Spring 2023     

Topics on Kinetic Collisional Theory:

Statistical Mechanics of Kinetic Interactive Systems in Mean Field Modeling

Analysis and Numerical Methods.
M 393C (# 55145)     and     CSE 396 (#)

 

Instructor:   Prof. Irene M. Gamba
Office: RLM 10.166, Phone: 471-7150

Class Webpage: UT Canvas
E-Mail: gamba@math.utexas.edu
Office hours: by appointment
Meeting Hours:   PMA 10.176  -  T-Th 2-3:15pm       

 

Statistical Mechanics of Kinetic Interactive Systems in Mean Field Modeling: Analysis and Numerical Methods.

This topics course covers broad issues on collisional theory arising from particle systems. We will discuss two class of systems:  the first one consists are Boltzmann and Landau sets equations for conservative and conservative systems and their connections to non-equilibrium statistical mechanics. The second one are related to weak turbulence models in plasma physic,  arising from perturbation of collisionless Vlasov-Poisson and Vlasov-Maxwell around bulk statistical equilibria resulting in a model reduction form coupling diffusion equations for the electron particle couple to spectral wave density from the Poisson or Maxwell equation in Fourier representation.

 

·       Introduction and elementary properties.  Binary elastic interaction, time irreversibility, conservation laws, H-theorem and energy inequalities for inelastic interactions.    The grazing collision limit for Coulomb interactions and the connection to the Landau Equation.  Small mean free path, Hilbert and Chapman expansions. Moment methods and connections to hydrodynamic models in fluid dynamics. 

·       Space homogeneous problems under special averaging regimes.

Collision systems: Angular averaging lemmas and gain of integrability properties for hard potentials. Existence and uniqueness properties in connection of moment inequalities. Carleman integral representation and comparisons principles for pointwise bounds to solutions. Summability of moments and exponential moments and tails. Solutions to the Cauchy problem by ODE in Banach spaces.  Convolution inequalities for collision operators. Fourier representation of the Boltzmann and relativistic Landau equations. W^m,p(R^d)-theory, p in [1, ∞].   Special case of kinetic equations of Maxwell type, special solutions in Fourier space.

Mean field coupled systems:  emerging particle-spectral wake energy systems in the modeling weak turbulence in plasma dynamics by small perturbation of bulk equilibrium states, under dynamics for null special averages.  Braking of symmetry and absence of Landau damping mechanisms. The quasilinear spectral wave system and  Balescu-Lenard models.

·      Numerical approximations to kinetic particle systems. Deterministic solvers for linear and non-linear collisional forms of Boltzmann and Landau type. Conservative spectral and FEM methods. Galerkin-Petrov schemes and moment methods.  Comparisons to Discrete Simulation Monte Carlo (DSMC) methods. Stability and error estimates. Applications to kinetic models for plasmas and charge transport as well as to inverse problem in nano-scale. The Boltzmann-Poisson system.       

 

We will have an extra hour discussion time when needed on a date and place to be set.

 

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, NY,1994.

·       Villani, C.; A review of Mathematical topics in collisional kinetic theory, Handbook of Fluid Mechanics, 2003.

·       Nicholson, D.R.; Introduction to Plasma Theory. John Wiley and Sons, 1982.

·       Sone, Y.; Kinetic Theory and Fluid Dynamics (Birkhäuser, 2002): Click here to download supplementary notes and errata

·       Sone, Y., Molecular Gas Dynamics (Birkhäuser, 2007): Click here to download supplementary notes and errata

·       Stix, T.H.; Waves in Plasmas, AIP, NY, 1992.

·       Thorne, K.S. and Blandford, R.D,;  Modern Classical Physics: Optics, Fluids, Plasmas, Elasticity, Relativity, and Statistical Physics, Princeton University Press, 2017.

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

Prerequisites: Some knowledge of Methods of Applied Analysis and Mathematics and Partial Differential Equations is beneficial

Testing and examination plan and policies:      Attendance at lectures is expected.   A 30-min prepared presentation on a topic to be discussed with the instructor.  

There will be neither exams nor tests for this course.

Evaluation:   The course and instructor will be evaluated at the end of the semester using the approved form. 

 

 

This course maybe viewed as complementary to CSE 397 / EM 397

 

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.