Portale della Didattica

Stochastic simulation methods in physics

01SPNPF

A.A. 2020/21

Course Language

Inglese

Degree programme(s)

Master of science-level of the Bologna process in Physics Of Complex Systems (Fisica Dei Sistemi Complessi) - Torino/Trieste/Parigi

Course structure

Teaching	Hours
Lezioni	80

Lecturers

Teacher	Status	SSD	h.Les	h.Ex	h.Lab	h.Tut	Years teaching
Pagnani Andrea	Professore Ordinario	PHYS-04/A	60	0	0	0	8

Co-lectures

Espandi

Teacher	Status	SSD	h.Les	h.Ex	h.Lab	h.Tut
Braunstein Alfredo	Professore Associato	PHYS-02/A	20	0	0	0

Context

SSD	CFU	Activities	Area context
FIS/02 FIS/03 FIS/03 MAT/06	3 2 1 2	B - Caratterizzanti B - Caratterizzanti F - Altre attività (art. 10) F - Altre attività (art. 10)	Discipline matematiche, fisiche e informatiche Discipline matematiche, fisiche e informatiche Altre conoscenze utili per l'inserimento nel mondo del lavoro Abilità informatiche e telematiche

Date d'appello

Orario delle lezioni

Statistiche superamento esami

Anno accademico di inizio validità

2020/21

Course description

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.

Expected Learning Outcomes

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.

Pre-requirements

Mathematical analysis, general physics, basic statistics.

Course topics

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.

1) Concepts of probability and statistics (2 credits) Random variables. Statistical description of data. Introduction to multivariate statistics. Relevant distributions. Maximum entropy estimation of the parameters of relevant probability distributions. Bayes theorem. Large numbers law, central limit theorem. Finite size correction to central limit theorem. Large deviations. Experimental Data Analysis. Random Walks, Wiener process, Master Equations. Fokker-Planck equation. 2) . Numerical methods (2 credits). Introduction to basic Unix/Linux commands. Introduction to the Julia language. Simple programs, plotting data, input and output. Finding roots of equations: bisection, regula falsi, secant and Newton's methods. Pseudorandom number. Defining a custom pdf sampler. The case of discrete variables. Monte Carlo (MC) integration. 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. Simple ordinary differential equation: the harmonic oscillator. Euler-Cromer, mid-point, leap-frog. Stability analysis. Least square regression. Finite dimensional lattices. Ising model in finite dimensional lattice. Phase transitions. 4) Improved Monte Carlo methods (2 credits). Sampling efficiently from complicated distributions. Faster than-the-clock Monte Carlo schemes. Rejectionless Monte Carlo. Wolff's algorithm. Hit-and-run Monte Carlo. Introduction to the final project.

Sustainable development goals

Additional information

Course structure

Frontal lectures and computer lab sessions. Both frontal and lab lectures will be also recorded online.

Reading materials

- 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).

Assessment and grading criteria for ONLINE exam

Exam: Compulsory oral exam; Individual project;

The two tests will be done using the politecnico online platform or other remote conference-call media. The exams is considered passed when both tests will be performed.

Assessment and grading criteria for BLENDED exam (online and onsite)

Exam: Compulsory oral exam; Individual project;

The two tests (theoretical part and individual project) will be either discussed in remote or in presence.