arXiv subjects

Thierry Paul

Publications and source records attributed to Thierry Paul.

At least 37 records · Page 2Linked to original sources

arXiv2021

Quantum and Semiquantum Pseudometrics and Applications

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.

math.AP↗

arXiv2021

Towards Optimal Transport for Quantum Densities

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]).

math-ph↗

arXiv2020

Semiclassical Evolution With Low Regularity

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.

math.AP↗

arXiv2020

Symbolic calculus for singular curve operators

We define a generalization of the T\''oplitz quantization, suitable for operators whose T\''oplitz symbols are singular. We then show that singular curve operators in Topological Quantum Fields Theory (TQFT) are precisely generalized T\''oplitz operators of this kind and we compute for some of them, and conjecture for the others, their main symbol, determined by the associated classical trace function.

math-ph↗

arXiv2019

The Random Batch Method for $N$-Body Quantum Dynamics

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 $p 1$ and independent of the Planck constant $\hbar$.

math.AP↗

arXiv2019

Mean-Field and Classical Limit for the N-Body Quantum Dynamics with Coulomb Interaction

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]].

math.AP↗

arXiv2019

On the Convergence of Time Splitting Methods for Quantum Dynamics in the Semiclassical Regime

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.

math.NA↗

arXiv2018

On the derivation of the Hartree equation in the mean field limit: Uniformity in the Planck constant

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.

math.AP↗

arXiv2018

On the size of chaos in the mean field dynamics

We consider the error arising from the approximation of an N-particle dynamics with its description in terms of a one-particle kinetic equation. We estimate the distance between the j-marginal of the system and the factorized state, obtained in a mean field limit as N $\rightarrow$ $\infty$. Our analysis relies on the evolution equation for the "correlation error" rather than on the usual BBGKY hierarchy. The rate of convergence is shown to be O(j 2 N) in any bounded interval of time (size of chaos), as expected from heuristic arguments. Our formalism applies to an abstract hierarchical mean field model with bounded collision operator and a large class of initial data, covering (a) stochastic jump processes converging to the homogeneous Boltzmann and the Povzner equation and (b) quantum systems giving rise to the Hartree equation.

math.AP↗

arXiv2017

Empirical Measures and Quantum Mechanics: Application to the Mean-Field Limit

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 Schrödinger 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 $(-Δ)^{3+d/2}V$ have integrable Fourier transforms.

math.AP↗

arXiv2017

Asymptotic expansion of the mean-field approximation

We established and estimate the full asymptotic expansion in integer powers of 1 N of the [ $\sqrt$ N ] first marginals of N-body evolutions lying in a general paradigm containing Kac models and non-relativistic quantum evolution. We prove that the coefficients of the expansion are, at any time, explicitly computable given the knowledge of the linearization on the one-body meanfield kinetic limit equation. Instead of working directly with the corresponding BBGKY-type hierarchy, we follows a method developed in [22] for the meanfield limit, dealing with error terms analogue to the v-functions used in previous works. As a by-product we get that the rate of convergence to the meanfield limit in 1 N is optimal.

quant-ph↗

arXiv2017

Wave Packets and the Quadratic Monge-Kantorovich Distance in Quantum Mechanics

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 Töplitz 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.

math.AP↗

arXiv2017

Normalization in Banach scale Lie algebras via mould calculus and applications

We study a perturbative scheme for normalization problems involving resonances of the unperturbed situation, and therefore the necessity of a non-trivial normal form, in the general framework of Banach scale Lie algebras (this notion is defined in the article). This situation covers the case of classical and quantum normal forms in a unified way which allows a direct comparison. In particular we prove a precise estimate for the difference between quantum and classical normal forms, proven to be of order of the square of the Planck constant. Our method uses mould calculus (recalled in the article) and properties of the solution of a universal mould equation studied in a preceding paper.

math.AP↗

arXiv2017

Rayleigh-Schrödinger series and Birkhoff decomposition

We derive new expressions for the Rayleigh-Schrödinger seriesdescribing the perturbation of eigenvalues of quantumHamiltonians. The method, somehow close to the so-called dimensionalrenormalization in quantum field theory, involves the Birkhoffdecomposition of some Laurent series built up out of explicit fullynon-resonant terms present in the usual expression of theRayleigh-Schrödinger series. Our results provide new combinational formulae and a new way \ff{of deriving} perturbation series in Quantum Mechanics. More generally we prove that such a decomposition provides solutions of general normal form problems in Lie algebras.

math.AP↗

arXiv2016

The Schrödinger Equation in the Mean-Field and Semiclassical Regime

In this paper, we establish (1) the classical limit of the Hartree equation leading to the Vlasov equation, (2) the classical limit of the $N$-body linear Schrödinger equation uniformly in N leading to the N-body Liouville equation of classical mechanics and (3) the simultaneous mean-field and classical limit of the N-body linear Schrödinger equation leading to the Vlasov equation. In all these limits, we assume that the gradient of the interaction potential is Lipschitz continuous. All our results are formulated as estimates involving a quantum analogue of the Monge-Kantorovich distance of exponent 2 adapted to the classical limit, reminiscent of, but different from the one defined in [F. Golse, C. Mouhot, T. Paul, Commun. Math. Phys. 343 (2016), 165-205]. As a by-product, we also provide bounds on the quadratic Monge-Kantorovich distances between the classical densities and the Husimi functions of the quantum density matrices.

math.AP↗

arXiv2016

Normalization in Lie algebras via mould calculus and applications

We establish Ecalle's mould calculus in an abstract Lie-theoretic setting and use it to solve a normalization problem, which covers several formal normal form problems in the theory of dynamical systems. The mould formalism allows us to reduce the Lie-theoretic problem to a mould equation, the solutions of which are remarkably explicit and can be fully described by means of a gauge transformation group. The dynamical applications include the construction of Poincar{é}-Dulac formal normal forms for a vector field around an equilibrium point, a formal infinite-order multiphase averaging procedure for vector fields with fast angular variables (Hamiltonian or not), or the construction of Birkhoff normal forms both in classical and quantum situations. As a by-product we obtain, in the case of harmonic oscillators, the convergence of the quantum Birkhoff form to the classical one, without any Diophantine hypothesis on the frequencies of the unperturbed Hamiltonians.

math.DS↗

arXiv2015

On the Mean Field and Classical Limits of Quantum Mechanics

The main result in this paper is a new inequality bearing on solutions of the $N$-body linear Schrödinger equation and of the mean field Hartree equation. This inequality implies that the mean field limit of the quantum mechanics of $N$ identical particles is uniform in the classical limit and provides a quantitative estimate of the quality of the approximation. This result applies to the case of $C^{1,1}$ interaction potentials. The quantity measuring the approximation of the $N$-body quantum dynamics by its mean field limit is analogous to the Monge-Kantorovich (or Wasserstein) distance with exponent $2$. The inequality satisfied by this quantity is reminiscent of the work of Dobrushin on the mean field limit in classical mechanics [Func. Anal. Appl. 13 (1979), 115-123]. Our approach of this problem is based on a direct analysis of the $N$-particle Liouville equation, and avoids using techniques based on the BBGKY hierarchy or on second quantization.

math.AP↗

arXiv2015

Quantum singular complete integrability

We consider some perturbations of a family of pairwise commuting linear quantum Hamiltonians on the torus with possibly dense pure point spectra. We prove that the Rayleigh-Schr{ö}dinger perturbation series converge near each unperturbed eigenvalue under the form of a convergent quantum Birkhoff normal form. Moreover the family is jointly diagonalised by a common unitary operator explicitly constructed by a Newton type algorithm. This leads to the fact that the spectra of the family remain pure point. The results are uniform in the Planck constant near $\hbar= 0$. The unperturbed frequencies satisfy a small divisors condition %(Bruno type condition (including the Diophantine case) and we explicitly estimate how this condition can be released when the family tends to the unperturbed one.

math-ph↗