Fall 2010            Kinetic Theory


M 393C and CAM 393C   
Unique# 55790 and 65075

Instructor: Prof. Irene M. Gamba
Office: RLM 10.166, Phone: 471-7150
E-Mail: gamba@math.utexas.edu
Office hours: by appointment

Meeting Hours: T-Th 12:30-1:45pm  RLM 10.176

Extra discussion hours by announcement: Wednesdays 6:00 - 7:15pm  RLM 9.176

class webpage : M 393C Kinetic theory  
Unique# M393C (55790) CAM 393C (65075)

This topics course covers issues on the Boltzmann and Smoluchowski type of equations for conservative and non-conservative systems and connection to non-equilibrium statistical mechanics.
More specifically, we will have first and introduction to elementary properties of the solutions, time irreversibility, conservation laws, H-theorem and energy inequalities.
Conservative and non-conservative kinetic problems for interacting kernels from variable potentials. Convolution inequalities for collision Operators. Space Homogeneous problems. Existence and uniqueness properties. Povzner type lemmas. Carleman integral. Moment inequalities. Comparisons for point-wise bounds to solutions.
The space inhomogeneous problem. The Kaniel/Shimbrot iteration method. Averaging lemmas and renormalized solutions.
Kinetic equations of Maxwell type, stationary and self-similar solutions for space homogeneous problems. Connections to dynamical scaling and connections to stable laws from continuous probability theory to non-Gaussian states. Applications to information propagation problems.
Non-conservative kinetic problems for interacting kernels from variable potentials. Existence and uniqueness  properties.  Comparisons  for point-wise bounds to solutions of Boltzmann equations.
 The  space inhomogeneous problem. Averaging lemmas and renormalized solutions.

Derivation of kinetic models for charge transport.
The Boltzmann-Poisson system. Modeling inhomogeneous small devices. Boltzmann-Poisson vs. Fokker-Plank-Poisson systems.
From kinetic to fluid dynamical models. Small mean free path, Hilbert and Chapman expansions. Moment Methods. Derivation of fluid level equations. Low field approximations: Drift-Diffusion models. High field approximations. Hydrodynamic models. The initial--boundary value problem.
Derivation of kinetic models for charge transport. The Boltzmann-Poisson system. Modeling inhomogeneous small devices. Boltzmann-Poisson vs. Fokker-Plank-Poisson systems.
From kinetic to fluid dynamical models. Small mean free path, Hilbert and Chapman expansions. Moment Methods. Derivation of fluid level equations. Low field approximations: Drift-Diffusion models. High field approximations. Hydrodynamic models.
Some topics on numerical simulations of particle kinetic systems: Direct Simulations of Monte-Carlo (DSMC) vs. deterministic solvers.

Prerequisites: Some knowledge of methods of applied mathematics and differential equations.

The following is a "Suggested Bibliography'':
Cercignani C., "The Boltzmann Equation and its Applications", Springer, New York, 1988.
Cercignani C., Illner, R. and Pulverenti, M., "The Mathematical Theory of Diluted Gases", Springer, New York, 1994.
Villani, C., A review of Mathematical topics in collisional kinetic theory, Handbook of fluid mechanics, Handbook of Fluid Mechanics, (2003).

Class notes and several recent papers to be distributed in class.

The University of Texas at Austin provides upon request appropriate academic accommodations for qualified students with disabilities. For more information, contact the Office of the Dean of Students at 471-6259, 471-4641 TTY.