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.