# TEACHING

### Ingegneria Matematica 2016/2017

#### Some notes on stochastic processes:

Download

Notes 1 on stochastic processes.

Download

Notes 2 on persistence times of a birth-death process.

#### 1: Introduction to Monte Carlo methods

* Generation of sequences of

**uniformly distributed** (pseudo)random numbers, using different simple congruent methods, illustrating their performance.

Source code.

* Generation of

**exponentially distributed** random numbers, using the inverse method.

Source code.

* Generation of

**Gaussian** random numbers, using the Box-Muller method.

Source code.

*

Exercises.

#### 2: Maxwell-Boltzmann distribution for ideal gases

* Calculation of the

**pressure from microscopic collisions** with the wall (Ex. 3.4 Sethna).

We run a simulation and compute the histogram of collisions in the right wall during an interval of time, as a function of the moment carried by the particles.

Source code (need to be improved!).

#### 3: Microscopic reversibility vs macroscopic irreversibility and 2nd Law of Thermodynamics

* Simulation: non-interacting point particles confined in a square box.

Source code.

*

Exercises.

**Bibliography:** J. Lebowitz, "From Time-symmetric Microscopic Dynamics to Time-asymmetric Macroscopic Behavior: An Overview".
#### 4: Monte Carlo simulation of equilibrium processes

* Simulation: Metropolis algorithm for the Ising model in a 2D-lattice.

Source code.

*

Exercises. Deadline: 5 May 2017

**Bibliography:** M. Newman and G. Barkema, "Monte Carlo Methods in Statistical Physics", Oxford University Press: New York, USA, 1999. Chapters 2-3.

#### 5: Artificial neural networks and Hopfield model

* Simulation: Implementation of the Hopfield model for image recognition.

Source code and some

images.

#### 6: Collective motion

* Starlings in Rome,

video.

* Vicsek model for flocking behavior,

article.

* Collective motion,

review article.

* Criticality in insect swarms,

viewpoint of

this other paper.

#### 7: Random walks

* Program to generate Random Walks (most of the code refers to Exercise 2.5 on Sethna's book),

Source code.

* Levy flights and human mobility,

video.

#### 8: Stochastic simulation of master equation systems

* Review:

D. Gillespie, "Stochastic simulation of chemical kinetics. Annu. Rev. Phys. Chem., 2007.

* Examples of systems described by a Master Equation. Radioactive decay (

code) and logistic growth (

code).

* Exercise:

Production-degradation process.

### Ingegneria dei Processi Industriali e dei Materiali - Elementi di Fisica

**Programma d'esame**

**Simulazione Appello - 1**

**Simulazione Appello - 2**

** Soluzioni 1 - Gennaio 2014**

** Soluzioni 2 - Gennaio 2014**

** Soluzioni 3 - Gennaio 2014**

** Soluzioni 4 - Gennaio 2014**

### Ingegneria Matematica 2015/2016

** Appello 29/6/16 - Soluzione Problema 1**

** Appello 29/6/16 - Soluzione Problema 2**

** Appello 29/6/16 - Soluzione Problema 3**