This course provides the students with tools in numerical analysis that are frequently used in modern theoretical physics and in the analysis of complex systems.

1. Learning the basic methodologies to simulate statistical systems. 2. Understanding the use of numerical methods to obtain approximate solutions to otherwise intractable problems. 3. Learn a high-level general-purpose scientific computing language (julia, jupyter notebook etc.) in a Linux/Unix environment. 4. Acquiring general-purpose data analysis and visualisation skills.

Mathematical analysis, general physics, basic statistics.

1) Concepts of probability and statistics (2 credits) Random variables. Statistical description of data. Numerical calculation of basic estimators: average, variance, correlations. Joint, conditional, marginal distributions. Bayes theorem. Large numbers law, central limit theorem. Notable probability distributions: binomial, Poisson, Gauss. Maximum entropy estimate of parameters. Large deviations. Experimental Data Analysis. Random Walks, Wiener process, Master Equations. Stochastic equations: Itô calculus, Fokker-Planck equation. Entropy and Information and their relation to statistical mechanics. 2) . Numerical methods (2 credits). Introduction to basic Unix/Linux. Introduction to Julia language. Simple programs, plotting data, input and output. Finding roots of equations: bisection, regula falsi, secant and Newton's methods. Gradient descent. Numerical integration: trapezoid and Simpson's rule. Numerical differentiation: forward- and centred-difference methods. First order ordinary differential equations (ODE), initial value problems (IVP). Random numbers: definition and properties of pseudo-random numbers, classes of uniform random number generators, non-uniform random numbers. Applications of random numbers: Monte Carlo (MC) integration, percolation, random walks. Basic algorithms for the numerical integration of stochastic differential equations. Colored noise. Numerical simulation of master equations. Rate equations. Gillespie algorithm. Modeling simple biological networks. 3) Simulation of discrete systems at equilibrium (2 credits). Sampling the canonical ensemble with Monte Carlo: Metropolis-Hastings rule, balance and detailed balance, hybrid Monte Carlo. Applications to phase transitions Critical phenomena. Finite-size scaling analysis. Simulated annealing. 4) Simulation of newtonian mechanics (2 credits). Basic concepts of Newtonian dynamics and Statistical Mechanics: energy conservation, time reversibility and phase-space incompressibility, Liouville Theorem, Ergodicity. Derivation of the microcanonical, canonical and grand-canonical statistical ensemble. Simple integration schemes for molecular dynamics and their relation to Monte Carlo methods.

Frontal lectures and computer lab sessions.

- Frederick Reif, Fundamentals of Statistical and Thermal Physics, McGraw-Hill - Crispin W. Gardiner - Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences (Springer Series in Synergetics 13) (1994); - C. Kittel, "Elementary statistical physics". Courier Corporation. (2004); - Kerson Huang, "Statistical Mechanics", Wiley (1987); - M. E. J. Newman G. T. Barkema - Monte Carlo Methods in Statistical Physics (1999); - Luciano M Barone, Enzo Marinari, Giovanni Organtini, Federico Ricci Tersenghi - Scientific Programming C-Language, Algorithms and Models in Science. World Scientific Publishing Company (2013); - Malvin H. Kalos, Paula A. Whitlock - Monte Carlo Methods , Wiley-VCH (2008); - Michael P. Allen, Dominic J. Tildesley - Computer Simulation of Liquids. Oxford University Press (2017).

Modalità di esame: Prova orale obbligatoria;

Exam: Compulsory oral exam;

... The exam is made of two tests of equal weight. The first is an individual numerical project to evaluate the problem-solving ability of the student. The project is to be presented as a short written report (5-10 pages). The second is an oral test where the individual project and theoretical topics will be discussed. The final exam is oral and consists in: 1) A discussion of the individual project 2) Up to three broad questions on the main topics of the course.

Gli studenti e le studentesse con disabilità o con Disturbi Specifici di Apprendimento (DSA), oltre alla segnalazione tramite procedura informatizzata, sono invitati a comunicare anche direttamente al/la docente titolare dell'insegnamento, con un preavviso non inferiore ad una settimana dall'avvio della sessione d'esame, gli strumenti compensativi concordati con l'Unità Special Needs, al fine di permettere al/la docente la declinazione più idonea in riferimento alla specifica tipologia di esame.