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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., Comment: 32 pages, 1 figure

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 $\dot{V}= -\frac{\kappa}{\epsilon} S(|V|) V$, where $V$ is the velocity of the point particle, $\kappa$ is a positive constant, $\epsilon$ 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 $\epsilon$ 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 $\delta$ function in $v=0$; this contribution guarantees the conservation of mass for the limit equation., Comment: 30 pages, 0 figures; added acknowledgement to Pessoa project, no changes in the scientific part

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.

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., Comment: 47 pages, 11 figures

We prove that the set of singular times for weak solutions of the homogeneous Boltzmann equation with very soft potentials constructed as in Villani (1998) has Hausdorff dimension at most $\frac{|\gamma+2s|}{2s}$ with $\gamma \in [-4s,-2s)$ and $s \in (0,1)$.

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., Comment: 53 pages, in French language, 3 figures. Bourbaki seminar, November 19th 2022

This text is a set of lecture notes for a 4.5-hour course given at the Erd\"os Center (R\'enyi 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., Comment: 81 pages, 7 figures

This short note gives a proof of the triangle inequality based on the Kantorovich duality formula for the Wasserstein distances of exponent $p\in[1,+\infty)$ in the case of a general Polish space. In particular it avoids the "glueing of couplings" procedure used in most textbooks on optimal transport., Comment: 10 pages, no figure

Most problems in uncertainty quantification, despite its ubiquitousness in scientific computing, applied mathematics and data science, remain formidable on a classical computer. For uncertainties that arise in partial differential equations (PDEs), large numbers M>>1 of samples are required to obtain accurate ensemble averages. This usually involves solving the PDE M times. In addition, to characterise the stochasticity in a PDE, the dimension L of the random input variables is high in most cases, and classical algorithms suffer from curse-of-dimensionality. We propose new quantum algorithms for PDEs with uncertain coefficients that are more efficient in M and L in various important regimes, compared to their classical counterparts. We introduce transformations that transfer the original d-dimensional equation (with uncertain coefficients) into d+L (for dissipative equations) or d+2L (for wave type equations) dimensional equations (with certain coefficients) in which the uncertainties appear only in the initial data. These transformations also allow one to superimpose the M different initial data, so the computational cost for the quantum algorithm to obtain the ensemble average from M different samples is then independent of M, while also showing potential advantage in d, L and precision in computing ensemble averaged solutions or physical observables.

To study the temperature in a gas subjected to electromagnetic radiations, one may use the Radiative Transfer equations coupled with the Navier-Stokes equations. The problem has 7 dimensions; however with minimal simplifications it is equivalent to a small number of integro-differential equations in 3 dimensions. We present the method and a numerical implementation using an H-matrix compression scheme. The result is a very fast: 50K physical points, all directions of radiation and 680 frequencies require less than 5 minutes on an Apple M1 Laptop. The method is capable of handling variable absorptioN and scattering functionS of spatial positions and frequencies. The implementation is done using htool, a matrix compression library interfaced with the PDE solver freefem++. Applications to the temperature in the French Chamonix valley is presented at different hours of the day with and without snow / clouds and with a variable absorption taken from the Gemini measurements. The result is precise enough to assert temperature differences due to increased absorption in the vibrational frequency subrange of greenhouse gasses.

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 $a_{ij} (z)=|z|^\gamma (|z|^2 \delta_{ij} - z_iz_j)$ for $\gamma \in [-3,-2)$. 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 $m_\ast:= \frac72 |2+\gamma|$. 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., Comment: 58 pages. Hausdorff dimension improved

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., Comment: 32 pages, no figure. In this version, the introduction has been slightly modified so as to better indicate the main results in the paper. Similarly, the beginning of section 8.2 has been slightly expanded so as to better explain the connection between quantum Klimontovich solutions and Pickl's functional (specifically, the equivalence between formulas (58) and (18))

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., Comment: 46 pages. Lecture notes of a course given at the CIRM (Centre International de Recherche Math\'ematique), Luminy (France), during the Research School "Scaling Limits from Microscopic to Macroscopic Physics", January 18th-22nd 2021, organized by the Jean-Morlet Chair, with Shi Jin (chair) and Mihai Bostan (local project leader)

We study the Radiative Transfer equations coupled with the time dependent temperature equation of a fluid: existence, uniqueness, a maximum principle are established. A short numerical section illustrates the pros and cons of the method., Comment: 20 pages 5 figures

In this note, we propose a slightly different proof of Gallavotti's theorem ["Statistical Mechanics: A Short Treatise", Springer, 1999, pp. 48--55] on the derivation of the linear Boltzmann equation for the Lorentz gas with a Poisson distribution of obstacles in the Boltzmann-Grad limit., Comment: 18 pages

Using theoretical and numerical arguments we discuss some of the commonly accepted approximations for the radiative transfer equations in climatology., Comment: 22 pages, 5 figures, one computer listing

We establish a Kantorovich duality for the pseudometric $\mathcal{E}_\hbar$ introduced in [F. Golse, T. Paul, Arch. Rational Mech. Anal. 223 (2017), 57--94], obtained from the usual Monge-Kantorovich distance $d_{MK,2}$ between classical densities by quantization of one of the two densities involved. We show several type of inequalities comparing $d_{MK,2}$, $\mathcal{E}_\hbar$ and $MK_\hbar$, a full quantum analogue of $d_{MK,2}$ introduced in [F. Golse, C. Mouhot, T. Paul, Commun. Math. Phys. 343 (2016), 165--205], including an up to $\hbar$ triangle inequality for $MK_\hbar$. Finally, we show that, when nice optimal Kantorovich potentials exist for $\mathcal{E}_\hbar$, optimal couplings induce classical/quantum optimal transports and the potentials are linked by a semiquantum Legendre type transform., Comment: 33 pages, no figure

We present a time dependent quantum perturbation result, uniform in the Planck constant, for perturbations of potentials whose gradients are Lipschitz continuous by potentials whose gradients are only bounded a.e.. Though this low regularity of the full potential is not enough to provide the existence of the classical underlying dynamics, at variance with the quantum one, our result shows that the classical limit of the perturbed quantum dynamics remains in a tubular neighbourhood of the classical unperturbed one of size of order of the square root of the size of the perturbation. We treat both Schr\"odinger and von Neumann-Heisenberg equations., Comment: 18 pages

We establish an observation inequality for the Schr\"odinger equation on $\mathbf{R}^d$, uniform in the Planck constant $\hbar\in[0,1]$. The proof is based on the pseudometric introduced in [F. Golse, T. Paul, Arch. Rational Mech. Anal. 223 (2017), 57-94]. This inequality involves only effective constants which are computed explicitly in their dependence in $\hbar$ and all parameters involved., Comment: 15 pages, no figure

An analogue of the quadratic Wasserstein (or Monge-Kantorovich) distance between Borel probability measures on $\mathbf{R}^d$ has been defined in [F. Golse, C. Mouhot, T. Paul: Commun. Math. Phys. 343 (2015), 165-205] for density operators on $L^2(\mathbf{R}^d)$, and used to estimate the convergence rate of various asymptotic theories in the context of quantum mechanics. The present work proves a Kantorovich type duality theorem for this quantum variant of the Monge-Kantorovich or Wasserstein distance, and discusses the structure of optimal quantum couplings. Specifically, we prove that optimal quantum couplings involve a gradient type structure similar to the Brenier transport map (which is the gradient of a convex function), or more generally, to the subdifferential of a l.s.c. convex function as in the Knott-Smith optimality criterion (see Theorem 2.12 in [C. Villani: "Topics in Optimal Transportation", Amer. Math. Soc. 2003])., Comment: 48 pages, no figure. This new version includes a more detailed discussion of the case of finite rank densities

We prove semiclassical estimates for the Schr\''odinger-von Neumann evolution with $C^{1,1}$ potentials and density matrices whose square root have either Wigner functions with low regularity independent of the dimension, or matrix elements between Hermite functions having long range decay. The estimates are settled in different weak topologies and apply to initial density operators whose square root have Wigner functions $7$ times differentiable, independently of the dimension. They also apply to the $N$ body quantum dynamics uniformly in $N$. In a appendix, we finally estimate the dependence in the dimension of the constant appearing on the Calderon-Vaillancourt Theorem.

This paper discusses a numerical method for computing the evolution of large interacting system of quantum particles. The idea of the random batch method is to replace the total interaction of each particle with the $N-1$ other particles by the interaction with $p1$ and independent of the Planck constant $\hbar$., Comment: 23 pages

This paper proves the validity of the joint mean-field and classical limit of the quantum $N$-body dynamics leading to the pressureless Euler-Poisson system for factorized initial data whose first marginal has a monokinetic Wigner measure. The interaction potential is assumed to be the repulsive Coulomb potential. The validity of this derivation is limited to finite time intervals on which the Euler-Poisson system has a smooth solution that is rapidly decaying at infinity. One key ingredient in the proof is an inequality from [S. Serfaty, with an appendix of M. Duerinckx arXiv:1803.08345v3 [math.AP]]., Comment: 35 pages, no figure

Consider the steady Boltzmann equation with slab symmetry for a monatomic, hard sphere gas in a half space. At the boundary of the half space, it is assumed that the gas is in contact with its condensed phase. The present paper discusses the existence and uniqueness of a uniformly decaying boundary layer type solution of the Boltzmann equation in this situation, in the vicinity of the Maxwellian equilibrium with zero bulk velocity, with the same temperature as that of the condensed phase, and whose pressure is the saturating vapor pressure at the temperature of the interface. This problem has been extensively studied first by Y. Sone, K. Aoki and their collaborators, by means of careful numerical simulations. See section 2 of [C. Bardos, F. Golse, Y. Sone: J. Stat. Phys. 124 (2006), 275-300] for a very detailed presentation of these works. More recently T.-P. Liu and S.-H. Yu [Arch. Rational Mech. Anal. 209 (2013), 869-997] have proposed an extensive mathematical strategy to handle the problems studied numerically by Y. Sone, K. Aoki and their group. The present paper offers an alternative, possibly simpler proof of one of the results discussed in [T.P. Liu, S.-H. Yu, loc. cit.]

We compare bipartite (Euclidean) matching problems in classical and quantum mechanics. The quantum case is treated in terms of a quantum version of the Wasserstein distance introduced in [F. Golse, C. Mouhot, T. Paul, Commun. Math. Phys. 343 (2016), 165-205]. We show that the optimal quantum cost can be cheaper than the classical one. We treat in detail the case of two particles: the equal mass case leads to equal quantum and classical costs. Moreover, we show examples with different masses for which the quantum cost is strictly cheaper than the classical cost.

By using the pseudo-metric introduced in [F. Golse, T. Paul: Archive for Rational Mech. Anal. 223 (2017) 57-94], which is an analogue of the Wasserstein distance of exponent $2$ between a quantum density operator and a classical (phase-space) density, we prove that the convergence of time splitting algorithms for the von Neumann equation of quantum dynamics is uniform in the Planck constant $\hbar$. We obtain explicit uniform in $\hbar$ error estimates for the first order Lie-Trotter, and the second order Strang splitting methods.

We prove that the set of singular times for weak solutions of the space homogeneous Landau equation with Coulomb potential constructed as in [C. Villani, Arch. Rational Mech. Anal. 143 (1998), 273-307] has Hausdorff dimension at most 1/2.

The Lohe matrix model is a continuous-time dynamical system describing the collective dynamics of group elements in the unitary group manifold, and it has been introduced as a toy model of a non abelian generalization of the Kuramoto phase model. In the absence of couplings, it reduces to the finite-dimensional decoupled free Schr\"{o}dinger equations with constant Hamiltonians. In this paper, we study a rigorous mean-field limit of the Lohe matrix model which results in a Vlasov type equation for the probability density function on the corresponding phase space. We also provide two different settings for the emergent synchronous dynamics of the kinetic Lohe equation in terms of the initial data and the coupling strength., Comment: 38 pages

In this paper, we define a quantum analogue of the notion of empirical measure in the classical mechanics of $N$-particle systems. We establish an equation governing the evolution of our quantum analogue of the $N$-particle empirical measure, and we prove that this equation contains the Hartree equation as a special case. Our main application of this new object to the mean-field limit of the $N$-particle Schr\"odinger equation is an $O(1/\sqrt{N})$ convergence rate in some dual Sobolev norm for the Wigner transform of the single-particle marginal of the $N$-particle density operator, uniform in $\hbar\in(0,1]$ (where $\hbar$ is the Planck constant) provided that $V$ and $(-\Delta)^{3+d/2}V$ have integrable Fourier transforms., Comment: 35 pages, no figure

Consider the linear Boltzmann equation of radiative transfer in a half-space, with constant scattering coefficient $\sigma$. Assume that, on the boundary of the half-space, the radiation intensity satisfies the Lambert (i.e. diffuse) reflection law with albedo coefficient $\alpha$. Moreover, assume that there is a temperature gradient on the boundary of the half-space, which radiates energy in the half-space according to the Stefan-Boltzmann law. In the asymptotic regime where $\sigma\to+\infty$ and $1-\alpha\sim C/\sigma$, we prove that the radiation pressure exerted on the boundary of the half-space is governed by a fractional diffusion equation. This result provides an example of fractional diffusion asymptotic limit of a kinetic model which is based on the harmonic extension definition of $\sqrt{-\Delta}$. This fractional diffusion limit therefore differs from most of other such limits for kinetic models reported in the literature, which are based on specific properties of the equilibrium distributions (heavy tails) or of the scattering coefficient as in [U. Frisch-H. Frisch: Mon. Not. R. Astr. Not. 181 (1977), 273-280]., Comment: 25 pages, no figure

In this paper, we extend the upper and lower bounds for the "pseudo-distance" on quantum densities analogous to the quadratic Monge-Kantorovich(-Vasershtein) distance introduced in [F. Golse, C. Mouhot, T. Paul, Commun. Math. Phys. 343 (2016) 165-205] to positive quantizations defined in terms of the family of phase space translates of a density operator, not necessarily of rank one as in the case of the T\"oplitz quantization. As a corollary, we prove that the uniform (for vanishing h) convergence rate for the mean-field limit of the N-particle Heisenberg equation holds for a much wider class of initial data than in [F. Golse, C. Mouhot, T. Paul, loc. cit.]. We also discuss the relevance of the pseudo-distance compared to the Schatten norms for the purpose of metrizing the set of quantum density operators in the semiclassical regime., Comment: 23 pages, no figure

In this short paper, we formally derive the thin spray equation for a steady Stokes gas, i.e. the equation consists in a coupling between a kinetic (Vlasov type) equation for the dispersed phase and a (steady) Stokes equation for the gas. Our starting point is a system of Boltzmann equations for a binary gas mixture. The derivation follows the procedure already outlined in [Bernard-Desvillettes-Golse-Ricci, arXiv:1608.00422 [math.AP]] where the evolution of the gas is governed by the Navier-Stokes equation., Comment: 23 pages. arXiv admin note: substantial text overlap with arXiv:1608.00422

This article proposes a derivation of the Vlasov-Navier-Stokes system for spray/aerosol flows. The distribution function of the dispersed phase is governed by a Vlasov-equation, while the velocity field of the propellant satisfies the Navier-Stokes equations for incompressible fluids. The dynamics of the dispersed phase and of the propellant are coupled through the drag force exerted by the propellant on the dispersed phase. We present a formal derivation of this model from a multiphase Boltzmann system for a binary gaseous mixture, involving the droplets/dust particles in the dispersed phase as one species, and the gas molecules as the other species. Under suitable assumptions on the collision kernels, we prove that the sequences of solutions to the multiphase Boltzmann system converge to distributional solutions to the Vlasov-Navier-Stokes equation in some appropriate distinguished scaling limit. Specifically, we assume (a) that the mass ratio of the gas molecules to the dust particles/droplets is small, (b) that the thermal speed of the dust particles/droplets is much smaller than that of the gas molecules and (c) that the mass density of the gas and of the dispersed phase are of the same order of magnitude., Comment: 35 pages

In this paper we study a perturbative approach to the problem of quantization of measures in the plane. Motivated by the fact that, as the number of points tends to infinity, hexagonal lattices are asymptotically optimal from an energetic point of view (see [Morgan, Bolton: Amer. Math. Monthly 109 (2002), 165-172]), we consider configurations that are small perturbations of the hexagonal lattice and we show that: (1) in the limit as the number of points tends to infinity, the hexagonal lattice is a strict minimizer of the energy; (2) the gradient flow of the limiting functional allows us to evolve any perturbed configuration to the optimal one exponentially fast. In particular, our analysis provides a solid mathematical justification of the asymptotic optimality of the hexagonal lattice among its nearby configurations., Comment: 24 pages, 6 figures

In this paper the Hartree equation is derived from the $N$-body Schr\''odinger equation in the mean-field limit, with convergence rate estimates that are uniform in the Planck constant $\hbar$. Specifically, we consider the two following cases:(a) T\''oplitz initial data and Lipschitz interaction forces, and (b) analytic initial data and interaction potential, over short time intervals independent of $\hbar$. The convergence rates in these two cases are $1/\sqrt{\log\log N}$ and $1/N$ respectively. The treatment of the second case is entirely self-contained and all the constants appearing in the final estimate are explicit. It provides a derivation of the Vlasov equation from the $N$-body classical dynamics using BBGKY hierarchies instead of empirical measures.

This paper aims to justify the Maxwell-Boltzmann approximation for electrons, preserving the dynamics of ions at the kinetic level. Under sufficient regularity assumption, we provide a precise scaling where the Maxwell-Boltzmann approximation is obtained. In addition, we prove that the reduced ions problem is well-posed globally in time.