Aims and scope

The course aims to derive the macroscopic properties of gaseous systems starting from

their microscopic (molecular dynamics) analysis.

In this perspective, analytical and numerical methods for the solution of the Boltzmann equation

are introduced and the influence of rarefied gas properties in micro-nano-technologies is

highlighted.

               Knowledge and understanding

At  the end of the course the students are expected to:

-know the basic mathematical properties of the Boltzmann equation

-be able to solve the Boltzmann equation both analytically and numerically

-be able to model various problems arising in the micro-technology environment

 by using the Boltzmann equation.

                       Program

1-Introduction to Statistical Mechanics

-The postulate of classical statistical mechanics.

-Microcanonical Ensemble.

-Canonical Ensemble.

-Grand Canonical Ensemble.

2-Thermodynamics and Kinetic Theory

-The laws of Thermodynamics.

-Entropy.

-The problem of Kinetic Theory.

3-The Boltzmann equation 

-Derivation (Lanford Theorem). Existence of solutions (Diperna-Lions Theorem).

-The BBGKY Hierarchy.

-Elementary properties of the collision operator.

-Boundary conditions and the gas-surface interaction.

-The H-Theorem and the problem of irreversibility.

-Kinetic Models.

4-Methods of solution of the Boltzmann equation

-Variational Methods.

-Transformation into Integral Equations.

-The method of Elementary Solutions.

-Numerical methods.

5-Application of the Boltzmann equation to Microfluidics

-Analytical and numerical investigation of microchannel flows (Poiseuille, Couette and

 Thermal-Creep problems; sound waves propagation).

-Slip boundary conditions for the Navier-Stokes equations.

-Kinetic modeling of gas mixtures.

6-Mathematical Models and Methods in Micro-Nano-Technologies

-Characterization of Micro-Electro-Mechanical Systems (MEMS) devices.

-Analysis of damping forces in low-frequency and radio-frequency MEMS devices.

7-Boltzmann-type equations in Socio-Economic applications

                  Preconditions

First level courses concerning Mathematical Analysis, Numerical Analysis

and Probability.

                  Examination

The final exam consists either in an oral discussion on the whole program,

or in a written project in which one of the topics, dealt with during the course,

is investigated thoroughly.