Meeting Hours: RLM
10.176, T-Th
12:30-2:00pm
We will have an extra hour discussion time when needed on a
date and place to be set.
This topics course covers
issues on the Boltzmann and Smoluchowski type
equations for conservative and non-conservative systems and connection to
non-equilibrium statistical mechanics.
More specifically, we
will discuss introduction and elementary properties associated to elastic
and inelastic collisional theory and to solutions of the Boltzmann transport
equation, time irreversibility, conservation laws, H-theorem and energy
inequalities. The grazing collision limit for Coulomb
interactions and the connection to the Landau Equation.
Topics to be cover are space
Homogeneous problems and Povzner type lemmas. Existence and uniqueness properties in connection of moment inequalities.
Carleman
integral representation and comparisons principles for pointwise bounds to solutions.
Convolution inequalities for collision Operators. Fourier representation of the Boltzmann equation. Kinetic
equations of Maxwell type, stationary and self-similar solutions for space
homogeneous problems.
Connections
to dynamical scaling and existence of stable laws from continuous probability
theory to non-Gaussian states. Applications to information
propagation problems.
The
space inhomogeneous problem. The space inhomogeneous
problem in all space. The Kaniel-Shimbrot
iteration method vs the Hamdache
method. Scattering effects for solutions in all space due
to dispersion vs dissipation.
Averaging lemmas and
renormalized DiPerna-Lions Solutions 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
Numerical approximations
to kinetic particle systems: deterministic solvers for linear and non-linear
collisional forms. Conservative spectral and FEM methods Topics on kinetic
models for plasmas and charge transport as well as to inverse problem in nano-scales. The Boltzmann-Poisson
system. Boltzmann-Poisson vs. Fokker-Plank-Poisson
systems.
The
Boltzmann-Poisson system. Boltzmann-Poisson vs.
Fokker-Plank-Poisson systems.
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 Pulvirenti, 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.