Project

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Derivation of the Boltzmann equation from classical and quantum dynamics

Applicant Saffirio Chiara
Number 161287
Funding scheme Ambizione
Research institution Institut für Mathematik Universität Zürich
Institution of higher education University of Zurich - ZH
Main discipline Mathematics
Start/End 01.08.2016 - 31.07.2019
Approved amount 379'388.00
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Keywords (6)

Quantum dynamics; Effective evolution equations; Scaling limits; Landau equation; Boltzmann equation; Hartree-Fock equation

Lay Summary (Italian)

Lead
Tipici sistemi di interesse in fisica, biologia e scienze applicate possono essere descritti da modelli a un gran numero di componenti. A causa dell'elevato numero di particelle, il loro comportamento microscopico è molto complicato. D'altra parte, l'osservatore è interessato al comportamento collettivo del sistema, emergente su una scala spazio-temporale più grande di quella caratterizzante la dinamica microscopica. Su questa scala macroscopica, il sistema appare più semplice e può essere descritto da equazioni alle derivate parziali.
Lay summary

L'obiettivo principale di questo progetto è studiare matematicamente l'emergere di una dinamica macroscopica effettiva da una microscopica.
In particolare ci proponiamo di: (1) dare una derivazione rigorosa dell'equazione di Boltzmann (utilizzata nell'ingegneria per descrivere la dinamica di gas rarefatti) nel caso in cui le particelle interagiscano su scala microscopica tramite un potenziale a lunga portata; (2) studiare il caso in cui l'interazione sia data da un potenziale di Coulomb, la cui evoluzione macroscopica è ipotizzata essere descritta dall'equazione di Landau (utilizzata nella fisica dei plasmi); (3) ottenere una derivazione rigorosa dell'equazione di Hartree-Fock nel caso di fermioni interagenti tramite un potenziale di Coulomb; (4) derivare l'equazione di Boltzmann quantistica per particelle con statistica di Fermi.  

La rilevanza di questo studio consiste nel giustificare rigorosamente l'emergere di teorie macroscopiche ottenute dai fisici su base empirica, derivandole dalla scala microscopica. Tale derivazione è cruciale per determinare l'affidabilità e identificare la scala di validità delle teorie effettive.   

Direct link to Lay Summary Last update: 18.05.2016

Lay Summary (English)

Lead
Typical systems if interest in physics, biology and applied science can be described by models with a large number of components. Because of the huge number of particles, the microscopic behavior is very complicated. On the other hand, observers are interested in the collective behaviour of the system, arising on space and time scales which are much larger than the ones characterizing the microscopic dynamics. On such macroscopic scales, the systems appear much simpler, and can be described by partial differential equations.
Lay summary

The goal of this project is to study the effective macroscopic dynamics emerging from the microscopic one.

In particular, we aim to: (1) give a rigorous derivation of the Boltzmann equation (used to describe the dynamics of rarefied gases) for particles interacting through a long range potential at a microscopic scale; (2) study the case in which the interaction is given by a Coulomb potential (here the Landau equation, used in plasma physics, is expected to hold); (3) obtain a rigorous derivation of the Hartree-Fock equation in the case of fermions interacting through a Coulomb potential; (4) do the groundwork to derive the quantum Boltzmann equation for fermions.

The relevance of this study is to justify the effective theories obtained empirically by physicists, by deriving them from a microscopic scale. The rigorous derivation is crucial to determine the reliability and to identify the validity scale of the effective theories.

Direct link to Lay Summary Last update: 18.05.2016

Responsible applicant and co-applicants

Employees

Publications

Publication
Semiclassical limit to the Vlasov equation with inverse power law potentials
SaffirioChiara (2019), Semiclassical limit to the Vlasov equation with inverse power law potentials, in Communications in Mathematical Physics, 1.
Lagrangian solutions to the Vlasov-Poisson system with a point charge
Crippa Gianluca, Ligabue Silvia, Saffirio Chiara (2018), Lagrangian solutions to the Vlasov-Poisson system with a point charge, in Kinetic & Related Models, 11(6), 1277-1299.
Mean-field evolution of fermions with singular interaction
Saffirio Chiara (2017), Mean-field evolution of fermions with singular interaction, in Macroscopic Limits of Quantum Systems. Munich, Germany, March 20 - April 1, 2017, Springer, Switzerland.

Collaboration

Group / person Country
Types of collaboration
Scuola Normale Superiore di Pisa Italy (Europe)
- in-depth/constructive exchanges on approaches, methods or results
- Exchange of personnel
Rutgers University United States of America (North America)
- in-depth/constructive exchanges on approaches, methods or results
Institute of Mathematics, University of Zurich Switzerland (Europe)
- in-depth/constructive exchanges on approaches, methods or results
- Publication
ENS Paris France (Europe)
- in-depth/constructive exchanges on approaches, methods or results
Lyon ENS France (Europe)
- in-depth/constructive exchanges on approaches, methods or results
- Publication
Dundee University Great Britain and Northern Ireland (Europe)
- in-depth/constructive exchanges on approaches, methods or results
- Publication
Università di Torino Italy (Europe)
- in-depth/constructive exchanges on approaches, methods or results
- Publication
Univeristy of Bonn Germany (Europe)
- in-depth/constructive exchanges on approaches, methods or results
- Publication
Paris 7 France (Europe)
- in-depth/constructive exchanges on approaches, methods or results
- Publication
University of Copenaghen Denmark (Europe)
- in-depth/constructive exchanges on approaches, methods or results
- Publication

Scientific events

Active participation

Title Type of contribution Title of article or contribution Date Place Persons involved
Workshop "Analytical and Computational Problems for Mixtures and Plasma Dynamics" Talk given at a conference From the many-body quantum dynamics to the Vlasov equation 17.06.2019 HCM Bonn, Germany Saffirio Chiara;
Seminar in Mathematical Physics Individual talk From the many-body quantum dynamics to the Vlasov equation 28.05.2019 Heidelberg, Germany Saffirio Chiara;
Seminar in Analysis Individual talk From the many-body quantum dynamics to the Vlasov equation 04.02.2019 Dundee, Great Britain and Northern Ireland Saffirio Chiara;
Nonlinear phenomena in Stockholm Talk given at a conference From the many-body quantum dynamics to the Vlasov equation 19.11.2018 KTH Stockholm, Sweden Saffirio Chiara;
Young Women in Mathematical Physics Talk given at a conference From the many-body quantum dynamics to the Vlasov equation 24.09.2018 Bonn, Germany Saffirio Chiara;
La mécanique quantique à N corps Talk given at a conference From many-body quantum dynamics to the Vlasov equation 10.09.2018 Montreal, Canada Saffirio Chiara;
Seminar in Mathematical Physics Individual talk Derivation of the Vlasov-Boltzmann equation 13.12.2017 Politecnico di Milano, Italy Saffirio Chiara;
Classical and Quantum Mechanical Models of Many-Particle Systems Talk given at a conference From the Hartree-Fock dynamics to the Vlasov equation with singular interactions 03.12.2017 MFO Oberwolfach, Germany Saffirio Chiara;
Particle systems and PDEs VI Talk given at a conference From the Hartree-Fock dynamics to the Vlasov Equation 27.11.2017 Nice, France Saffirio Chiara;
Collisionless Boltzmann (Vlasov) equation and modeling of self-gravitating systems and plasmas Talk given at a conference Lagrangian solutions to the Vlasov-Poisson system with a point charge 30.10.2017 CIRM Luminy, France Saffirio Chiara;
Workshop on the Analysis of Effective One-Particle Equations and their Derivation Talk given at a conference Derivation of the linear Bolzmann-Vlasov equation from particle system 31.07.2017 LMU Munich, Germany Saffirio Chiara;
7TH INTERNATIONAL WORKSHOP ON KINETIC THEORY & APPLICATIONS Talk given at a conference Derivation of the linear Bolzmann-Vlasov equation from particle system 19.06.2017 University of Karlstad, Sweden Saffirio Chiara;
Macroscopic Limits of Quantum Systems Talk given at a conference Mean Field Evolution of Fermions with Coulomb Interaction 30.03.2017 TU Munich, Germany Saffirio Chiara;
Seminar in Statistical Mechanics Individual talk Mean Field Evolution of Fermions with Coulomb Interaction 09.03.2017 University of Warwick , Great Britain and Northern Ireland Saffirio Chiara;
Seminar in Mathematical Physics Individual talk Mean Field Evolution of Fermions with Coulomb Interaction 02.02.2017 SISSA Trieste, Italy Saffirio Chiara;
Mean-field dynamics of many particle systems Talk given at a conference Mean Field Evolution of Fermions with Coulomb Interaction 14.12.2016 Pauli Institut, Vienna , Austria Saffirio Chiara;
Mathematical Physics Talk given at a conference Mean Field Evolution of Fermions with Coulomb Interaction 28.11.2016 ETH Zürich, Switzerland Saffirio Chiara;
Kinetic Theory and its neighbours Talk given at a conference Mean field evolution of fermions with Coulomb interaction 19.10.2016 L'Aquila, Italy Saffirio Chiara;
Weakly seminar Individual talk Derivation of effective equations from many body dynamics 11.10.2016 Politecnico Torino, Italy Saffirio Chiara;


Self-organised

Title Date Place
Workshop "Effective equations: frontiers in classical and quantum systems" 24.06.2019 HCM Bonn, Germany
Oberwolfach mini-workshop "Lorentz gas dynamics: particle systems and scaling limits" 04.03.2019 MFO Oberwolfach, Germany

Communication with the public

Communication Title Media Place Year
Talks/events/exhibitions What is... the arrow of time? German-speaking Switzerland 2018

Awards

Title Year
SNSF Eccellenza Professorial Fellow 2018

Associated projects

Number Title Start Funding scheme
181153 From Newton and Schrödinger many-body dynamics to the Boltzmann equation 01.08.2019 Eccellenza

Abstract

The subject of this project is the rigorous derivation of effective macroscopic equations starting from the fundamental laws of classical and quantum physics. This is a central question in non-equilibrium statistical mechanics. In particular, we are interested in the Boltzmann equation arising from classical Hamiltonian mechanics for long range potentials and in the emergence of the Hartree-Fock equation and of the quantum Boltzmann equation from the many body Schroedinger evolution.
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