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From Particle Systems to Partial Differential Equations

PSPDE X, Braga, Portugal, June 2022

Editors: Eric Carlen, Patrícia Gonçalves, Ana Jacinta Soares

Publisher: Springer International Publishing

Book Series : Springer Proceedings in Mathematics & Statistics

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

Table of Contents

About this book

This book presents the proceedings of the international conference Particle Systems and Partial Differential Equations X, which was held at the University of Minho, Braga, Portugal, from 2022. It includes papers on mathematical problems motivated by various applications in physics, engineering, economics, chemistry, and biology.

Frontmatter

Necessary and Sufficient Conditions for Strong Stability of Explicit Runge–Kutta Methods

Abstract

Strong stability is a property of time integration schemes for ODEs that preserve temporal monotonicity of solutions in arbitrary (inner product) norms. It is proved that explicit Runge–Kutta schemes of order \(p\in 4{\mathbb {N}}\) with \(s=p\) stages for linear autonomous ODE systems are not strongly stable, closing an open stability question from [Z. Sun and C.-W. Shu, SIAM J. Numer. Anal. 57 (2019), 1158–1182]. Furthermore, for explicit Runge–Kutta methods of order \(p\in {\mathbb {N}}\) and \(s>p\) stages, we prove several sufficient as well as necessary conditions for a less restrictive notion of strong stability. These conditions involve both the stability function and the hypocoercivity index of the ODE system matrix. This index is a structural property combining the Hermitian and skew-Hermitian part of the system matrix.

Franz Achleitner, Anton Arnold, Ansgar Jüngel

Radial Laplacian on Rotation Groups

Abstract

The Laplacian on the rotation group is invariant by conjugation. Hence, it maps class functions to class functions. A maximal torus consists of block diagonal matrices whose blocks are planar rotations. Class functions are determined by their values on this maximal torus. Hence, the Laplacian induces a second order operator on the maximal torus called the radial Laplacian. In this paper, we derive the expression of the radial Laplacian. Then, we use it to find the eigenvalues of the Laplacian, using that characters are class functions whose expressions are given by the Weyl character formula. Although this material is familiar to Lie-group experts, we gather it here in a synthetic and accessible way which may be useful to non experts who need to work with these concepts.

Pierre Degond

Some Remarks About the Link Between the Fisher Information and Landau or Landau-Fermi-Dirac Entropy Dissipation

Abstract

We present in this work variants of existing estimates for the Landau or Landau-Fermi-Dirac entropy dissipation, in terms of Fisher information, in the hard potential case. The specificity of those variants is that the entropy is never used in the estimates (in order to control possible concentrations on a zero measure set). The proofs are significantly simplified with respect to previous papers on the subject.

Laurent Desvillettes

The Entropic Journey of Kac’s Model

Abstract

The goal of this paper is to review the advances that were made during the last few decades in the study of the entropy, and in particular the entropy method, for Kac’s many particle system.

Amit Einav

Entropy Production Bounds for the Kac Model are Uniform in the Number of Particles

Abstract

In this paper, we prove fully quantitative entropy power law bounds for Kac’s collision model, uniform in the number of particles, by making use of the HWI inequality and Wasserstein distance power laws obtained by Rousset [34] via a Markovian coupling method. In turn, these bounds allow us to derive entropy power laws for the spatially homogeneous Boltzmann equation, akin to the results by Villani [40], by relatively elementary methods and requiring only moment and mild integrability conditions on the initial distribution.

Luís Simão Ferreira

Quantum Wasserstein and Observability for Quantum Dynamics

Abstract

In recent years, there have been several extensions of various tools and methods of optimal transport to the quantum setting. In particular, a pseudometric analogous to the Wasserstein distance of exponent 2 has been defined in [F. Golse, T. Paul, Arch. Rational Mech. Anal. 223 (2017) 57–94] for the purpose of comparing probability densities defined on \(\textbf{R}^d\times \textbf{R}^d\) with density operators on \(L^2(\textbf{R}^d)\). This pseudometric is particularly convenient if one seeks a quantitative error estimate for the classical limit of quantum dynamics. In this talk, we explain how to use this tool in order to study the observability problem for the Schrödinger or the von Neumann equations. Our analysis of this problem uses the quantum analogue of the Wasserstein distance, together with a geometric condition on the classical trajectories corresponding to the quantum dynamics under a condition analogous to the Bardos-Lebeau-Rauch geometric condition for the exact controllability of the wave equation [C. Bardos, G. Lebeau, J. Rauch, SIAM J. Control Opti. 30 (1992) 1024–1065]. The material presented in this paper is a review of a series of joint works with T. Paul, especially [Math. Models Methods Appl. Sciences, 32 (2022) 941–963].

François Golse

Results About the Free Kawasaki Dynamics of Continuous Particle Systems in Infinite Volume: Long-time Asymptotics and Hydrodynamic Limit

Abstract

An infinite particle system of independent jumping particles in infinite volume is considered. Their construction is recalled, further properties are derived, the relation with hierarchical equations, Poissonian analysis, and second quantization are discussed. The hydrodynamic limit for a general initial distribution satisfying a mixing condition is derived. The long-time asymptotics is computed under an extra assumption. The relation with constructions based on infinite volume limits is discussed.

Yuri G. Kondratiev, Tobias Kuna, Maria João Oliveira, José Luís da Silva, Ludwig Streit

Approach to Equilibrium for the Kac Model

Abstract

This is a review on the Kac Master Equation. Various issues will be presented such as the resolution of Kac’s conjecture about the gap for the three dimensional hard sphere gas, entropic propagation of chaos and other topics such as systems coupled to reservoirs and thermostats. The discussion is informal with few proofs and those who are presented are only sketched.

Federico Bonetto, Eric A. Carlen, Lukas Hauger, Michael Loss

Hydrodynamic Limit from the Boltzmann Equation in a Slightly Compressible Regime

Abstract

We discuss the hydrodynamic limit for the Boltzmann equation under the diffusive scaling with initial and/or boundary non homogeneous conditions for density and temperature with gradients of order 1.

Rossana Marra

Intemperate Lévy Processes and White Noises

Abstract

The distributional support of Lévy processes is important for the construction of sparse statistical models, integration in infinite dimensions and the existence of generalized solutions of stochastic partial differential equations driven by Lévy white noise. The white noise associated to a class of Lévy processes without support in \(\mathscr {S}^{\prime }\), the space of tempered distributions, is constructed as a generalized random process. This construction also provides measures on the distribution spaces that support the paths of the Lévy processes.

Rui Vilela Mendes

Connecting the Deep Quench Obstacle Problem with Surface Diffusion via Their Steady States

Abstract

In modeling phase transitions, it is useful to be able to connect diffuse interface descriptions of the dynamics with corresponding limiting sharp interface motions. In the case of the deep quench obstacle problem (DQOP) and surface diffusion (SD), while a formal connection was demonstrated many years ago, rigorous proof of the connection has yet to be established. In the present note, we show how information regarding the steady states for both these motions can provide insight into the dynamic connection, and we outline tools that should enable further progress. For simplicity, we take both motions to be defined on a planar disk.

Eric A. Carlen, Amy Novick-Cohen, Lydia Peres Hari

Mean Field Limit for the Kac Model and Grand Canonical Formalism

Abstract

We consider the classical Kac’s model for the approximation of the Boltzmann equation, and study the correlation error measuring the defect of propagation of chaos in the mean field limit. This contribution is inspired by a recent paper of the same authors [23] where a large class of models, including quantum systems, are considered. Here we outline the main ideas in the context of grand canonical measures, for which both the evolution equations and the proof simplify.

Thierry Paul, Mario Pulvirenti, Sergio Simonella

Magnetic Reconnection: Some Basic Ideas and a Review on a Collisionless Model

Abstract

Basic ideas and definitions are surveyed concerning the theory of magnetic reconnection in plasmas, a phenomenon occurring under rather various and different conditions, both in astrophysical systems and in laboratory experiments. Approaches to the description of this phenomenon are reviewed, discussing in the last part a linearized analysis for collisionless regimes.

Valeria Ricci, Bruno Coppi, Renato Spigler

Uniqueness Criteria for the Vlasov–Poisson System and Applications to Semiclassical Analysis

Abstract

We review some uniqueness criteria for the Vlasov–Poisson system, emerging as corollaries of stability estimates in strong or weak topologies, and show how they serve as a guideline to solve problems arising in semiclassical analysis. More specifically, we look at the Schatten norms and at the quantum analogue of the Wasserstein distance and prove the semiclassical limit from the Hartree equation with Coulomb interaction to the Vlasov–Poisson system in these topologies by mimicking weak-strong uniqueness principles for the Vlasov–Poisson equation in the respective classical topologies. Different topologies allow to treat different classes of quantum states.

Laurent Lafleche, Chiara Saffirio

The Extended Phase Space Method in Kinetic Theory

Abstract

This article describes the extended phase space technique in the study of kinetic equations and provides a review of some relevant contributions appeared in the literature.

Francesco Salvarani

Mesoscale Mode Coupling Theory for the Weakly Asymmetric Simple Exclusion Process

Abstract

The asymmetric simple exclusion process and its analysis by mode coupling theory (MCT) is reviewed. To treat the weakly asymmetric case at large space scale \(x\varepsilon ^{-1}\), large time scale \(t \varepsilon ^{-\chi }\) and weak hopping bias \(b \varepsilon ^{\kappa }\) in the limit \(\varepsilon \rightarrow 0\) we develop a mesoscale MCT that allows for studying the crossover at \(\kappa =1/2\) and \(\chi =2\) from Kardar-Parisi-Zhang (KPZ) to Edwards-Wilkinson (EW) universality. The dynamical structure function is shown to satisfy for all \(\kappa \) an integral equation that is independent of the microscopic model parameters and has a solution that yields a scale-invariant function with the KPZ dynamical exponent \(z=3/2\) at scale \(\chi =3/2+\kappa \) for \(0\le \kappa <1/2\) and for \(\chi =2\) the exact Gaussian EW solution with \(z=2\) for \(\kappa >1/2\). At the crossover point it is a function of both scaling variables which converges at macroscopic scale to the conventional MCT approximation of KPZ universality for \(\kappa <1/2\). This fluctuation pattern confirms long-standing conjectures for \(\kappa \le 1/2\) and is in agreement with mathematically rigorous results for \(\kappa >1/2\) despite the numerous uncontrolled approximations on which MCT is based.

Gunter M. Schütz

The Two Dimensional Lorentz Gas in the Kinetic Limit: Theoretical and Numerical Results

Abstract

The Lorentz gas is a model for the motion of electrons in a metal, where the motion is dominated by collisions of the electrons with immobile atomic nuclei, the scatterers. The motion depends on the distribution of scatterers, and we focus here on modifications of periodic scatterer distributions in two dimensions, and in the low density, or Boltzmann Grad, limit. Some theoretical results are complemented with numerical illustrations including modified periodic scatterer distributions, and scatterer distributions given by a quasi-crystal and by the zero set of a Gaussian analytic function.

Bernt Wennberg

Title: From Particle Systems to Partial Differential Equations
Editors: Eric Carlen
Patrícia Gonçalves
Ana Jacinta Soares
Publisher: Springer International Publishing
Electronic ISBN: 978-3-031-65195-3
Print ISBN: 978-3-031-65194-6
DOI: https://doi.org/10.1007/978-3-031-65195-3