Searcharxiv⌕ Search

arXiv subjects

Patrick Bernard

Publications and source records attributed to Patrick Bernard.

At least 19 recordsLinked to original sources
arXiv2026

Identification of Solid-Electrolyte Interphase Species by Joint Characterization of Li-ion Battery Chemistry by Mass Spectrometry and Electro-Chemical Reaction Networks

The formation and stability of the solid electrolyte interphase (SEI) play a central role in determining the long-term performance and safety of modern electrochemical energy storage systems. Despite decades of research, the SEI's heterogeneous, dynamic, and multi-phase nature has defied comprehensive molecular-level characterization, creating a critical knowledge gap that limits rational battery design. In this work, we introduce a computational-experimental framework that integrates high-throughput quantum chemistry calculations, data-driven electro-chemical reaction networks (eCRNs), stochastic algorithms, and Laser Desorption/Ionization Fourier Transform Ion Cyclotron Resonance Mass Spectrometry (LDI-FTICR-MS) to unravel SEI formation in carbonate-based electrolytes without imposing predefined mechanisms. We constructed the most comprehensive eCRN to date, spanning over 10,000 species and 209 million reactions. Through stochastic network analysis, we successfully recovered 27 species that were previously reported in literature and predicted 28 novel SEI species, nearly doubling our scientific knowledge in this area. Each new species was rigorously confirmed through advanced mass spectra analysis of its distinct molecular and isotopic signatures. We kinetically refined the formation pathways for a select set of both previously reported and novel SEI products, revealing kinetically feasible elementary reaction mechanisms with activation barriers below 1 eV. This computational-experimental approach deepens our molecular-level understanding of SEI chemistry and supports the rational design of advanced electrolytes and engineered interphases for next-generation lithium-based batteries.

physics.chem-ph↗
arXiv2023

The Siegel-Bruno linearization Theorem

The purpose of this paper is to provide a short and self-contained account on Siegel's Theorem, as improved by Bruno, which states that a holomorphic map f of C which fixes 0 can be locally linearized, under certain conditions on the multiplier.

math.DS↗
arXiv2020

Normal form near orbit segments of convex Hamiltonian systems

In the study of Hamiltonian systems on cotangent bundles, it is natural to perturb Hamiltoni-ans by adding potentials (functions depending only on the base point). This led to the definition of Ma{\~n}{\'e} genericity: a property is generic if, given a Hamiltonian H, the set of potentials u such that H + u satisfies the property is generic. This notion is mostly used in the context of Hamiltonians which are convex in p, in the sense that $\partial$ 2 pp H is positive definite at each points. We will also restrict our study to this situation. There is a close relation between perturbations of Hamiltonians by a small additive potential and perturbations by a positive factor close to one. Indeed, the Hamiltonians H + u and H/(1 -- u) have the same level one energy surface, hence their dynamics on this energy surface are reparametrisation of each other, this is the Maupertuis principle. This remark is particularly relevant when H is homogeneous in the fibers (which corresponds to Finsler metrics) or even fiberwise quadratic (which corresponds to Riemannian metrics). In these cases, perturbations by potentials of the Hamiltonian correspond, up to parametrisation, to conformal perturbations of the metric. One of the widely studied aspects is to understand to what extent the return map associated to a periodic orbit can be perturbed by adding a small potential. This kind of question depend strongly on the context in which they are posed. Some of the most studied contexts are, in increasing order of difficulty, perturbations of general vector fields, perturbations of Hamiltonian systems inside the class of Hamiltonian systems, perturbations of Riemannian metrics inside the class of Riemannian metrics, Ma{\~n}{\'e} perturbations of convex Hamiltonians. It is for example well-known that each vector field can be perturbed to a vector field with only hyperbolic periodic orbits, this is part of the Kupka-Smale theorem, see [5, 13]. There is no such result in the context of Hamiltonian vector fields, but it remains true that each Hamiltonian can be perturbed to a Hamiltonian with only non-degenerate periodic orbits (including the iterated ones), see [11, 12]. The same result is true in the context of Riemannian metrics: every Riemannian metric can be perturbed to a Riemannian metric with only non-degenerate closed geodesics, this is the bumpy metric theorem, see [4, 2, 1]. The question was investigated only much more recently in the context of Ma{\~n}{\'e} perturbations of convex Hamiltonians, see [9, 10]. It is proved in [10] that the same result holds : If H is a convex Hamiltonian and a is a regular value of H, then there exist arbitrarily small potentials u such that all periodic orbits (including iterated ones) of H + u at energy a are non-degenerate. The proof given in [10] is actually rather similar to the ones given in papers on the perturbations of Riemannian metrics. In all these proofs, it is very useful to work

math.DS↗
arXiv2017

Smoothing causal functions

We describe, in the general setting of closed cone fields, the set of causal functions which can be approximated by smooth Lyapunov. We derive several consequences on causality theory. Dans le contexte g\'en\'eral des champs de cones ferm\'es, on d\'ecrit l'ensemble des fonctions causales qui peuvent \^etre approch\'ees par des fonctions de Lyapunov lisses. On en d\'eduit quelques cons\'equence en th\'eorie de la causalit\'e.

math.DG↗
arXiv2017

Arnold diffusion in arbitrary degrees of freedom and normally hyperbolic invariant cylinders

We prove a form of Arnold diffusion in the a priori stable case. Let H0(p) + $ε$H1($θ$, p, t), $θ$ $\in$ T n , p $\in$ B n , t $\in$ T = R/T be a nearly integrable system of arbitrary degrees of freedom n 2 with a strictly convex H0. We show that for a "generic" $ε$H1, there exists an orbit ($θ$, p)(t) satisfying p(t) -- p(0) {\textgreater} l(H1) {\textgreater} 0, where l(H1) is independent of $ε$. The diffusion orbit travels along a co-dimension one resonance , and the only obstruction to our construction is a finite set of additional resonances. For the proof we use a combination geometric and variational methods, and manage to adapt tools which have recently been developed in the a priori unstable case.

math.DS↗
arXiv2016

Une propriété de transfert en approximation diophantienne

Given a vector $ω\in \mathbb{R}^n$,the sequence $T_i$ of periods is defined as the sequence of times of best returns near the origin of the translation $x \longmapsto x+ω$ on the torus $\mathbb{T}^n$. In the present paper, we study how the Diophantine properties of $ω$ can be expressed considering the sequence of its periods. More precisely, we prove that, if the vector $ω$ is not resonant,and if the sequence of periods satisfy the inequality$T_{i+1} \leq CT_i^{1+τ}$ with$τ<(n-1)^{-1}$, then the vector $ω$ is Diophantine.

math.DS↗
arXiv2015

Lyapounov Functions of closed Cone Fields: from Conley Theory to Time Functions

We propose a theory "a la Conley" for cone fields using a notion of relaxed orbits based on cone enlargements, in the spirit of space time geometry. We work in the setting of closed (or equivalently semi-continuous) cone fields with singularities. This setting contains (for questions which are parametrization independent such as the existence of Lyapounov functions) the case of continuous vector-fields on manifolds, of differential inclusions, of Lorentzian metrics, and of continuous cone fields. We generalize to this setting the equivalence between stable causality and the existence of temporal functions. We also generalize the equivalence between global hyperbolicity and the existence of a steep temporal functions.

math.DG↗
arXiv2014

Semi-concave singularities and the Hamilton-Jacobi equation

We study the Cauchy problem for the Hamilton-Jacobi equation with a semiconcave initial condition. We prove an inequality between two types of weak solutions emanating from such an initial condition (the variational and the viscosity solution).We also give conditions for an explicit semi-concave function to be a viscosity solution. These conditions generalize the entropy inequality characterizing piecewise smooth solutions of scalar conservation laws in dimension one.

math.DS↗
arXiv2014

Homoclinic connections with many loops near a $0^2 iw$ resonant fixed point for Hamiltonian systems

In this paper we study the dynamics near the equilibrium point of a family of Hamiltonian systems in the neighborhood of a $0^2 iw$ resonance. The existence of a family of periodic orbits surrounding the equilibrium is well-known and we show here the existence of homoclinic connections with several loops for every periodic orbit close to the origin, except the origin itself. To prove this result, we first show a Hamiltonian normal form theorem inspired by the Elphick-Tirapegui-Brachet-Coullet-Iooss normal form. We then use a local Hamiltonian normalization relying on a result of Moser. We obtain the result of existence of homoclinic orbits by geometrical arguments based on the low dimension and with the aid of a KAM theorem which allows to confine the loops. The same problem was studied before for reversible non Hamiltonian vectorfields, and the splitting of the homoclinic orbits lead to exponentially small terms which prevent the existence of homoclinic connections to exponentially small periodic orbits. The same phenomenon occurs here but we get round this difficulty thanks to geometric arguments specific to Hamiltonian systems and by studying homoclinic orbits with many loops.

math.DS↗
arXiv2012

Some remarks on Thom's Transversality Theorem

We study Thom Transversality Theorem using a point of view, suggested by Gromov, which allows to avoid the use of Sard Theorem and gives finer informations on the structure of the set of non-transverse maps.

math.DG↗
arXiv2012

The Lax-Oleinik semi-group: a Hamiltonian point of view

The Weak KAM theory was developed by Fathi in order to study the dynamics of convex Hamiltonian systems. It somehow makes a bridge between viscosity solutions of the Hamilton-Jacobi equation and Mather invariant sets of Hamiltonian systems, although this was fully understood only a posteriori. These theories converge under the hypothesis of convexity, and the richness of applications mostly comes from this remarkable convergence. In the present course, we provide an elementary exposition of some of the basic concepts of weak KAM theory. In a companion lecture, Albert Fathi exposes the aspects of his theory which are more directly related to viscosity solutions. Here on the contrary, we focus on dynamical applications, even if we also discuss some viscosity aspects to underline the connections with Fathi's lecture. The fundamental reference on Weak KAM theory is the still unpublished book of Albert Fathi \textit{Weak KAM theorem in Lagrangian dynamics}. Although we do not offer new results, our exposition is original in several aspects. We only work with the Hamiltonian and do not rely on the Lagrangian, even if some proofs are directly inspired from the classical Lagrangian proofs. This approach is made easier by the choice of a somewhat specific setting. We work on $\Rm^d$ and make uniform hypotheses on the Hamiltonian. This allows us to replace some compactness arguments by explicit estimates. For the most interesting dynamical applications however, the compactness of the configuration space remains a useful hypothesis and we retrieve it by considering periodic (in space) Hamiltonians. Our exposition is centered on the Cauchy problem for the Hamilton-Jacobi equation and the Lax-Oleinik evolution operators associated to it. Dynamical applications are reached by considering fixed points of these evolution operators, the Weak KAM solutions. The evolution operators can also be used for their regularizing properties, this opens a second way to dynamical applications.

math.DS↗
arXiv2012

Some remarks on the continuity equation

We describe some relations between the properties of the Cauchy problem for an ODE and the properties of the Cauchy problem for the associated continuity equation in the class of measures.

math.DS↗