the finite size effects around this first order phase transition and analyze the phase coexistence between the active and inactive dynamical phases in dimension 1. In higher dimensions, we show that the finite size effects are also determined by the dimension and the choice of boundary conditions.

Communications in Mathematical Physics - 2012

From Hard Sphere Dynamics to the Stokes–Fourier Equations: An Analysis of the Boltzmann–Grad Limit

Thierry Bodineau, Isabelle Gallagher, Laure Saint-Raymond

We derive the linear acoustic and Stokes-Fourier equations as the limiting dynamics of a system of N hard spheres of diameter $\epsilon$ in two space dimensions, when N $\rightarrow$ $\infty$, $\epsilon$ $\rightarrow$ 0, N $\epsilon$ = $\alpha$ $\rightarrow$ $\infty$, using the linearized Boltzmann equation as an intermediate step. Our proof is based on Lanford's strategy [18], and on the pruning procedure developed in [5] to improve the convergence time to all kinetic times with a quantitative control which allows us to reach also hydrodynamic time scales. The main novelty here is that uniform L 2 a pri-ori estimates combined with a subtle symmetry argument provide a weak version of chaos, in the form of a cumulant expansion describing the asymptotic decorrelation between the particles. A refined geometric analysis of recollisions is also required in order to discard the possibility of multiple recollisions.

Annals of PDE - 2017-02

Derivation of an ornstein-uhlenbeck process for a massive particle in a rarified gas of particles

Thierry Bodineau, Isabelle Gallagher, Laure Saint-Raymond

We consider the statistical motion of a convex rigid body in a gas of N smaller (spherical) atoms close to thermodynamic equilibrium. Because the rigid body is much bigger and heavier, it undergoes a lot of collisions leading to small deflections. We prove that its velocity is described, in a suitable limit, by an Ornstein-Uhlenbeck process. The strategy of proof relies on Lanford's arguments [17] together with the pruning procedure from [3] to reach diffusive times, much larger than the mean free time. Furthermore, we need to introduce a modified dynamics to avoid pathological collisions of atoms with the rigid body: these collisions, due to the geometry of the rigid body, require developing a new type of trajectory analysis.

Annales de l'Institut Henri Poincare Physique Theorique - 2018

Quantitative estimates for the flux of TASEP with dilute site disorder

We prove that the flux function of the totally asymmetric simple exclusion process (TASEP) with site disorder exhibits a flat segment for sufficiently dilute disorder. For high dilution, we obtain an accurate description of the flux. The result is established under a decay assumption of the maximum current in finite boxes, which is implied in particular by a sufficiently slow power tail assumption on the disorder distribution near its minimum. To circumvent the absence of explicit invariant measures, we use an original renormalization procedure and some ideas inspired by homogenization.

Electronic Journal of Probability - 2018-01-07

Current large deviations for Asymmetric Exclusion Processes with open boundaries

Thierry Bodineau, Bernard Derrida

We study the large deviation functional of the current for the Weakly Asymmetric Simple Exclusion Process in contact with two reservoirs. We compare this functional in the large drift limit to the one of the Totally Asymmetric Simple Exclusion Process, in particular to the Jensen-Varadhan functional. Conjectures for generalizing the Jensen-Varadhan functional to open systems are also stated.

Journal of Statistical Physics - 2006-04-21

Translation invariant Gibbs states for the Ising model

Thierry Bodineau

We prove that all the translation invariant Gibbs states of the Ising model are a linear combination of the pure phases $\mu^+_\gb,\mu^-_\gb$ for any $\gb \not = \gb_c$. This implies that the average magnetization is continuous for $\gb >\gb_c$. Furthermore, combined with previous results on the slab percolation threshold this shows the validity of Pisztora's coarse graining up to the critical temperature.

- 2004-10-08

THE BROWNIAN MOTION AS THE LIMIT OF A DETERMINISTIC SYSTEM OF HARD-SPHERES

Thierry Bodineau, Isabelle Gallagher, Laure Saint-Raymond

We provide a rigorous derivation of the brownian motion as the limit of a deterministic system of hard-spheres as the number of particles N goes to infinity and their diameter ε simultaneously goes to 0, in the fast relaxation limit α = N ε d−1 → ∞ (with a suitable diffusive scaling of the observation time). As suggested by Hilbert in his sixth problem, we rely on a kinetic formulation as an intermediate level of description between the microscopic and the fluid descriptions: we use indeed the linear Boltzmann equation to describe one tagged particle in a gas close to global equilibrium. Our proof is based on the fundamental ideas of Lanford. The main novelty here is the detailed study of the branching process, leading to explicit estimates on pathological collision trees.

Inventiones Mathematicae - 2015

A MICROSCOPIC VIEW ON THE FOURIER LAW

Thierry Bodineau, Isabelle Gallagher, Laure Saint-Raymond

The Fourier law of heat conduction describes heat diffusion in macroscopic systems. This physical law has been experimentally tested for a large class of physical systems. A natural question is to know whether it can be derived from the microscopic models using the fundamental laws of mechanics.

- 2019-12-08

A microscopic view of the Fourier law

Thierry Bodineau, Isabelle Gallagher, Laure Saint-Raymond

Comptes Rendus. Physique - 2019-08-31

Log-Sobolev Inequality for the Continuum Sine-Gordon Model

Roland Bauerschmidt, Thierry Bodineau

We derive a multiscale generalisation of the Bakry-Émery criterion for a measure to satisfy a log-Sobolev inequality. Our criterion relies on the control of an associated PDE well-known in renormalisation theory: the Polchinski equation. It implies the usual Bakry-Émery criterion, but we show that it remains effective for measures that are far from log-concave. Indeed, using our criterion, we prove that the massive continuum sine-Gordon model with β < 6π satisfies asymptotically optimal log-Sobolev inequalities for Glauber and Kawasaki dynamics. These dynamics can be seen as singular SPDEs recently constructed via regularity structures, but our results are independent of this theory. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.

Commun.Pure Appl.Math. - 2021

Dynamics of dilute gases at equilibrium: from the atomistic description to fluctuating hydrodynamics

Thierry Bodineau, Isabelle Gallagher, Laure Saint-Raymond, Sergio Simonella

We derive linear fluctuating hydrodynamics as the low density limit of a deterministic system of particles at equilibrium. The proof builds upon previous results of the authors where the asymptotics of the covariance of the fluctuation field is obtained, and on the proof of the Wick rule for the fluctuation field.

- 2022-10-20

Exact Hydrodynamic Description of Active Lattice Gases

Mourtaza Kourbane-Houssene, Clément Erignoux, Thierry Bodineau, Julien Tailleur

We introduce a class of lattice gas models of active matter systems whose hydrodynamic description can be derived exactly. We illustrate our approach by considering two systems exhibiting two of the most studied collective behaviours in active matter: the motility-induced phase separation and the transition to collective motion. In both cases, we derive coupled partial differential equation describing the dynamics of the local density and polarization fields and show how they quantitatively predict the emerging properties of the macroscopic lattice gases.

Physical Review Letters - 2018-06-29

Universal current fluctuations in the symmetric exclusion process and other diffusive systems

Eric Akkermans, Thierry Bodineau, Bernard Derrida, Ohad Shpielberg

We show, using the macroscopic fluctuation theory of Bertini, De Sole, Gabrielli, Jona-Lasinio, and Landim, that the statistics of the current of the symmetric simple exclusion process (SSEP) connected to two reservoirs are the same on an arbitrary large finite domain in dimension $d$ as in the one dimensional case. Numerical results on squares support this claim while results on cubes exhibit some discrepancy. We argue that the results of the macroscopic fluctuation theory should be recovered by increasing the size of the contacts. The generalization to other diffusive systems is straightforward.

EPL - Europhysics Letters - 2013-07-29

One-sided convergence in the Boltzmann-Grad limit

Thierry Bodineau, Isabelle Gallagher, Laure Saint-Raymond, Sergio Simonella

We review various contributions on the fundamental work of Lanford deriving the Boltzmann equation from hard-sphere dynamics in the low density limit. We focus especially on the assumptions made on the initial data and on how they encode irreversibility. The impossibility to reverse time in the Boltzmann equation (expressed for instance by Boltzmann's H-theorem) is related to the lack of convergence of higher order marginals on some singular sets. Explicit counterexamples single out the microscopic sets where the initial data should converge in order to produce the Boltzmann dynamics.

Annales de la Faculté des Sciences de Toulouse. Mathématiques. - 2018