Kinetic Theory: transport models for interactive particle systems
M393C (60445) CAM 393C (66500)
We will discuss issues on the Boltzmann or Smoluchowski type of equations for
conservative and non-conservative systems and connection to non-equilibrium
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
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.
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.