Quantum optimal transport and weak topologies

Several extensions of the classical optimal transport distances to the quantum setting have been proposed. In this paper, we investigate the pseudometrics introduced by Golse, Mouhot and Paul in [Commun Math Phys 343:165-205, 2016] and by Golse and Paul in [Arch Ration Mech Anal 223:57-94, 2017]. These pseudometrics serve as a quantum analogue of the Monge--Kantorovich--Wasserstein distances of order $2$ on the phase space. We prove that they are comparable to negative Sobolev norms up to a small term in the semiclassical approximation, which can be expressed using the Wigner--Yanase Skew information. This enables us to improve the known results in the context of the mean-field and semiclassical limits by requiring less regularity on the initial data.Comment: 25 page

Optimal Semiclassical Regularity of Projection Operators and Strong Weyl Law

Projection operators arise naturally as one particle density operators associated to Slater determinants in fields such as quantum mechanics and the study of determinantal processes. In the context of the semiclassical approximation of quantum mechanics, projection operators can be seen as the analogue of characteristic functions of subsets of the phase space, which are discontinuous functions. We prove that projection operators indeed converge to characteristic functions of the phase space and that in terms of quantum Sobolev spaces, they exhibit the same maximal regularity as characteristic functions. This can be interpreted as a semiclassical asymptotic on the size of commutators in Schatten norms. Our study answers a question raised in [J. Chong, L. Lafleche, C. Saffirio, arXiv:2103.10946 [math.AP]] about the possibility of having projection operators as initial data, and also gives a strong convergence result for the Weyl law.Comment: 18 pages, 2 figures. v2: context on the semiclassical mean-field limit added, estimate of Remark 2.2 improved, bibliography update

A mean-field game economic growth model

Here, we examine a mean-field game (MFG) that models the economic growth of a population of non-cooperative rational agents. In this MFG, agents are described by two state variables - the capital and consumer goods they own. Each agent seeks to maximize their utility by taking into account statistical data of the total population. The individual actions drive the evolution of the players, and a market-clearing condition determines the relative price of capital and consumer goods. We study the existence and uniqueness of optimal strategies of the agents and develop numerical methods to compute these strategies and the equilibrium price

From many-body quantum dynamics to the Hartree-Fock and Vlasov equations with singular potentials

We obtain the combined mean-field and semiclassical limit from the $N$-body Schr\"{o}dinger equation for fermions interacting via singular potentials. To obtain the result, we first prove the uniformity in Planck's constant $h$ propagation of regularity for solutions to the Hartree\unicode{x2013}Fock equation with singular pair interaction potentials of the form $\pm |x-y|^{-a}$, including the Coulomb and gravitational interactions. In the context of mixed states, we use these regularity properties to obtain quantitative estimates on the distance between solutions to the Schr\"{o}dinger equation and solutions to the Hartree\unicode{x2013}Fock and Vlasov equations in Schatten norms. For $a\in(0,1/2)$, we obtain local-in-time results when $N^{-1/2} \ll h \leq N^{-1/3}$. In particular, it leads to the derivation of the Vlasov equation with singular potentials. For $a\in[1/2,1]$, our results hold only on a small time scale, or with an $N$-dependent cutoff.Comment: 75 pages; introduction improved and some errors in the propagation of regularity part correcte

Fractional hypocoercivity

This research report is devoted to kinetic equations without confinement. We investigate the large time behaviour induced by collision operators with fat tailed local equilibria. Such operators have an anomalous diffusion limit. In the appropriate scaling, the macroscopic equation involves a fractional diffusion operator so that the optimal decay rate is determined by a fractional Nash inequality. At kinetic level we develop an L 2 hypocoercivity approach and establish a rate of decay compatible with the anomalous diffusion limit

Quantum Optimal Transport and Weak Topologies

25 pages. v3: added new Inequality (12) (and Prop 2.2) and Theorem 1.2Several extensions of the classical optimal transport distances to the quantum setting have been proposed. In this paper, we investigate the pseudometrics introduced by Golse, Mouhot and Paul in [Commun Math Phys 343:165-205, 2016] and by Golse and Paul in [Arch Ration Mech Anal 223:57-94, 2017]. These pseudometrics serve as a quantum analogue of the Monge-Kantorovich-Wasserstein distances of order $2$ on the phase space. We prove that they are comparable to negative Sobolev norms up to a small term due to a positive "self-distance" in the semiclassical approximation, which can be bounded above using the Wigner-Yanase skew information. This enables us to improve the known results in the context of the mean-field and semiclassical limits by requiring less regularity on the initial data