ICERM Semester Program on "Kinetic Theory and Computation"
(September 7, 2011 - December 9, 2011)

 

Organizing Committee

Introduction

Kinetic theory plays a central role in many areas of mathematical physics, from nanoscales to continuum mechanics. It is an indispensable tool in the mathematical description of applications in physical and social sciences, from its origin in dilute gases, to wide applications such as semi-conductors, polymers, cells, plasma, galaxies, traffic networking, and swarming. The number of particles is typically more than 1020.

On the one hand, kinetic models provide more detailed and accurate description of regimes where hydrodynamic equations are either invalid or simply not available. On the other hand, because modern computers are still inadequate in simulating the molecular or even quantum dynamics in emerging industrial needs in micro- and nanotechnology, kinetic equations provide models that can capture important features of microscopic or quantum phenomena with a manageable computational cost. Kinetic theory is at the core of multiscale modeling, which connects fundamental invisible microscopic models with macroscopic models. Many challenges remain in both the analysis and efficient computational techniques for such problems. This semester-long program in kinetic theory and computation will provide the participants with an introduction to a broad range of analytical and computational aspects of kinetic theory. The program will be centered around three broad topics, for each of which an international workshop will be held.



 Workshops:

Vlasov Models in Kinetic Theory (September 19-23, 2011)

Tutorial Week (September 12-16, 2011)

Organizing Committee
Images

vlasov

 

[Image courtesy of NASA, ESA and Hubble Heritage Team]

 

Description

Vlasov-type models deal with continua of particles where the electric charges dominate the collisions, so that the collisions are ignored. They occur in physical plasmas, including astrophysical plasmas and fusion reactors. There are many examples of astrophysical plasmas of this type, such as the solar wind. When a fusion reactor is very hot, the relevant times scales are so short that collisions can be ignored. Vlasov theory also models systems where the dominant force is gravity, such as clusters of stars or galaxies.

Problems

 

Problem 1: Stability of Equilibria. There are a lot of steady states in the Vlasov theory and their stability has been a major focus of interest. Stability requires both very long-time computations as well as theoretical analysis in order to make progress. A fundamental open problem in Vlasov theory is the question of whether there are any periodic BGK modes that are stable under arbitrary perturbations of the same period. Another open problem is to find unstable galaxy configurations in stellar dynamics. Collaborations in these problems between numerical and theoretical researchers are particularly crucial.

Problem 2: Boundary Effects. Boundary effects play an major role physical plasmas. Many effects can occur at boundaries, for instance, chemical reactions with the surface material. An important focus of our program will be to design numerical schemes to capture possible singularity formation at boundaries and the propagation of singularities in a general 3D domain.

Problem 3: Landau Damping. The concept of Landau damping has been a major source of controversy in the physics community for decades. In principle it could be checked numerically except it requires very long-time computations. The need for great accuracy is a challenge to the numerical community. Recent work has established Landau damping for the case of analytic data for the Vlasov-Poisson system. A central question is to investigate a possible Landau damping effect in the presence of a magnetic field.

Problem 4: Well-posedness. A major open problem is whether the three- dimensional Vlasov-Maxwell system is globally well-posed as a Cauchy problem. All that is known is, on the one hand, global existence but not uniqueness of weak solutions and, on the other, well-posedness and regularity of solutions assuming either some symmetry or almost neutrality. Computation of asymmetric solutions should shed light on the general problem.

Novel Applications of Kinetic Theory and Computations (October 17-21, 2011)

Tutorial Week (October 10-14, 2011)

Organizing Committee

Birds Swarm

 

[Image courtesy of Christoffer A Rasmussen]

 

 

[Author: Vedexent at en wikipedia]

 

Description

There are several fundamental applications involving kinetic theory and computations. They range from semiconductor modeling involving kinetic and quantum charged transport, radiative transfer in cosmology, conservative and dissipative phenomena in rarefied gas dynamics in mixtures, and grain and polymer flows.

 

Issues to be addressed involve the derivation and multi-scale modeling due to different scales of effective constants, spatial heterogeneities and strength of boundary conditions. Because the basic drift-diffusion, hydrodynamic and quantum models may interact through interfaces, a basic understanding of boundary conditions as well as phase transitions are critical. An example of such modeling problem appears naturally in semiconductors devices where the electron and holes density flows through a highly heterogeneous crystal lattice. It is well established that drift-diffusion models are currently inadequate for the simulations of submicron devices where effective fields become very strong. As a consequence, kinetic transport modeling and even quantum modeling corrections are necessary to accurately model the current flow through devices. Mathematically it is critical to address the analytical and approximating properties of hydrodynamic and kinetic models of Euler and Boltzmann type coupled to Poisson's equation, as well as the Schrödinger and quantum Boltzmann equations that become relevant in different scaling regimes.

Recently, there have been new applications to biological systems, chain supply dynamics and quantitative finance, where statistical methods for multi-agent systems have given raised to of extension of Boltzmann equation to models for particle swarms, networks or the dynamics of information. This is a mathematical area that is not as well developed as its semiconductor counterpart. Our program will pay special attention to these new developments in an attempt to set basic benchmarks of terms of analytical as well as numerical modeling.

Problems

 

Problem 1: Boundary Effects. A major open area is to solve a hydrodynamic model in two or three dimensions with boundary conditions of contact type. So far this has been accomplished only in one dimension and for some reduced stationary models in two dimensions. These issues have raised important open questions about how to design numerical schemes for such hydrodynamic models.

 

Problem 2: Computational Issues in Quantum Modeling. For quantum-based computations of resonant tunneling diodes in semiconductors, high-dimensional computations are very expensive because of the high oscillations. However, in the most effective designs of devices the highest oscillations occur along preferred directions which naturally select appropriate homogenized model reductions. This is an example where the mathematics can efficiently reduce the solution structure to make the computations feasible.

Problem 3: Quantum Boltzmann Theory. Despite its importance, there has been very little work on quantum Boltzmann equations because of their severe nonlinearity. Our program will attempt to numerically compute and analytically construct global-in-time solutions near a Bose-Einstein distribution and to investigate the phenomenon of Bose-Einstein condensation.

Problem 4: Statistical Multi-agent Modeling. Another area of focus will be the modeling of swarms, information percolation, Pareto tail distributions and chain supply dynamics. These models exhibit a new sort of difficulty; in fact, their stationary states are not Maxwellian. New approaches to reduced dimensionality via hydrodynamic limits or moment methods are being considered. In addition, some social-biological interactions are modeled by systems of kinetic equations which remain broadly unaddressed.

Boltzmann Models in Kinetic Theory (November 7-11, 2011)

Tutorial Week (October 31 - November 4, 2011)

Organizing Committee
Description

The celebrated Boltzmann equation is the foundation of the kinetic theory for dilute collections of particles, which undergo elastic binary collisions. The Boltzmann theory is at the center of a series of multi-scaled physical models that connect microscopic multiparticle models to macroscopic fluid models such as the Navier-Stokes equations:

 

Particles → Boltzmann → Fluids

The first arrow refers the Boltzmann-Grad limit, while the second arrow refers to various hydrodynamic limits which lead to the fundamental equations of fluids. The Boltzmann theory therefore provides a practical tool and machinery for deriving macroscopic models in broad physical applications. Due to its importance, there has been an explosion of mathematical studies, both theoretical and numerical, for the Boltzmann equation. A major open problem that remains is to determine whether or not smooth initial data would lead to a unique global-in-time solution of the Boltzmann equation. Nevertheless, there have been exciting new developments in recent years. The focus of the program is to bring computational and theoretical people together to investigate problems of fundamental importance.


Problems

 

Problem 1: Boundary Effects. Boundary effects play an important role in the dynamics of particles confined in a bounded region. Yet its mathematical study is at an early stage. This is due to the fact that solutions to the Boltzmann equation in general will develop singularities. The focus is to investigate the formation and propagation of singularities, both from numerical and theoretical points of view.

Problem 2: Hydrodynamic Limits. There have been lots of studies of hydrodynamic limits of the Boltzmann equation. The focus in our program will be on error estimates and higher-order expansions of hydrodynamic limits both from the theoretical point of view and from the point of view of numerical simulation. Boundary and initial layer analysis for hydrodynamic limits, which has been barely studied, is an important area that is ready for investigation.

Problem 3: Boltzmann-Grad Limit. There has been little mathematical work in this direction since the work of Lanford. The focus will be on boundary effects in the Boltzmann-Grad limit, and on the application of Lanford?s proof to establish the Boltzmann-Grad limit for other particle systems of physical importance. Numerical simulations at the particle level will play an key role.

Kinetic Theory and Computation - Seminars


This page will show upcoming seminars that will be scheduled by organizers, speakers, and participants of the Fall 2011 semester program Kinetic Theory and Computation. Walk-ins are welcomed and encouraged for these seminars. Please check back regularly for updates.

Monday November 28th 2011
Time Description Speaker Location Abstracts
3:00 - 3:30 Coffee and Tea Break 11th Floor Collaborative Space
Tuesday November 29th 2011
Time Description Speaker Location Abstracts
12:00 - 1:00 Reserved Lecture Time TBA 11th Floor Lecture Hall
3:00 - 3:30 Coffee and Tea Break 11th Floor Collaborative Space
4:30 - 5:30 Professional development roundtable discussion: Open Question & Answer Session 11th Floor Lecture Hall
Wednesday November 30th 2011
Time Description Speaker Location Abstracts
2:00 - 3:00 ICERM Young Researcher Seminar 11th Floor Lecture Hall
3:00 - 3:30 Coffee and Tea Break 11th Floor Collaborative Space
Thursday December 1st 2011
Time Description Speaker Location Abstracts
2:00 - 3:00 ICERM Young Researcher Seminar 11th Floor Lecture Hall
3:00 - 3:30 Coffee and Tea Break 11th Floor Collaborative Sapce
Friday December 2nd 2011
Time Description Speaker Location Abstracts
1:00 - 2:00 Interfacial Instabilities in Hele-Shaw Flows Andong He, Pennsylvania State University 11th Floor Lecture Hall
PDF
2:00 - 2:30 Coffee and Tea Break 11th Floor Collaborative Space
week12_1.php
Monday December 5th 2011
Time Description Speaker Location Abstracts
3:00 - 3:30 Coffee & Tea Break   11th Floor Collaborative Space
Tuesday December 6th 2011
Time Description Speaker Location Abstracts
1:00 - 2:00 Reserved Lecture Time TBA 11th Floor Lecture Hall
3:00 - 3:30 Coffee & Tea Break   11th Floor Collaborative Space
4:00 - 5:00 Mathematics, Computation and Politics Philip Davis, Professor Emeritus of Applied Mathematics, Brown University 11th Floor Lecture Hall
5:00 - 6:00 Social Hour   11th Floor Collaborative Space
Wednesday December 7th 2011
Time Description Speaker Location Abstracts
11:00 - 12:00 Relating Global Maxwellians to Conserved Quantities of Classical Kinetic Equations over the Whole Space C. David Levermore, University of Maryland 11th Floor Lecture Hall
2:00 - 3:00 ICERM Young Researcher Seminar   11th Floor Lecture Hall
3:00 - 3:30 Coffee & Tea Break   11th Floor Collaborative Space
Thursday December 8th 2011
Time Description Speaker Location Abstracts
2:00 - 3:00 ICERM Young Researcher Seminar   11th Floor Lecture Hall
3:00 - 3:30 Coffee & Tea Break   11th Floor Collaborative Space
Friday December 9th 2011
Time Description Speaker Location Abstracts
10:00 - 11:00 Xtreme Geometry Seminar Ian Agol, UC Berkeley 11th Floor Lecture Hall
3:00 - 3:30 Coffee & Tea Break   11th Floor Collaborative Space