# MODELLING AND NUMERICAL SIMULATIONS

## Learning outcomes of the course unit

The course aims at providing an elementary introduction to modeling and

numerical simulations techniques, which are of common usage in

Computational Physics. Though born in the framework of scientific

applications, these techniques are as a matter of fact of general

relevance. As a results, they proved to be effective in a variety of fields

(economics and finance, computer networks, computational biophysics).

In view of this, the course will be to a large extent a collection of topics

presented in a seminar style. On top of providing conceptual and

technical tools, the course will aim in the end at tackling a project, in

which the students will finalize one of the simulations introduced during

the classes. This activity will be the subject of the final examination.

This is also intended to improve the presentation skills of the students

(they should be able to argue in public while presenting a project of their

own). After having attended the course, students should master the very

bases of probability theory and put those at work when it comes to statistics.

In particular, they should be able to estimate errors on sample means,

recognizing autocorrelation effects.

## Prerequisites

No prerequisites

## Course contents summary

First of all, we will aim at introducing the basics of probability theory and

statistics, with an emphasis on numerical techniques (probability

functions generation, data analysis). Data analysis will give the chance to

introduce modeling in the simple form of data fitting. A large fraction of

the course will be devoted to applications of Markov processes theory.

Modeling of queues will be the main application of the formalism. Simple

examples of dynamic MonteCarlo will be proposed to Physics students (if

any). An elementary introduction to stochastic differential equations will

be proposed to Maths students (if any), in particular facing Langevin

equation (brownian motion and tree-cutting problem). Basics of

percolation theory will be introduced as an example of how a simple

model can model a variety of phenomena.

## Course contents

- A short account of combinatorial analysis.

- Introduction to probability theory. Events, sum and product of events and their probabilty. Dependent and independent events. Short account of Bayes formula. Sequences of independent (yes/no) experiments. Discrete and continuous random variables and distribution laws; distribution function, probability density. Mean value, median, mode, moments, variance.

- Distribution laws: constant density, binomial, Poisson distribution, gaussian.

- Games and hypergeometric distribution.

- Probability distributions for two or more variables, covariance.

- Cebysev inequality, variance of arithmetic mean and law of large numbers. Estimating distribution parameters from samples: correct, unbiased, efficient estimates (variance as a case of study). Brief account of central limit theorem. Errors for independent experimental samples.

- Hypothesis verification.

- A brief account of pseudo random numbers generation. Generation of distributions, Von Neumann method. Hit or miss integration. How to generate tha gaussian distribution.

- Generalities on stocastic processes. Morkov chains, Markov matrices and their spectral properties. Stationary distribution of a Markov process. Herenfest process: a few words on periodic processes.

- Modelling a queue as a Markov process. Existence of stationary (equilibrium) distribution, variance and mean waiting time. An algoritm for simulating a queue.

- Autocorrelation effects for Markov processes, autocorrelation times and error estimation.

- A brief account of percolation theory (on a square, bidimensional lattice), cluster-finding algorithms; various observables for finding the percolation threshold (errors).

- Supplementals (upon request) for students in Mathematics (if any): a brief account of stochastic differential equations. Langevin equation and Fokker-Plank equation.

- Supplementals (upon request) for students in Physics (if any): a brief introduction to dynamic MonteCarlo methods as Markov processes.

## Recommended readings

Notes provided by the lecturer.

## Teaching methods

Style will be mostly informal, giving emphasis to problem solving. In view

of this, every subject will be tackled also via numerical experiments.

## Assessment methods and criteria

Evaluation will be in part in itinere, via the assignement of numerical

exercises. At the end every student will be assigned a problem to be

solved via numerical simulations. Students will present their solution

toghether with a report.