# Course Details for A.Y. 2019/2020

#### Name:

**Kinetic and hydrodynamic models / Kinetic and hydrodynamic models**

### Basic information

##### Credits:

*:* Master Degree in Mathematics 6 CFU (c)

*:* Master Degree in Mathematical Engineering 6 CFU (b)

##### Degree(s):

Master Degree in Mathematics 2^{nd} anno curriculum Generale Elective

Master Degree in Mathematical Engineering 2^{nd} anno curriculum Comune Compulsory

##### Language:

English

### Course Objectives

This course provides an introduction to the classical kinetic theory of gases and the derivation of hydrodynamic equations.
The course also offers an overview on the Monte Carlo simulation techniques.
On successful completion of this module the student has the knowledge on the basic principles and the simulation strategies of kinetic models.

### Course Content

- Phase space description, Liouville Theorem and the BBGKY hierarchy.
- The Boltzmann Equation and the H-Theorem: the onset of irreversibility.
- Methods of reduced description to derive hydrodynamics from kinetic theory: the Hilbert and the Chapman-Enskog expansions.
- Hydrodynamic modes, the dispersion relation and the spectrum of density fluctuations.
- Basic principles of Monte Carlo simulations.
- The Kac ring model

### Learning Outcomes (Dublin Descriptors)

On successful completion of this course, the student should

- have a clear overview on how to obtain hydrodynamic conservation laws starting from the Boltzmann equation and should also be able to build up his/her own Monte Carlo code to run stochastic simulations.

### Prerequisites and Learning Activities

Mathematical Analysis

### Assessment Methods and Criteria

Written exam with a few general questions regarding the topics treated in the class and presentation of a numerical code for Monte Carlo simulation of a particle system, developed under the guidance of the lecturer.

### Textbooks

- L. E. Reichl, A Modern Course in Statistical Physics, 4th Ed. , John Wiley & Sons,. 2016.
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- Neal Madras, Lectures on Monte Carlo Methods , American Mathematical Society. 2002 .
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- C. Cercignani, The Boltzmann equation and its Applications , Springer-Verlag, New York, . 1988.
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- M. E. J. Newman and G. T. Barkema, Monte Carlo Methods in Statistical Physics , Oxford University Press. 2001.
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### Notes

- This course provides an overview on the mathematical techniques that allow to bridge the gap between the kinetic description of a many-particle system and the hydrodynamic description.
Moreover, a review of some basic Monte Carlo simulation techniques will also be offered.

### Course page updates

This course page is available (with possible updates) also for the following academic years:

** To read the *** current * information on this course, if it is still available, go to the university course catalogue .
*Course information last updated on: 01 marzo 2018, 09:27*