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François Golse

Publications and source records attributed to François Golse.

At least 19 recordsLinked to original sources
arXiv2026

Velocity Averaging for the Wigner Equation with Electromagnetic Fields

We discuss the application of $L^2$-based velocity averaging to the Wigner kinetic equation governing the quantum evolution of the Wigner transform of a density operator of a particle subjected to an external electromagnetic field. This extends the $L^2$-based averaging lemma from [F. Golse, J. Möller: Commun. Math. Sci. (2026)] to electromagnetic fields. We make use of the magnetic or gauge-invariant Wigner transform due to Stratonovich [Sov. Phys. D., (1)414-418, 1956] which differs from the usual definition by a phase factor involving the circulation of the magnetic vector potential along a line segment. The magnetic Wigner transform obeys the Wigner equation in the generalized phase space variables $(x,v=ξ-A(t,x))$, which is a more natural setting for the application of velocity averaging. Since the magnetic Wigner equation contains a first order derivative in the spatial variable, we obtain compactness in $L^2$ but no gain in regularity.

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arXiv2026

The mean-field limit of the Schrödinger-Lohe model and emergent dynamics

The Schrödinger-Lohe (SL) model is a coupled system of nonlinear Schrödinger equations describing the temporal-spatial evolution of the component wave functions, and it corresponds to the infinite-dimensional counterpart of the Lohe matrix model for quantum synchronization. In this paper, we study a rigorous mean-field limit of the SL model and provide a quantitative estimate on the fluctuations between one-marginal distribution and one particle distribution in 2-Wasserstein distance. For the derived kinetic SL equation, we present sufficient conditions leading to the complete and practical synchronizations.

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arXiv2026

Velocity Averaging for the Wigner Kinetic Equation in the Semiclassical Regime

This paper discusses the possibility of applying the velocity averaging theorems in [F. Golse, P.-L. Lions, B. Perthame, R. Sentis: J. Funct. Anal. 76(1):110--125, 1988] to the Wigner equation governing the quantum evolution of the Wigner transform of quantum density operators. Our first main results address the case of the Wigner function of a special class of density operators associated to mixed states, whose Hilbert-Schmidt norm is of order $\hbar^{d/2}$, where $d$ is the space dimension and $\hbar$ the reduced Planck constant. In space dimension $d=1$, we prove that the density function belongs to the Sobolev space $H^s(\mathbb R)$ for some $s>0$. In the case of pure states, we first obtain a characterization of the Wigner transform of rank-one quantum density operators, and apply this characterization (1) to analyze a rather general setting in which velocity averaging cannot apply to the Wigner functions of a family of rank-one density operators whose evolution is governed by the von Neumann equation, and (2) to obtain a quick derivation of Madelung's system of quantum hydrodynamic equations. This derivation provides a physical explanation of one key assumption used in the proof of the negative result (1) described above.

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arXiv2026

On the approximation of the von Neumann equation in the semiclassical limit. Part II : numerical analysis

This paper is devoted to the numerical analysis of the Hermite spectral method proposed in [14], which provides, in the semiclassical limit, an asymptotic preserving approximation of the von Neumann equation. More precisely, it relies on the use of so-called Weyl's variables to effectively address the stiffness associated to the equation. Then by employing a truncated Hermite expansion of the density operator, we successfully manage this stiffness and provide error estimates by leveraging the propagation of regularity in the exact solution.

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arXiv2025

The Semi-Classical Limit from the Dirac Equation with Time-Dependent External Electromagnetic Field to Relativistic Vlasov Equations

We prove the mathematically rigorous (semi-)classical limit $\hbar \to 0$ of the Dirac equation with time-dependent external electromagnetic field to relativistic Vlasov equations with Lorentz force for electrons and positrons. In this limit antimatter and spin remain as intrinsically relativistic effects on a classical level. Our global-in-time results use Wigner transforms and a Lagrange multiplier viewpoint of the matrix-valued Wigner equation. In particular, we pass to the limit in the ''full" Wigner matrix equation without projecting on the eigenspaces of the matrix-valued symbol of the Dirac operator. In the limit, the Lagrange multiplier maintains the constraint that the Wigner measure and the symbol of the Dirac operator commute and vanishes when projected on the electron or positron eigenspace. This is a different approach to the problem as discussed in [P. Gérard, P. Markowich, N.J. Mauser, F. Poupaud: Comm. Pure Appl. Math. 50(4):323--379, 1997], where the limit is taken in the projected Wigner equation. By explicit calculation of the remainder term in the expansion of the Moyal product we are able to generalize to time-dependent potentials with much less regularity. We use uniform $L^2$ bounds for the Wigner transform, which are only possible for a special class of mixed states as initial data.

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arXiv2025

Velocity Averaging Lemmas: Classical, Quantum and Semi-Classical

Averaging lemmas were introduced as a tool of the mathematical analysis of kinetic equations, i.e. PDEs for functions in phase space $(x,v)$ containing a transport ("advection") term. By integrating over $v$ in velocity space $\mathbb{R}_v^d$ (velocity averaging), one gains regularity for the density in position space $\mathbb{R}_x^d$. The concept was invented independently by V.I. Agoshkov and by F. Golse, B. Perthame, R. Sentis and P.-L. Lions, and successfully applied to the analysis of Vlasov or Boltzmann equations in "classical kinetic theory". In "quantum kinetic theory", the Schrödinger equation for the complex-valued "wave function" in the physical space is converted into the Wigner equation for the real-valued Wigner function in phase space (which can take negative values). The Wigner ("Quantum Vlasov") equation contains the transport term of classical kinetic equations plus a pseudo-differential operator containing the potential. We give answers to the long standing question of whether and to which extent averaging lemmas apply to the "quantum" case of the Wigner equation. The hard part are the "semi-classical" averaging lemmas, where one considers the asymptotics of vanishing Planck constant towards the non-negative Wigner measure. In that context "pure vs. mixed states" play a crucial role, as well as the connection between the Schrödinger equation and Quantum Hydrodynamics (QHD). We present the results for the classical and quantum cases and sketch the "semi-classical" case which is worked out in full detail in a follow-up article.

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arXiv2024

Local Conservation Laws and Entropy Inequality for Kinetic Models with Delocalized Collision Integrals

This article presents a common setting for the collision integrals $\mathrm{St}$ appearing in the kinetic theory of dense gases. It includes the collision integrals of the Enskog equation, of (a variant of) the Povzner equation, and of a model for soft sphere collisions proposed by Cercignani [Comm. Pure Appl. Math. 36 (1983), 479-494]. All these collision integrals are delocalized, in the sense that they involve products of the distribution functions of gas molecules evaluated at positions whose distance is of the order of the molecular radius. Our first main result is to express these collision integrals as the divergence in $v$ of some mass current, where $v$ is the velocity variable, while $v_i\mathrm{St}$ and $|v|^2\mathrm{St}$ are expressed as the phase space divergence (i.e divergence in both position and velocity) of appropriate momentum and energy currents. This extends to the case of dense gases an earlier result by Villani [Math. Modelling Numer. Anal. M2AN 33 (1999), 209-227] in the case of the classical Boltzmann equation (where the collision integral is involves products of the distribution function of gas molecules evaluated at different velocities, but at the same position. Applications of this conservative formulation of delocalized collision integrals include the possibility of obtaining the local conservation laws of momentum and energy starting from this kinetic theory of denses gases. Similarly a local variant of the Boltzmann H Theorem, involving some kind of free energy instead of Boltzmann's H function, can be obtained in the form of an expression for the entropy production in terms of the phase space divergence of some phase space current, and of a nonpositive term.

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arXiv2024

On the approximation of the von Neumann equation in the semi-classical limit. Part I : numerical algorithm

We propose a new approach to discretize the von Neumann equation, which is efficient in the semi-classical limit. This method is first based on the so called Weyl's variables to address the stiffness associated with the equation. Then, by applying a truncated Hermite expansion of the density operator, we successfully handle this stiffness. Additionally, we develop a finite volume approximation for practical implementation and conduct numerical simulations to illustrate the efficiency of our approach. This asymptotic preserving numerical approximation, combined with the use of Hermite polynomials, provides an efficient tool for solving the von Neumann equation in all regimes, near classical or not.

math.AP↗
arXiv2024

A model for slowing particles in random media

We present a simple model in dimension $d\geq 2$ for slowing particles in random media, where point particles move in straight lines among and inside spherical identical obstacles with Poisson distributed centres. When crossing an obstacle, a particle is slowed down according to the law $\dot{V}= -\fracκε S(|V|) V$, where $V$ is the velocity of the point particle, $κ$ is a positive constant, $ε$ is the radius of the obstacle and $S(|V|)$ is a given slowing profile. With this choice, the slowing rate in the obstacles is such that the variation of speed at each crossing is of order $1$. We study the asymptotic limit of the particle system when $ε$ vanishes and the mean free path of the point particles stays finite. We prove the convergence of the point particles density measure to the solution of a kinetic-like equation with a collision term which includes a contribution proportional to a $δ$ function in $v=0$; this contribution guarantees the conservation of mass for the limit equation.

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arXiv2024

The Physisorbate-Layer Problem Arising in Kinetic Theory of Gas-Surface Interaction

A half-space problem of a linear kinetic equation for gas molecules physisorbed close to a solid surface, relevant to a kinetic model of gas-surface interactions and derived by Aoki et al. [K.~Aoki et al., in: Phys. Rev. E 106:035306, 2022], is considered. The equation contains a confinement potential in the vicinity of the solid surface and an interaction term between gas molecules and phonons. It is proved that a unique solution exists when the incoming molecular flux is specified at infinity. This validates the natural observation that the half-space problem serves as the boundary condition for the Boltzmann equation. It is also proved that the sequence of approximate solutions used for the existence proof converges exponentially fast. In addition, numerical results showing the details of the solution to the half-space problem are presented.

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arXiv2024

Pickl's Proof of the Quantum Mean-Field Limit and Quantum Klimontovich Solutions

This paper discusses the mean-field limit for the quantum dynamics of $N$ identical bosons in $\mathbf R^3$ interacting via a binary potential with Coulomb type singularity. Our approach is based on the theory of quantum Klimontovich solutions defined in [F. Golse, T. Paul, Commun. Math. Phys. 369 (2019), 1021-1053]. Our first main result is a definition of the interaction nonlinearity in the equation governing the dynamics of quantum Klimontovich solutions for a class of interaction potentials slightly less general than those considered in [T. Kato, Trans. Amer. Math. Soc. 70 (1951), 195-211]. Our second main result is a new operator inequality satisfied by the quantum Klimontovich solution in the case of an interaction potential with Coulomb type singularity. When evaluated on an initial bosonic pure state, this operator inequality reduces to a Gronwall inequality for a functional introduced in [P. Pickl, Lett. Math. Phys. 97 (2011), 151-164], resulting in a convergence rate estimate for the quantum mean-field limit leading to the time-dependent Hartree equation.

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arXiv2024

Local regularity for the space-homogeneous Landau equation with very soft potentials

This paper deals with the space-homogenous Landau equation with very soft potentials, including the Coulomb case. This nonlinear equation is of parabolic type with diffusion matrix given by the convolution product of the solution with the matrix $a_{ij} (z)=|z|^γ(|z|^2 δ_{ij} - z_iz_j)$ for $γ\in [-3,-2)$. We derive local truncated entropy estimates and use them to establish two facts. Firstly, we prove that the set of singular points (in time and velocity) for the weak solutions constructed as in [C. Villani, Arch. Rational Mech. Anal. 143 (1998), 273-307] has zero $\mathscr{P}^{m_\ast}$ parabolic Hausdorff measure with $m_\ast:= \frac72 |2+γ|$. Secondly, we prove that if such a weak solution is axisymmetric, then it is smooth away from the symmetry axis. In particular, radially symmetric weak solutions are smooth away from the origin.

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arXiv2023

Validité de la théorie cinétique des gaz: au-delà de l'équation de Boltzmann [d'après T. Bodineau, I. Gallagher, L. Saint-Raymond, S. Simonella]

Obtaining a rigorous justification of the kinetic theory of gases from Newton's second law of dynamics for a large number of identical spheres interacting by elastic binary collisions, is a problem formulated by Hilbert in 1900 (Hilbert's 6th problem). In 1975, Lanford demonstrated the validity of the Boltzmann equation over a very short time interval, of the order of a fraction of the average time between two successive collisions experienced by the same particle. This result of Lanford can be interpreted as a kind of law of large numbers as the number of particles tends to infinity. This point of view raises several questions. First, the core of the argument used by Boltzmann to arrive at the equation bearing his name is the assumption that two particles about to collide are statistically almost independent. This suggests to examine the validity of this hypothesis by studying the dynamics of correlations between particles. On the other hand, the interpretation of the Boltzmann equation as a law of large numbers leads to study precisely the fluctuations of the phase space empirical measure of the particle system about its mean (whose evolution is described by the Boltzmann equation). A series of recent papers by T. Bodineau, I. Gallagher, L. Saint-Raymond and S. Simonella answers these various questions and allows going beyond the Boltzmann equation in the understanding of the kinetic theory of gases.

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arXiv2023

Quantum Optimal Transport: Quantum Couplings and Many-Body Problems

This text is a set of lecture notes for a 4.5-hour course given at the Erdös Center (Rényi Institute, Budapest) during the Summer School "Optimal Transport on Quantum Structures" (September 19th-23rd, 2023). Lecture I introduces the quantum analogue of the Wasserstein distance of exponent $2$ defined in [F. Golse, C. Mouhot, T. Paul: Comm. Math. Phys. 343 (2016), 165-205], and in [F. Golse, T. Paul: Arch. Ration. Mech. Anal. 223 (2017) 57-94]. Lecture II discusses various applications of this quantum analogue of the Wasserstein distance of exponent $2$, while Lecture III discusses several of its most important properties, such as the triangle inequality, and the Kantorovich duality in the quantum setting, together with some of their implications.

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arXiv2022

Mean-Field Limits in Statistical Dynamics

These lectures notes are aimed at introducing the reader to some recent mathematical tools and results for the mean-field limit in statistical dynamics. As a warm-up, lecture 1 reviews the approach to the mean-field limit in classical mechanics following the ideas of W. Braun, K. Hepp and R.L. Dobrushin, based on the notions of phase space empirical measures, Klimontovich solutions and Monge-Kantorovich-Wasserstein distances between probability measures. Lecture 2 discusses an analogue of the notion of Klimontovich solution in quantum dynamics, and explains how this notion appears in Pickl's method to handle the case of interaction potentials with a Coulomb type singularity at the origin. Finally, lecture 3 explains how the mean-field and the classical limits can be taken jointly on quantum $N$-particle dynamics, leading to the Vlasov equation. These lectures are based on a series of joint works with C. Mouhot and T. Paul.

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