Kinetic Theory: transport models for interactive particle systems
M393C (60445) CAM 393C (66500)
TTH 12:30-2
RLM 10.176

We will discuss issues on the Boltzmann or Smoluchowski type of equations for conservative and non-conservative systems and connection to non-equilibrium statistical mechanics.
Topics  include: Elementary properties of the solutions, time irreversibility, conservation  laws, H-theorem and energy inequalities.
Kinetic equations of Maxwell type, stationary and self-similar solutions for space homogeneous problems as well as connections to dynamical scaling and connections to stable laws from continuous probability theory to non-Gaussian states.

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.

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.