Math 796/896: Introduction to Kinetic Theory and Mesoscopic Methods for Computational Mechanics I. and II.

Time:
Tuesdays and Thursdays, 1:30pm-2:45pm, Aug. 26 - Dec. 8, 2006
Place:
Room 2120 in ECS BLDG
Reference:
An Introduction to the Theory of the Boltzmann Equation by S. Harris and Notes by the instructor
Synopsis:

The goal of this course is to provide a rudimentary introduction to kinetic theory and nonequilibrium statistical mechanics, which bridges the microscopic theories and the macroscopic, continuum theories of flows. The Boltzmann equation and its solutions are the central part of kinetic theory and nonequilibrium statistical mechanics. The Boltzmann equation is relevant to micros/nano-flows and other thermo-chemically nonequilibrium flows (e.g., rarefied gases, micro-/nano-flows, and hypersonic flows). The course consists of two parts and each part is for one semester.

This course is suitable for the graduate students who are interested in kinetic theory, mesoscopic methods, nonequilibriam statistical mechanics, and computational sciences.

There will be no exam for the course, but there will be a set of take-home assignments. The final grade depends the grades of the assignments.

Part I. Math 796

Prerequisite: Classical Mechanics (PHYS 451), Statistical Mechanics (PHYS 454), and Partial Differential Equations (MATH 401)

The first part covers basic kinetic theory including following topics:

  • Liouville equation for the N particle distribution function FN in phase space and its equivalence to the N particle Hamiltonian system;
  • Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy of equations for the reduced distribution function FM for M < N;
  • The Boltzmann equation, binary collisions, the properties of the collision operator, and collisional invariants;
  • the moments of the single distribution function f:=F1 and hydrodynamic equations;
  • Linearized collision operator and its eigen theory;
  • The H-theorem and irreversibility;
  • Normal solutions of the Boltzmann equation, the Fredholm theorems for integral equations, and Hilbert's uniqueness theorem;
  • Chapman-Enskog procedure;
  • Calculation of the transport coefficients.

    • Part II. Math 896

    Prerequisite: Math 796

    The second part of the course covers models of the Boltzmann equation and numerical solution techniques for hydrodynamic equations (Euler and Navier-Stokes equations) and the Boltzmann equation. Specific topics are:

  • Non-normal and Moment method solutions of the Boltzmann equation, Maxwell's moment method;
  • Grad's moment method, and Grad's 13-moment equations;
  • Model Boltzmann equations, BGK model equation, generalized model equations and Gross-Jackson procedure;
  • Special solutions of the Boltzmann equation, boundary conditions in kinetic theory, the plane Poiseuille flow, and the Couette flow;
  • Gas mixtures, transport phenomena in mixtures, and linearized and model Boltzmann equations for mixtures;
  • Generalized Boltzmann equations, the Wang Chang-Uhlenbeck equation, and Enskog equation for dense gases;
  • The lattice Boltzmann equation for incompressible flows;
  • The gas-kinetic scheme for compressible flows;
  • The Direct Simulation Monte Carlo (DSMC) method;
  • Some examples of nonequilibrium flows.