This is a searchable list of papers that are directly related to my research. It is still in construction.

- R. Adami, F. Golse, A. Teta. Rigorous Derivation of the Cubic NLS in Dimension One. Journal of Statistical Physics, 127(6):1193-1220, 2007. Abstract.
We derive rigorously the one-dimensional cubic nonlinear Schrödinger equation from a many-body quantum dynamics. The interaction potential is rescaled through a weak-coupling limit together with a short-range one. We start from a factorized initial state, and prove propagation of chaos with the usual two-step procedure: in the former step, convergence of the solution of the BBGKY hierarchy associated to the many-body quantum system to a solution of the BBGKY hierarchy obtained from the cubic NLS by factorization is proven; in the latter, we show the uniqueness for the solution of the infinite BBGKY hierarchy.
- R. Agarwal, F. Tcherimmissine. Computation of Hypersonic Shock Wave Flows of Diatomic Gases and Gas Mixtures using the Generalized Boltzmann Equation. In 48th AIAA Aerospace Sciences Meeting, 2012. Abstract.
Hypersonic flows about space vehicles produce flow fields in thermodynamic non-equilibrium with local Knudsen numbers Kn =λ/L (where λ is the mean free path of gas molecules and L is a characteristic length) which may lie in all the three regimes-continuum, transition and rarefied. Flows in continuum regime can be modeled accurately by the Navier-Stokes (NS) equations; however the flows in transition and rarefied regimes require a kinetic approach such as the Direct Simulation Monte Carlo (DSMC) method or the solution of the Boltzmann equation. This paper describes the development of a computational methodology and a code for computing hypersonic non-equilibrium shockwave flows of diatomic gases using the Generalized Boltzmann Equation (GBE) at Knudsen numbers in transitional and rarefied flow regimes. The GBE solver has been validated by computing the 1D shock structure in nitrogen for Rotational-Translational (R-T) relaxations and comparing the numerical results with the experimental data for Mach numbers up to 15. The solver has been exercised successfully for computing the 2D blunt body flowsin nitrogenand 3D flow from a rectangular jet of nitrogen in vacuum for R-T relaxations. The issuesof stability of the algorithm and the possibility of reducing the number of rotational levels in the computations without compromising the accuracy of the solutions have been rigorously addressed.A new two-level kinetic model has been developed for computing the RT relaxations in a diatomic gas and has been validated by comparing the results with the solutions of complete GBE. The model is about twenty times more efficient than the GBE in computing the shock structure. It should be noted that the model is different than the BGK model; it accounts for both elastic and inelastic collisions. The computational methodology has been extendedto compute the hypersonic shock structure in diatomic gases including both the RT and Vibrational-Translational (V-T) relaxations. 1-D shock structure in nitrogen has been computed including both R-T and V-T relaxations and has been validated by comparing the results with the experimental data. A computational methodology has also been developed to compute the hypersonic shock structure in a non-reactive mixture of two diatomic gases. 1-D shock structure has been computed in an inert mixture of nitrogen and oxygen for R-T relaxations. To accomplish this, the GBE is formulated and solved in “impulse space” instead of velocity space.
- R. Alexander. Time Evolution for Infinitely Many Hard Spheres. Communications in Mathematical Physics, Vol. 49, pages 217-232, 1976. Abstract.
We construct the time evolution for infinitely many particles in R^v interacting by the hard-sphere potential. Because there are abundant examples of hard-sphere configurations with more than one solution to the Newtonian equations of motion, we introduce the concept of a regular solution, in which the growth of velocities and crowding of particles at infinity are limited. We prove that (1) regular solutions exist with probability one in every equilibrium state, and (2) any configuration of the infinite system is the initial point of at most one regular solution. Equilibrium states are invariant under the time-evolution.
- Z. Ammari, F. Nier. Mean Field Propagation of Wigner Measures and BBGKY Hierarchies for General Bosonic States. Journal de Mathématiques Pures et Appliquées. 95(6): 585-626, 2011. Abstract.
Contrary to the finite dimensional case, Weyl and Wick quantizations are no more asymptotically equivalent in the infinite dimensional bosonic second quantization. Moreover neither the Weyl calculus defined for cylindrical symbols nor the Wick calculus defined for polynomials are preserved by the action of a nonlinear flow. Nevertheless taking advantage carefully of the information brought by these two calculuses in the mean field asymptotics, the propagation of Wigner measures for general states can be proved, extending to the infinite dimensional case a standard result of semiclassical analysis.
- A. Athanassoulis, T. Paul, F. Pezzotti, M. Pulvirenti. Semiclassical Propagation of Coherent States for the Hartree Equation. Annales Henri Poincaré, 12(8): 1613-1634, 2011. Abstract.
In this paper we consider the nonlinear Hartree equation in presence of a given external potential, for an initial coherent state. Under suitable smoothness assumptions, we approximate the solution in terms of a time dependent coherent state, whose phase and amplitude can be determined by a classical flow. The error can be estimated in $L^2$ by $C \sqrt {\var}$, $\var$ being the Planck constant. Finally we present a full formal asymptotic expansion.
- G. Bai, U. Koley, S. Mishra, R. Molinaro. Physics Informed Neural Networks (PINNs) for Approximating Nonlinear Dispersive PDEs. arXiv.org, 2022. Abstract.
We propose a novel algorithm, based on physics-informed neural networks (PINNs) to efficiently approximate solutions of nonlinear dispersive PDEs such as the KdV-Kawahara, Camassa-Holm and Benjamin-Ono equations. The stability of solutions of these dispersive PDEs is leveraged to prove rigorous bounds on the resulting error. We present several numerical experiments to demonstrate that PINNs can approximate solutions of these dispersive PDEs very accurately.
- C. Bardos, F. Golse, A. D. Gottlieb, N. J. Mauser. Mean Field Dynamics of Fermions and the Time-Dependent Hartree-Fock Equation. Journal de Mathématiques Pures et Appliquées, 82(6): 665-683, 2003. Abstract.
The time-dependent Hartree-Fock equations are derived from the N-particle Schrödinger equation with mean-field scaling in the infinite particle limit, for initial data that are like Slater determinants. Only the case of bounded interaction potentials is treated in this work. We prove that, in the infnite particle limit, the first partial trace of the N-particle density operator approaches the solution of the time-dependent Hartree-Fock equations in the trace norm.
- J. Bedrossian, N. Masmoudi, C. Mouhot. Landau Damping: Paraproducts and Gevrey Regularity. Annals of PDE, 2(4), 2016. Abstract.
We give a new, simpler, proof of nonlinear Landau damping on T^d in Gevrey-1/s regularity (s > 1/3) which matches the regularity requirement predicted by the formal analysis of Mouhot and Villani in the original proof of Landau damping [Acta Mathematica 2011]. Our proof combines in a novel way ideas from the original proof of Landau damping and the proof of inviscid damping in 2D Euler [arXiv:1306.5028]. As in the work on 2D Euler, we use paraproduct decompositions and controlled regularity loss to replace the Newton iteration scheme employed in the original proof. We perform time-response estimates adapted from the original proof to control the plasma echoes and couple them to energy estimates on the distribution function in the style of the work on 2D Euler.
- N. Benedikter, V. Jakšić, M. Porta, C. Saffirio, B. Schlein. Mean-Field Evolution of Fermionic Mixed States. Communications on Pure and Applied Mathematics, 69(12):2250-2303, 2016. Abstract.
In this paper we study the dynamics of fermionic mixed states in the mean-field regime. We consider initial states that are close to quasi-free states and prove that, under suitable assumptions on the initial data and on the many-body interaction, the quantum evolution of such initial data is well approximated by a suitable quasi-free state. In particular, we prove that the evolution of the reduced one-particle density matrix converges, as the number of particles goes to infinity, to the solution of the time-dependent Hartree-Fock equation. Our result holds for all times and gives effective estimates on the rate of convergence of the many-body dynamics towards the Hartree-Fock evolution.
- N. Benedikter, M. Porta, C. Saffirio, B. Schlein. From the Hartree Dynamics to the Vlasov Equation. Archive for Rational Mechanics and Analysis, 221(1): 272-334, 2016. Abstract.
We consider the evolution of quasi-free states describing N fermions in the mean field limit, as governed by the nonlinear Hartree equation. In the limit of large N, we study the convergence towards the classical Vlasov equation. For a class of regular interaction potentials, we establish precise bounds on the rate of convergence.
- N. Benedikter, M. Porta, B. Schlein. Mean-Field Dynamics of Fermions with Relativistic Dispersion. Journal of Mathematical Physics, 55(2), 2014. Abstract.
We extend the derivation of the time-dependent Hartree-Fock equation recently obtained in [2] to fermions with a relativistic dispersion law. The main new ingredient is the propagation of semiclassical commutator bounds along the pseudo-relativistic Hartree-Fock evolution.
- J. Berner, P. Grohs, G. Kutynick, P. Peterson. The Modern Mathematics of Deep Learning. Cambridge University Press, pages 1-111, 2022. Abstract.
We describe the new field of mathematical analysis of deep learning. This field emerged around a list of research questions that were not answered within the classical framework of learning theory. These questions concern: the outstanding generalization power of overparametrized neural networks, the role of depth in deep architectures, the apparent absence of the curse of dimensionality, the surprisingly successful optimization performance despite the non-convexity of the problem, understanding what features are learned, why deep architectures perform exceptionally well in physical problems, and which fine aspects of an architecture affect the behavior of a learning task in which way. We present an overview of modern approaches that yield partial answers to these questions. For selected approaches, we describe the main ideas in more detail.
- C. Bianca, C. Dogbe. On the Boltzmann Gas Mixture Equation: Linking the Kinetic and Fluid Regimes. Communications in Nonlinear Science and Numerical Simulation, 29:240-256, 2015. Abstract.
This paper aims at developing a new connection between the Boltzmann equation and the Navier-Stokes equation. Specifically the paper deals with the derivation of the macroscopic equations from asymptotic limits of the Boltzmann equation for a binary gas mixture of hard-sphere gases. By extending the methodology of the single-component gases case and by employing different time and space scalings, we show that it is possible to recover, under suitable technical assumptions, various fluid dynamics equations like the incompressible linearized and nonlinear Navier-Stokes equations, the incompressible linearized and nonlinear Euler equations. The novelty of this paper is the method that we propose, which differs from the Hilbert and Chapman-Enskog expansions. Future research directions are also discussed in the last section of the paper with special attention at the different scalings that can be employed in order to obtain equations presenting a ghost effect.
- A.B. Bobylev, I. M. Gamba. Boltzmann Equations for Mixtures of Maxwell Gases: Exact Solutions and Power Like Tails. Journal of Statistical Physics, 124: 497-516, 2006. Abstract.
We consider the Boltzmann equations for mixtures of Maxwell gases. It is shown that in certain limiting case the equations admit self-similar solutions that can be constructed in explicit form. More precisely, the solutions have simple explicit integral representations. The most interesting solutions have finite energy and power like tails. This shows that power like tails can appear not just for granular particles (Maxwell models are far from reality in this case), but also in the system of particles interacting in accordance with laws of classical mechanics. In addition, non-existence of positive self-similar solutions with finite moments of any order is proven for a wide class of Maxwell models.
- L. Boltzmann. Weitere Studien über das Wärmengleichgewicht unter Gasmolekülen. Sitzungsberichte Akad. Wiss., Vienna (II), 66:275-370, 1872. Abstract.
Nach der mechanischen Wärmetheorie gehorchen die thermischen Eigenschaften von Gasen und anderen Stoffen trotz der Tatsache, daß diese Stoffe aus einer großen Anzahl von Molekülen zusammengesetzt sind, die sich in schneller ungleichmäßiger Bewegung befinden, wohlbestimmten Gesetzen. Die Erklärung dieser Eigenschaften muß auf die Wahrscheinlichkeitsrechnung gegründet werden, und dazu muß man die Verteilungsfunktion kennen, die zu jedem Zeitpunkt die Anzahl von Molekülen in jedem Zustand bestimmt. Um diese Verteilungsfunktion f(x,t) = Anzahl der Moleküle, die zur Zeit t die Energie x haben, zu ermitteln, wird für f eine partielle Differentialgleichung hergeleitet, indem untersucht wird, wie sich f während eines kleinen Zeitintervalls infolge von Stößen zwischen den Molekülen ändert.

According to the mechanical theory of heat, the thermal properties of gases and other substances obey well-determined laws, despite the fact that these substances are composed of a large number of molecules in rapid, uneven motion. The explanation of these properties must be based on probability theory, and for this one must know the distribution function that determines the number of molecules in each state at any given time. In order to determine this distribution function f(x,t) = number of molecules that have energy x at time t, a partial differential equation is derived for f by examining how f changes during a small time interval as a result of collisions between the molecules changes. - W. Braun, K. Hepp. The Vlasov Dynamics and its Fluctuations in the 1/N Limit of Interacting Classical Particles. 56(2):101-113, 1977. Abstract.
For classical N-particle systems with pair interaction ... the Vlasov dynamics is shown to be the w*-limit as $N \rightarrow \infty$. Propagation of molecular chaos holds in this limit, and the fluctuations of intensive observables converge to a Gaussian stochastic process.
- M. Briant, E. Daus. The Boltzmann Equation for a Multi-species Mixture Close to Global Equilibrium. Archive for Rational Mechanics and Analaysis, 222: 1367-1443, 2016. Abstract.
We study the Cauchy theory for a multi-species mixture, where the different species can have different masses, in a perturbative setting on the three dimensional torus. The ultimate aim of this work is to obtain the existence, uniqueness and exponential trend to equilibrium of solutions to the multi-species Boltzmann equation in , where is a polynomial weight. We prove the existence of a spectral gap for the linear multi-species Boltzmann operator allowing different masses, and then we establish a semigroup property thanks to a new explicit coercive estimate for the Boltzmann operator. Then we develop an theory à la Guo for the linear perturbed equation. Finally, we combine the latter results with a decomposition of the multi-species Boltzmann equation in order to deal with the full equation. We emphasize that dealing with different masses induces a loss of symmetry in the Boltzmann operator which prevents the direct adaptation of standard mono-species methods (for example Carleman representation, Povzner inequality). Of important note is the fact that all methods used and developed in this work are constructive. Moreover, they do not require any Sobolev regularity and the framework is dealt with for any, recovering the optimal physical threshold of finite energy in the particular case of a multi-species hard spheres mixture with the same masses.
- J. Bruna, B. Perstorfer, E. Vanden-Eijnden. Neural Galerkin Scheme with Active Learning for High-Dimensional Evolution Equations. arXiv.org, 2022. Abstract.
Deep neural networks have been shown to provide accurate function approximations in high dimensions. However, fitting network parameters requires training data that may not be available beforehand, which is particularly challenging in science and engineering applications where often it is even unclear how to collect new informative training data in the first place. This work proposes Neural Galerkin schemes based on deep learning that generate training data samples with active learning for numerically solving high-dimensional partial differential equations. Neural Galerkin schemes train networks by minimizing the residual sequentially over time, which enables adaptively collecting new training data in a self-informed manner that is guided by the dynamics described by the partial differential equations, which is in stark contrast to many other machine learning methods that aim to fit network parameters globally in time without taking into account training data acquisition. Our finding is that the active form of gathering training data of the proposed Neural Galerkin schemes is key for numerically realizing the expressive power of networks in high dimensions. Numerical experiments demonstrate that Neural Galerkin schemes have the potential to enable simulating phenomena and processes with many variables for which traditional and other deep-learning-based solvers fail, especially when features of the solutions evolve locally such as in high-dimensional wave propagation problems and interacting particle systems described by Fokker-Planck and kinetic equations.
- A. Campo, M. M. Papari, E. Abu-Nada. Estimation of the Minimum Prandtl Number for the Binary Gas Mixtures Formed with Light Helium and Certain Heavier Gases: Application to Thermoacoustic Refrigerators. Applied Thermal Engineering, 31(16):3142-3146, 2011. Abstract.
This paper addresses a detailed procedure for the accurate estimation of low Prandtl numbers of selected binary gas mixtures. In this context, helium (He) is the light primary gas and the heavier secondary gases are nitrogen (N2), oxygen (O2), xenon (Xe), carbon dioxide (CO2), methane (CH4), tetrafluoromethane or carbon tetrafluoride (CF4) and sulfur hexafluoride (SF6). The three thermophysical properties forming the Prandtl number of binary gas mixtures Prmix are heat capacity at constant pressure Cp,mix (thermodynamic property), viscosity ηmix (transport property) and thermal conductivity λmix (transport property), which in general depend on temperature T and molar gas composition w. The precise formulas for the calculation of the trio Cp,mix, ηmix, and λmix are gathered from various dependable sources. When the set of computed Prmix values for the seven binary gas mixtures He + N2, He + O2, He + Xe, He + CO2, He + CH4, He + CF4, He + SF6 at atmospheric conditions T = 300 K, p = 1 atm is plotted against the molar gas composition w on the w-domain [0,1], the family of Prmix(w) curves exhibited distinctive concave shapes. In the curves format, all Prmix(w) curves initiate with Pr ≈ 0.7 at w = 0 (associated with light primary He). Forthwith, each Prmix(w) curve descends to a unique minimum and thereafter ascend back to Pr ≈ 0.7 at the terminal point w = 1 (connected to heavier secondary gases). Overall, it was found that among the seven binary gas mixtures tested, the He + Xe gas mixture delivered the absolute minimum Prandtl number Prmix,min = 0.12 at the optimal molar gas composition wopt = 0.975.
- S. Chapman, T. G. Cowling. The Mathematical Theory of Non-Uniform Gases. Cambridge University Press, London, 1952. Abstract.
This classic book, now reissued in paperback, presents a detailed account of the mathematical theory of viscosity, thermal conduction, and diffusion in non-uniform gases based on the solution of the Maxwell-Boltzmann equations. The theory of Chapman and Enskog, describing work on dense gases, quantum theory of collisions, and the theory of conduction and diffusion in ionized gases in the presence of electric and magnetic fields is also included in the later chapters. This reprint of the third edition, first published in 1970, includes revisions that take account of extensions of the theory to fresh molecular models and of new methods used in discussing dense gases and plasmas.
- L. Chen, J. O. Lee, B. Schlein. Rate of Convergence Towards Hartree Dynamics. Journal of Statistical Physics, 144(4):872-903, 2011. Abstract.
We consider a system of N bosons interacting through a two-body potential with, possibly, Coulomb-type singularities. We show that the difference between the many-body Schrödinger evolution in the mean-field regime and the effective nonlinear Hartree dynamics is at most of the order 1/N, for any fixed time. The N-dependence of the bound is optimal.
- T. Chen, C. Hainzl, N. Pavlović, R. Seiringer. Unconditional Uniqueness for the Cubic Gross-Pitaevskii Hierarchy via Quantum de Finetti. Communications on Pure and Applied Mathematics, 68(10):1845-1884, 2015. Abstract.
We present a new, simpler proof of the unconditional uniqueness of solutions to the cubic Gross-Pitaevskii hierarchy in $\R^3$. One of the main tools in our analysis is the quantum de Finetti theorem. Our uniqueness result is equivalent to the one established in the celebrated works of Erdös, Schlein and Yau.
- S. Choi, S. Kwon. Modified Scattering for the Vlasov-Poisson System. Nonlinearity, 29(9):2755-2774, 2016. Abstract.
We study the asymptotic behavior of dispersing solutions to the Vlasov-Poisson system. Due to long interaction range, we do not expect linear scattering (Choi S-H and Ha S-Y 2011 SIAM J. Math. Anal. 43 2050-77). Instead, we prove a modified scattering result (or long range scattering result) of small and dispersing solutions. We find a quasi-free forward trajectory so that along the trajectory, the solution has an asymptotic limit. We extract the logarithmic growth part of the Duhamel term, and absorb it into the quasi-free trajectory, then the remaining part enjoys faster decay so as to obtain the asymptotic state.
- J. J. Chong, L. Lafleche, C. Saffirio. From Many-Body Quantum Dynamics to the Hartree-Fock and Vlasov Equations with Singular Potentials. arXiv.org, 2023. Abstract.
We obtain the combined mean-field and semiclassical limit from the N-body Schrödinger 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-Fock equation with singular pair interaction potentials of the form ±|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 Schrödinger equation and solutions to the Hartree-Fock and Vlasov equations in Schatten norms. For a∈(0,1/2), we obtain local-in-time results when N-1/2≪h≤N-1/3. In particular, it leads to the derivation of the Vlasov equation with singular potentials. For a∈[1/2,1], our results hold only on a small time scale, or with an N-dependent cutoff.
- S. Cuomo, V.S. Di Cola, F. Giampaolo, G. Rozza, M. Raissi, F. Piccialli. Scientific Machine Learning Through Physics-Informed Neural Networks: Where We are and What's Next. Journal of Scientific Computing, 92(88), 2022. Abstract.
Physics-Informed Neural Networks (PINN) are neural networks (NNs) that encode model equations, like Partial Differential Equations (PDE), as a component of the neural network itself. PINNs are nowadays used to solve PDEs, fractional equations, integral-differential equations, and stochastic PDEs. This novel methodology has arisen as a multi-task learning framework in which a NN must fit observed data while reducing a PDE residual. This article provides a comprehensive review of the literature on PINNs: while the primary goal of the study was to characterize these networks and their related advantages and disadvantages. The review also attempts to incorporate publications on a broader range of collocation-based physics informed neural networks, which stars form the vanilla PINN, as well as many other variants, such as physics-constrained neural networks (PCNN), variational hp-VPINN, and conservative PINN (CPINN). The study indicates that most research has focused on customizing the PINN through different activation functions, gradient optimization techniques, neural network structures, and loss function structures. Despite the wide range of applications for which PINNs have been used, by demonstrating their ability to be more feasible in some contexts than classical numerical techniques like Finite Element Method (FEM), advancements are still possible, most notably theoretical issues that remain unresolved.
- A. Davies, P. Veličković, L. Buesing, S. Blackwell, D. ZHeng, N. Tomašev, R. Tanburn, P. Battaglia, C. Blundell, A. Juhász, M. Lackenby, G. Williamson, D. Hassabis, P. Kohli. Advancing Mathematics by Guiding Human Intuition with AI. Nature (London), 600(7887):70-74, 2021. Abstract.
The practice of mathematics involves discovering patterns and using these to formulate and prove conjectures, resulting in theorems. Since the 1960s, mathematicians have used computers to assist in the discovery of patterns and formulation of conjectures1, most famously in the Birch and Swinnerton-Dyer conjecture2, a Millennium Prize Problem3. Here we provide examples of new fundamental results in pure mathematics that have been discovered with the assistance of machine learning—demonstrating a method by which machine learning can aid mathematicians in discovering new conjectures and theorems. We propose a process of using machine learning to discover potential patterns and relations between mathematical objects, understanding them with attribution techniques and using these observations to guide intuition and propose conjectures. We outline this machine-learning-guided framework and demonstrate its successful application to current research questions in distinct areas of pure mathematics, in each case showing how it led to meaningful mathematical contributions on important open problems: a new connection between the algebraic and geometric structure of knots, and a candidate algorithm predicted by the combinatorial invariance conjecture for symmetric groups4. Our work may serve as a model for collaboration between the fields of mathematics and artificial intelligence (AI) that can achieve surprising results by leveraging the respective strengths of mathematicians and machine learning.
- E. de la Canal, I. M. Gamba, M. Pavić-Čolić. Propogation of $L^p_\beta$-norm, $1 < p \leq \infty$, for the System of Boltzmann Equations for Monoatomic Gas Mixtures. arXiv.org, 2020. Abstract.
With the existence and uniqueness of a vector value solution for the full non-linear homogeneous Boltzmann system of equations describing multi-component monatomic gas mixtures for binary interactions proved [8], we present in this manuscript several properties for such a solution. We start by proving the gain of integrability of the gain term of the multispecies collision operator, extending the work done previously for the single species case [3]. In addition, we study the integrability properties of the multispecies collision operator as a bilinear form, revisiting and expanding the work done for a single gas [2]. With these estimates, together with a control by below for the loss term of the collision operator as in [8], we develop the propagation for the polynomially and exponentially β-weighted Lpβ-norms for the vector value solution. Finally, we extend such Lpβ-norms propagation property to p=∞.
- C. Dietze, J. Lee. Uniform in Time Convergence to Bose-Einstein Condensation for a Weakly Interacting Bose Gas with an External Potential. arXiv.org, 2022. Abstract.
We consider a gas of weakly interacting bosons in three dimensions subject to an external potential in the mean field regime. Assuming that the initial state of our system is a product state, we show that in the trace topology of one-body density matrices, the dynamics of the system can be described by the solution to the corresponding Hartree type equation. Using a dispersive estimate for the Hartree type equation, we obtain an error term that is uniform in time. Moreover, the dependence of the error term on the particle number is optimal. We also consider a class of intermediate regimes between the mean field regime and the Gross-Pitaevskii regime, where the error term is uniform in time but not optimal in the number of particles.
- R. Dobrušin. Vlasov Equations. Funktsional. Anal. i Prilozhen, 13(2):48-58, 1979. Abstract.
In Russian.
- M. Duda, X. Chen, A. Schindewolf, R. Bause, J. von Milczewski, R. Schmidt, I. Bloch, X. Luo. Transition from a Polaronic Condensate to a Degenerate Fermi Gas of Heteronuclear Molecules. Nature Physics, 19(5):720-725, 2023. Abstract.
The interplay of quantum statistics and interactions in atomic Bose-Fermi mixtures leads to a phase diagram markedly different from pure fermionic or bosonic systems. However, investigating this phase diagram remains challenging when bosons condense due to the resulting fast interspecies loss. Here we report observations consistent with a phase transition from a polaronic to a molecular phase in a density-matched degenerate Bose-Fermi mixture. The condensate fraction, representing the order parameter of the transition, is depleted by interactions, and the build-up of strong correlations results in the emergence of a molecular Fermi gas. The features of the underlying quantum phase transition represent a new phenomenon complementary to the paradigmatic Bose-Einstein condensate/Bardeen-Cooper-Schrieffer crossover observed in Fermi systems. By driving the system through the transition, we produce a sample of sodium-potassium molecules exhibiting a large molecule-frame dipole moment in the quantum-degenerate regime.
- M. Duerinckx. On the Size of Chaos via Glauber Calculus in the Classical Mean-Field Dynamics. Communications in Mathematical Physics, 382(1):613-653, 2021. Abstract.
We consider a system of classical particles, interacting via a smooth, long-range potential, in the mean-field regime, and we optimally analyze the propagation of chaos in form of sharp estimates on many-particle correlation functions. While approaches based on the BBGKY hierarchy are doomed by uncontrolled losses of derivatives, we propose a novel non-hierarchical approach that focusses on the empirical measure of the system and exploits a Glauber type calculus with respect to initial data in form of higher-order Poincaré inequalities for cumulants. This main result allows to rigorously truncate the BBGKY hierarchy to an arbitrary precision on the mean-field timescale, thus justifying the Bogolyubov corrections to mean field. As corollaries, we also deduce a quantitative central limit theorem for fluctuations of the empirical measure, and we partially justify the Lenard-Balescu limit for a spatially homogeneous system away from thermal equilibrium.
- W. E. Principles of Multiscale Modeling. Cambridge University Press, 2011. Abstract.
Physical phenomena can be modeled at varying degrees of complexity and at different scales. Multiscale modeling provides a framework, based on fundamental principles, for constructing mathematical and computational models of such phenomena, by examining the connection between models at different scales. This book, by a leading contributor to the field, is the first to provide a unified treatment of the subject, covering, in a systematic way, the general principles of multiscale models, algorithms and analysis. After discussing the basic techniques and introducing the fundamental physical models, the author focuses on the two most typical applications of multiscale modeling: capturing macroscale behavior and resolving local events. The treatment is complemented by chapters that deal with more specific problems. Throughout, the author strikes a balance between precision and accessibility, providing sufficient detail to enable the reader to understand the underlying principles without allowing technicalities to get in the way.
- A. Elgart, L. Erdős, B. Schlein, H. T. Yau. Non-Linear Hartree Equation as the Mean Field Limit of Weakly Coupled Fermions. Journal de Mathématiques Pures et Appliquées, 83(10):1241-1273, 2004. Abstract.
We consider a system of N weakly interacting fermions with a real analytic pair interaction. We prove that for a general class of initial data there exists a fixed time T such that the difference between the one particle density matrix of this system and the solution of the non-linear Hartree equation is of order 1/N for any time t less or equal T.
- L. Erdős, B. Schlein, H. T. Yau. Derivation of the Cubic Non-Linear Schrödinger Equation from Quantum Dynamics of Many-Body Systems. Inventiones Mathematicae, 167(3):515-614, 2007. Abstract.
We prove rigorously that the one-particle density matrix of three dimensional interacting Bose systems with a short-scale repulsive pair interaction converges to the solution of the cubic non-linear Schrödinger equation in a suitable scaling limit. The result is extended to k-particle density matrices for all positive integer k.
- L. Erdős, B. Schlein, H. T. Yau. Rigorous Derivation of the Gross-Pitaevskii Equation. Physical Review Letters, 98(4), 2007. Abstract.
The time-dependent Gross-Pitaevskii equation describes the dynamics of initially trapped Bose-Einstein condensates. We present a rigorous proof of this fact starting from a many-body bosonic Schrödinger equation with a short-scale repulsive interaction in the dilute limit. Our proof shows the persistence of an explicit short-scale correlation structure in the condensate.
- L. Erdős, B. Schlein, H. T. Yau. Rigorous Derivation of the Gross-Pitaevskii Equation with a Large Interaction Potential. Journal of the American Mathematical Society, 22(4):1099-1156, 2009. Abstract.
Consider a system of $N$ bosons in three dimensions interacting via a repulsive short range pair potential $N^2 V(N(x_i- x_j))$, where $x=(x_1, ..., x_N)$ denotes the positions of the particles. Let $H_N$ denote the Hamiltonian of the system and let $\psi_{N,t}$ be the solution to the Schrödinger equation. Suppose that the initial data $\psi_{N,0}$ satisfies the energy condition $\langle \psi_{N,0},H_N \psi_{N,0} \rangle \leq CN$ and that the one-particle density matrix converges to a projection as $N \rightarrow \infty$. Then, we prove that the k-particle density matrices of $\psi_{N,t}$ factorize in the limit $N \rightarrow \infty$. Moreover, the one particle orbital wave function solves the time-dependent Gross-Pitaevskii equation, a cubic non-linear Schrödinger equation with the coupling constant proportional to the scattering length of the potential $V$. In \cite{ESY}, we proved the same statement under the condition that the interaction potential $V$ is sufficiently small; in the present work we develop a new approach that requires no restriction on the size of the potential.
- L. Erdős, B. Schlein, H. T. Yau. Derivation of the Gross-Pitaevskii Equation for the Dynamics of Bose-Einstein Condensate. Annals of Mathematics, 172(1):291-370, 2010. Abstract.
Consider a system of N bosons in three dimensions interacting via a repulsive short range pair potential $N^2 V(N(x_i - x_j))$, where $x=(x_1, ..., x_N)$ denotes the positions of the particles. Let $H_N$ denote the Hamiltonian of the system and let $\psi_{N,t}$ be the solution to the Schrödinger equation. Suppose that the initial data 𝜓𝑁,0 satisfies the energy condition $\langle \psi_{N,0},H_N^k \psi_{N,0}\rangle \leq C^k N^k$ for $k = 1,2, ...$. We also assume that the $k$-particle density matrices of the initial state are asymptotically factorized as $N \rightarrow \infty$. We prove that the $k$-particle density matrices of $\psi_{N,t}$ are also asymptotically factorized and the one particle orbital wave function solves the Gross-Pitaevskii equation, a cubic nonlinear Schrödinger equation with the coupling constant given by the scattering length of the potential $V$. We also prove the same conclusion if the energy condition holds only for $k=1$ but the factorization of $\psi_{N,0}$ is assumed in a stronger sense.
- L. Erdős, H. T. Yau. Derivation of the Nonlinear Schrödinger Equation from a Many-Body Coulomb System. Adv. Theor. Math. Phys., 5:1169-1206, 2001. Abstract.
We consider the time evolution of N bosonic particles interacting via a mean field Coulomb potential. Suppose the initial state is a product wavefunction. We show that at any finite time the correlation functions factorize in the limit $N \rightarrow \infty$. Furthermore, the limiting one particle density matrix satisfies the nonlinear Hartree equation. The key ingredients are the uniqueness of the BBGKY hierarchy for the correlation functions and a new apriori estimate for the many-body Schrödinger equations.
- E. Faou, F. Rousset. Landau Damping in Sobolev Spaces for the Vlasov-HMF Model. Archive for Rational Mechanics and Analysis, 219(2):887-902, 2016. Abstract.
We consider the Vlasov-HMF (Hamiltonian Mean-Field) model. We consider solutions starting in a small Sobolev neighborhood of a spatially homogeneous state satisfying a linearized stability criterion (Penrose criterion). We prove that these solutions exhibit a scattering behavior to a modified state, which implies a nonlinear Landau damping effect with polynomial rate of damping.
- A. Fawzi, M. Balog, A. Huang, T. Hubert, B. Romera-Paredes, M. Barekatain,, A. Novikov, F. Ruiz, J. Schrittwieser, G. Swirszcz, D. Silver, D. Hassabis, P. Kohli. Discovering Faster Multiplication Algorithms with Reinforcement Learning. Nature, 610:47-53, 2022. Abstract.
Improving the efficiency of algorithms for fundamental computations can have a widespread impact, as it can affect the overall speed of a large amount of computations. Matrix multiplication is one such primitive task, occurring in many systems—from neural networks to scientific computing routines. The automatic discovery of algorithms using machine learning offers the prospect of reaching beyond human intuition and outperforming the current best human-designed algorithms. However, automating the algorithm discovery procedure is intricate, as the space of possible algorithms is enormous. Here we report a deep reinforcement learning approach based on AlphaZero1 for discovering efficient and provably correct algorithms for the multiplication of arbitrary matrices. Our agent, AlphaTensor, is trained to play a single-player game where the objective is finding tensor decompositions within a finite factor space. AlphaTensor discovered algorithms that outperform the state-of-the-art complexity for many matrix sizes. Particularly relevant is the case of 4 × 4 matrices in a finite field, where AlphaTensor’s algorithm improves on Strassen’s two-level algorithm for the first time, to our knowledge, since its discovery 50 years ago2. We further showcase the flexibility of AlphaTensor through different use-cases: algorithms with state-of-the-art complexity for structured matrix multiplication and improved practical efficiency by optimizing matrix multiplication for runtime on specific hardware. Our results highlight AlphaTensor’s ability to accelerate the process of algorithmic discovery on a range of problems, and to optimize for different criteria.
- A. S. Fernandes, W. Marques. Sound Propagation in Binary Gas Mixtures from a Kinetic Model of the Boltzmann Equation. Physica A, 322:29-46, 2004. Abstract.
TODO
- I. Fernandes, M. Delehaye, S. Laurent, A. T. Grier, M. Pierce, B. S. Rem, F. Chevy, C. Salomon. A Mixture of Bose and Fermi Superfluids. Science (American Association for the Advancement of Science), 345(6200):1035-1038, 2014. Abstract.
TODO
- P. Flynn, Z. Ouyang, B. Pausader, K. Widmayer. Scattering Map for the Vlasov-Poisson System. Peking Mathematical Journal 6(2):365-392, 2023. Abstract.
TODO
- S. Fournais, M. Lewin, J. P. Solovej. The Semi-Classical Limit of Large Fermionic Systems. Calculus of Variations and Partial Differential Equations, 57(4):1-42, 2018. Abstract.
TODO
- L. Fresta, M. Porta, B. Schlein. Effective Dynamics of Extended Fermi Gases in the High-Density Regime. Communications in Mathematical Physics, 401(2):1701-1751, 2023. Abstract.
TODO
- J. Frohlich, A. Knowles. A Microscopic Derivation of the Time-Dependent Hartree-Fock Equation with Coulomb Two-Body Interaction. Journal of Statistical Physics, 145(1):23-50, 2011. Abstract.
TODO
- J. Frohlich, A. Knowles, S. Schwarz. On the Mean-Field Limit of Bosons with Coulomb Two-Body Interaction. Communications in Mathematical Physics, 288(3):1023-1059, 2009. Abstract.
TODO
- I. Gallagher, L. Saint-Raymond, B. Texier. From Newton to Boltzmann: Hard Spheres and Short-Range Potentials. Zurich Lectures in Advanced Mathematics (European Mathematical Society), 2013. Abstract.
TODO
- I. M. Gamba, M. Pavic-Colic. On Existence and Uniqueness to Homogeneous Boltzmann Flows of Monoatomic Gas Mixtures. Archive for Rational Mechanics and Analysis, 235:723-781, 2020. Abstract.
TODO
- I. Gasser, R. Illner, P. A. Markowich, C. Schmeiser. Semiclassical $t \rightarrow \infty$ Asymptotics and Dispersive Effects for Hartree-Fock Systems. Modelization Mathematique et Analyse Numerique 32(6):699-713, 1998. Abstract.
TODO
- J. Gibbons. Collisionless Boltzmann Equations and Integrable Moment Equations. Physica D, 3(3):503-511, 1981. Abstract.
TODO
- H. Glockner. Applications of Hypocontinuous Bilinear Maps in Infinite-Dimensional Differential Calculus. Generalized Lie Theory in Mathematics, Physics, and Beyond (Springer), pages 171-186 , 2009. Abstract.
TODO
- F. Golse, C. Mouhot, T. Paul. On the Mean-Field and Classical Limits of Quantum Mechanics. Communications in Mathematical Physics, 343(1): 165-205, 2016. Abstract.
TODO
- F. Golse, C. Mouhot, V. Ricci. Empirical Measures and Vlasov Hierarchies. Kinetic & Related Models, 6(4):919-943, 2013. Abstract.
TODO
- F. Golse, T. Paul. The Schrodinger Equation in the Mean-Field and Semiclassical Regime. Archive for Rational Mechanics and Analysis, 223(1): 57-94, 2017. Abstract.
TODO
- F. Golse, T. Paul. Semiclassical Evolution with Low Regularity. Journal de Mathematiques Pures et Appliquees, 151:257-311, 2021. Abstract.
TODO
- F. Golse, T. Paul. Optimal Transport Pseudometrics for Quantum and Classical Densities. Journal of Functional Analysis, 282(9): ?? , 2022. Abstract.
TODO
- E. Grenier, T. T. Nguyen, I. Rodnianski. Landau Damping for Analytic and Gevrey Data. Mathematical Research Letters, 28(6):1679-1702, 2021. Abstract.
TODO
- E. Grenier, T. T. Nguyen, I. Rodnianski. Plasma Echoes Near Stable Penrose Data. SIAM Journal on Mathematical Analysis, 54(1):940-953, 2022. Abstract.
TODO
- M. Grillakis, M. Machedon. Pair Excitations and the Mean-Field Approximation of Interacting Bosons. Communications in Mathematical Physics, 324(2):601-636, 2013. Abstract.
TODO