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Thierry Bodineau

Publications and source records attributed to Thierry Bodineau.

At least 19 recordsLinked to original sources
arXiv2026

The Ding-Song-Sun inequality for a class of even ferromagnets

Ding-Song-Sun proved the remarkable correlation inequality that the truncated two-point correlation function of ferromagnetic Ising models with arbitrary mixed-sign external field is largest when the field vanishes. This inequality has various important consequences such as log-Sobolev inequalities for Ising and $φ^4$ models up to and at the critical point. We extend the DSS inequality to the class of single spin measures satisfying the GHS condition. This is an important ingredient in our proof of the log-Sobolev inequality for the massless sinh-Gordon model on $\mathbb{R}^2$ in a forthcoming article.

math.PR↗
arXiv2026

System driven out-of equilibrium by weak contacts with reservoirs

The non-equilibrium behavior of particle systems driven by reservoirs has been extensively studied in recent years. In one dimension, various regimes have been explored depending on the coupling strength to the reservoirs. In this paper, we investigate the role of the dimension and of the geometry of the contacts with the reservoirs. For the symmetric simple exclusion process with point contact reservoirs, we show that in dimension 2, as in one dimension, three different regimes occur depending on the coupling strength. On the other hand in dimensions 3 and higher, there exists only a weak coupling regime which is very sensitive to the microscopic structure of the contacts. We then argue that for reservoirs with mesoscopic size contacts the macroscopic fluctuation theory remains in force and we propose an extension of the additivity principle for multiple mesoscopic reservoirs.

cond-mat.stat-mech↗
arXiv2026

Linear Landau equation as a limit of a tagged particle in mean field interaction with a free gas

We consider a tagged particle in mean field interaction with a free gas of density N at equilibrium. In dimensions $d\geq4$, we prove the convergence of its trajectory, as N goes to infinity, to the one of a diffusion process associated with the linear Landau equation. The proof of the convergence of the martingale problem relies on two key ingredients: long time stability results of the microscopic dynamics, and controls on the probability of particle recollisions.

math.PR↗
arXiv2026

Derivation of the Boltzmann equation from hard-sphere dynamics (after Y. Deng, Z. Hani, and X. Ma)

Consider a microscopic system of $N$ hard spheres that are initially independent (modulo the exclusion condition on particle positions) and identically distributed in $\mathbb{R}^3$. When the number $N$ of particles goes to infinity and the diameter $\varepsilon$ of the particles goes to zero, and under the weak density assumption $N\varepsilon^2=1$, it has been known since the work of Lanford that the empirical measure for the particles converges to the solution of the Boltzmann equation in a short time interval. In particular, the particles remain dynamically independent, in this limit and in the short time interval where the correlations induced by their collisions remain under control. In a recent work, Y. Deng, Z. Hani and X. Ma successfully obtained the same convergence result in arbitrary large time; more precisely, the convergence result holds for any time interval on which the Boltzmann equation has a regular solution. In this note, we explain a few elements of their proof.

math.AP↗
arXiv2024

A perturbative approach to the macroscopic fluctuation theory

In this paper, we study the stationary states of diffusive dynamics driven out of equilibrium by reservoirs. For a small forcing, the system remains close to equilibrium and the large deviation functional of the density can be computed perturbatively by using the macroscopic fluctuation theory. This applies to general domains in $\mathbb{R}^d$ and diffusive dynamics with arbitrary transport coefficients. As a consequence, one can analyse the correlations at the first non trivial order in the forcing and show that, in general, all the long range correlation functions are not equal to 0, in contrast to the exactly solvable models previously known.

cond-mat.stat-mech↗
arXiv2024

Kawasaki dynamics beyond the uniqueness threshold

Glauber dynamics of the Ising model on a random regular graph is known to mix fast below the tree uniqueness threshold and exponentially slowly above it. We show that Kawasaki dynamics of the canonical ferromagnetic Ising model on a random $d$-regular graph mixes fast beyond the tree uniqueness threshold when $d$ is large enough (and conjecture that it mixes fast up to the tree reconstruction threshold for all $d\geq 3$). This result follows from a more general spectral condition for (modified) log-Sobolev inequalities for conservative dynamics of Ising models. The proof of this condition in fact extends to perturbations of distributions with log-concave generating polynomial.

math.PR↗
arXiv2024

Large deviations for out of equilibrium correlations in the symmetric simple exclusion process

For finite size Markov chains, the Donsker-Varadhan theory fully describes the large deviations of the time averaged empirical measure. We are interested in the extension of the Donsker-Varadhan theory for a large size non-equilibrium system: the one-dimensional symmetric simple exclusion process connected with reservoirs at different densities. The Donsker-Varadhan functional encodes a variety of scales depending on the observable of interest. In this paper, we focus on the time-averaged two point correlations and investigate the large deviations from the steady state behaviour. To control two point correlations out of equilibrium, the key input is the construction of a simple approximation to the invariant measure. This approximation is quantitative in time and space as estimated through relative entropy bounds building on the work of Jara and Menezes arXiv:1810.09526.

math.PR↗
arXiv2024

Stochastic dynamics and the Polchinski equation: an introduction

This introduction surveys a renormalisation group perspective on log-Sobolev inequalities and related properties of stochastic dynamics. We also explain the relationship of this approach to related recent and less recent developments such as Eldan's stochastic localisation and the Föllmer process, the Boué--Dupuis variational formula and the Barashkov--Gubinelli approach, the transportation of measure perspective, and the classical analogues of these ideas for Hamilton--Jacobi equations which arise in mean-field limits.

math.PR↗
arXiv2022

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

We present a mathematical theory of dynamical fluctuations for the hard sphere gas in the Boltzmann-Grad limit. We prove that: (1) fluctuations of the empirical measure from the solution of the Boltzmann equation, scaled with the square root of the average number of particles, converge to a Gaussian process driven by the fluctuating Boltzmann equation, as predicted in [67]; (2) large deviations are exponentially small in the average number of particles and are characterized, under regularity assumptions, by a large deviation functional as previously obtained in [61] for dynamics with stochastic collisions. The results are valid away from thermal equilibrium, but only for short times. Our strategy is based on uniform a priori bounds on the cumulant generating function, characterizing the fine structure of the small correlations.

math.AP↗
arXiv2022

Cluster expansion for a dilute hard sphere gas dynamics

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.

math.AP↗
arXiv2022

Dynamics of dilute gases: a statistical approach

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.

math.AP↗
arXiv2022

Long-time derivation at equilibrium of the fluctuating Boltzmann equation

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].

math.AP↗
arXiv2020

Long-time correlations for a hard-sphere gas at equilibrium

It has been known since Lanford [19] 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 duality method coupled with a pruning argument to prove that the covariance of the fluctuations 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.

math.AP↗
arXiv2020

Fluctuation theory in the Boltzmann--Grad limit

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.

math.AP↗
arXiv2020

Log-Sobolev inequality for the continuum sine-Gordon model

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 which 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.

math.PR↗
arXiv2019

A microscopic view on the Fourier law

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.

math.AP↗
arXiv2019

Spectral gap critical exponent for Glauber dynamics of hierarchical spin models

We develop a renormalisation group approach to deriving the asymptotics of the spectral gap of the generator of Glauber type dynamics of spin systems with strong correlations (at and near a critical point). In our approach, we derive a spectral gap inequality for the measure recursively in terms of spectral gap inequalities for a sequence of renormalised measures. We apply our method to hierarchical versions of the $4$-dimensional $n$-component $|φ|^4$ model at the critical point and its approach from the high temperature side, and of 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 $|φ|^4$ model), the scaling limit of these models in equilibrium.

math-ph↗