Math 855/856:
Introduction to Kinetic Theory and Mesoscopic Methods for
Computational Mechanics I. and II.
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 855
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:
Classical Hamiltonian system of N particles;
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.
The grading is based on the assignments.
Assigments:
No. 1 (Chapter 1, page 10-11): Problems 1-1, 1-2, 1-4, 1-6, 1-7, 1-8.
No. 2 (Chapter 2, pages 28-29): Problems 2-4, 2-5, 2-6, 2-7, 2-8, 2-9.
No. 3 (Chapter 3, pages 50-51): Problems 3-1, 3-2, 3-6, 3-8, 3-9.
No. 4 (Chapter 4, pages 64-65): Problems 4-1, 4-2, 4-7, 4-8.
No. 5 (Chapter 5, pages 77-78): Problems: 5-2, 5-2, 5-3, 5-4, 5-5, 5-6.
No. 6 (Chapter 6, pages 101-102): Problems 6-1, 6-5.
No. 7 (Chapter 7, pages 118-119): Problems 7-1, 7-6.
No. 8 (Chapter 8, pages 129-130): Problems 8-3, 8-4, 8-5.
Part II. Math 856
Prerequisite: Math 855
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.
The grading is based on the assignments and project.
Assigments:
No. 1 (Chapter 7, pages 118-119): Problems 7-1, 7-4, 7-6, 7-7, 7-8.
No. 2 (Chapter 8, pages 129-130): Problems 8-3, 8-4, 8-5.
No. 3 (Chapter 9, pages 157-159): Problems 9-1, 9-10.
No. 4 (Chapter 10, pages 175-176): Problems 10-1, 10-2, 10-3.
No. 5 (Chapter 11, pages 191-193): Problems: 11-9, 11-10.