# 87450 - MODELS AND NUMERICAL METHODS IN PHYSICS

## Learning outcomes

At the end of the course the student will acquire the tools to build up dynamical models for the evolution of the classical physical systems formed by interacting particles under the influence of external fields. He/she will be able to use numerical techniques for the solution of the corresponding differential equation even in the case of fluctuating fields. In particular, in the limit of a large number of particles the kinetic and the fluid approximations will be developed; in the case of long range interactions the average field equations will be considered, together with self-consistent solutions and collision models based on stochastic processes.

## Course contents

Basic notion of Dynamical Systems

Concept of phase flow associated to a dynamical system. Basic properties of ordinary differential equations. Concept of Stability and classifications of the fixed point in the phase space. Solution of linear differential equations. Hamiltonian systems: integrable and chaotic systems. Liuoville equation. Introduction to dissipative systems and existence of stange attractors.

Basic numerical methods

Recurrences: Newton's and bisection methods. Interpolation and quadratures. Rational approximation, orthogonal polynomials and Gauss quadratures. Systems of linear equations. Integration methods for ordinary differential equations. Symplectic integrators for Hamiltonian systems.

Stochastic processes

Probability measures and stochastic processes. Introduction to ergodic theory. Markov models: the random walk and the Wiener process. Langevin equation and Stochastic dynamical systems. The stochastic Liouville equation and the Fokker-Planck equation.

Integration methods for stochastic differential equation and partial differential equation. Finite difference methods for partial differential equations.

Models

Discrete dynamical systems and the Master equations. The transition from discrete to continuum models. Continuity and momentum equations. Oscillators system and elastic media. Kinetic and fluid description for a gas and a plasma.

G. Turchetti Appunti di metodi numerici (http://www.physycom.unibo.it/turchetti_modelli_numerici.html)

G. Turchetti Dinamica Classica dei Sistemi Fisici Ed. Zanichelli (2002)

D. Potter Computational Physics Ed. J. Wiley & Sons (1977)

## Teaching methods

Frontal Lessons and computer pratise and implementation of simple models

## Assessment methods

At the end of classes an oral exam or a short paper and a working numerical model on a subject proposed by the candidate and approved by the teacher. The paper must include a in depth analysis of the subject and a discussion of the faced problems.

## Teaching tools

computer exercises to implement algorithms

## Office hours

See the website of Sandro Rambaldi