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From Newton and Schrödinger many-body dynamics to the Boltzmann equation

Applicant Saffirio Chiara
Number 181153
Funding scheme Eccellenza
Research institution Departement Mathematik und Informatik Universität Basel
Institution of higher education University of Basel - BS
Main discipline Mathematics
Start/End 01.08.2019 - 31.07.2024
Approved amount 1'585'377.00
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Keywords (8)

Scaling limits; Kinetic Theory; Effective evolution equations; Vlasov-Poisson equation; Boltzmann equation; Hartree-Fock equation; Landau equation ; Quantum dynamics

Lay Summary (Italian)

Lead
Molti fenomeni fisici sono descritti da sistemi composti da molte particelle interagenti. Questi modelli sono molto complessi e tipicamente molto difficili da studiare perché il numero di particelle è molto grande.L'idea di base delle teorie cinetiche è che colui che osserva il fenomeno fisico non sia interessato al comportamento della singola componente, quanto al comportamento collettivo del sistema.Tale comportamento collettivo emerge su scale di spazio e tempo più grandi rispetto a quelle che caratterizzano la dinamica microscopica. È proprio su questa scala macroscopica che il sistema può essere descritto da un'equazione alle derivate parziali, la cui analisi qualitativa è molto più semplice. Il progetto in questione è incentrato sul dimostrare che, nel caso di gas classici e quantistici, la descrizione macroscopica è una buona approssimazione della dinamica microscopica.
Lay summary
L'equazione di Boltzmann è stata introdotta da Boltzmann e Maxwell alla fine del XIX secolo per cercare di dare una descrizione del comportamento di un gas rarefatto su scala macroscopica a partire dalle leggi microscopiche della meccanica classica, fornendo pertanto la prima giustificazione del secondo principio della termodinamica.
La correttezza di tale descrizione è ad oggi largamente accettata dalla comunità scientifica, come manifestato dall'utilizzo dell'equazione di Boltzmann in molte applicazioni. Nonostante ciò, una comprensione matematica completa della validità dell'equazione di Boltzmann come approssimazione di un sistema di particelle interagenti in un opportuno limite di scala è ad oggi assente. 
Questo progetto ha lo scopo di affrontare alcuni tra i problemi aperti nel campo, tra cui la modellizzazione di interazioni a lunga portata, di domini con bordo e di effetti quantistici. Particolare attenzione è dedicata ai metodi quantitativi, in grado di fornire stime esplicite sull'errore commesso quando si approssima la dinamica di particelle interagenti su scala macroscopica con l'equazione di Boltzmann a livello macroscopico.
Direct link to Lay Summary Last update: 12.06.2019

Lay Summary (English)

Lead
Many phenomena of physical interest are described by systems of many interacting particles. As the number of particles is typically very big, this models are usually very difficult to study.The underlying idea of kinetic theory is that what matters is the collective behaviour of the system and not the motion of each single component. Such a collective behaviour arises on space and time scales which are much larger than the ones characterising the microscopic dynamics. At macroscopic scales, the systems can be described by partial differential equations. The core of this project addresses the validity of such a reduction from microscopic to macroscopic scale in the framework of classical and quantum gases.
Lay summary
The Boltzmann equation was introduced at the end of the XIX century by Boltzmann and Maxwell in the attempt of modelling the evolution in time of a rarefied gas at a macroscopic scale starting from the fundamental laws of classical mechanics, thus providing for the first time a justification to the second principle of thermodynamics. 
The validity of such a description is nowadays endorsed by the scientific community, as manifested by its large use in applications. Despite that, a fully rigorous mathematical comprehension is still missing. This project aims to deal with some of the major open problems in the field, such as modelling long range interactions, boundaries and quantum effects. Special relevance is given to quantitative methods which allow for explicit bounds of the errors produced when considering Boltzmann-like equations at the macroscopic scale instead of interacting particles at a microscopic scale. 
Direct link to Lay Summary Last update: 12.06.2019

Responsible applicant and co-applicants

Employees

Associated projects

Number Title Start Funding scheme
161287 Derivation of the Boltzmann equation from classical and quantum dynamics 01.08.2016 Ambizione

Abstract

This proposal aims at investigating the rigorous derivation of effective macroscopic equations starting from the microscopic laws of classical and quantum dynamics. Albeit considerable progress has been done in last years, several problems still claim for solutions. Among them, questions concerning the validity of the Boltzmann Eq.n, as well as its quantum version, are certainly of primary interest both from a physical and mathematical viewpoint. Major open questions deal with the long time derivation of the Boltzmann Eq.n; its validity for long range interactions; the role of boundary conditions; the derivation of the quantum Boltzmann Eq.n from a system of quantum particles. In light of recent mathematical developments, we propose a research agenda to contribute in the aforementioned items:a) Derivation of the Boltzmann Eq.n for long range potentials. Our goal is to deeply understand the onset of instability due to the long tail effects, first of all looking at the linear Boltzmann Eq.n. We believe techniques developed in a work in preparation joint with L. Desvillettes and S. Simonella will be successful ingredients and draw the lines for a long term investigation in the nonlinear case.b) Derivation of the Boltzmann Eq.n with inelastic collisions. When collisions occurring at microscopic scale are inelastic, a loss of energy in the scattering process appears. It can be seen as a boundary effect. We plan to rigorously derive the Boltzmann Eq.n for granular gases, by adapting techniques developed by Gallagher, Saint-Raymond, Texier and Pulvirenti, Saffirio, Simonella. The result would be relevant for applications and provide the first step to understand the role of boundaries in the validity of the Boltzmann Eq.n.c) Derivation of the Landau Eq.n. When the long-range interaction is the Coulomb potential, the Boltzmann collision operator is no longer well defined and has to be replaced by the Landau operator. We are interested in the rigorous derivation of the Landau Eq.n from a Hamiltonian system of interacting particles in the weak-coupling limit. We plan to combine the result obtained by Bobylev, Pulvirenti, Saffirio with the mathematical methods recently developed by Velázquez and Winter, thus providing an equivalent of Lanford’s result in the weak-coupling context.d) Evolution of fermionic systems with Coulomb interaction. In light of recent developments obtained by Benedikter, Porta, Saffirio, Schelin, we plan to provide an explicit rate for the semiclassical limit of the Hartree-Fock dynamics towards the Vlasov-Poisson Eq.n. We expect techniques developed here to be useful for point e).e) Derivation of the quantum Boltzmann Eq.n. We plan to perform a weak-coupling scaling in the many-body Schr odinger Eq.n and to obtain the quantum Boltzmann Eq.n in the limit. We believe that the choice of the initial data will play a crucial role. In particular, we will construct two-particle reduced density matrices that take into account correlations between two particles in the spirit of Benedikter, De Oliveira, Schlein.
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