Portale della Didattica

Stochastic simulation methods in physics

01SPNPF

A.A. 2024/25

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 Altre conoscenze utili per l'inserimento nel mondo del lavoro

Statistiche superamento esami

Anno accademico di inizio validità

2023/24

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. The use of numerical methods, and statistical analysis, is fundamental for the research in theoretical physics. Numerically simulating a complex system is often the only way to get some intuition about its behavior. The course will provide first an introduction to multivariate statistics, then will be the theoretical background to build a theory of stochastic processes, such as Markov Process with discrete/continuous time and state evolution (e.g. Monte Carlo Markov Chains, Master Equation, Fokker-Plank Equation). The concepts acquired in the first part of the course will be then used in the second part of the course to develop a theory of numerical techniques to analyze concrete complex systems (e.g. spin models for phase transitions, epidemic modeling, etc.)

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.

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).

Study materials

Lecture slides; Lecture notes; Lab exercises; Lab exercises with solutions; Video lectures (previous years); Simulation tools;

Taking the exam before attending the course

You can take this exam before attending the course

Assessment and grading criteria

Exam: Compulsory oral exam;

Expected Results Understanding of the topics covered and calculation skills in using the related mathematical tools introduced. Ability to recognize and use appropriate mathematical tools in engineering disciplines. Ability to build a logical path, using the mathematical tools introduced. Ability to implement and deploy a numerical simulation of complex systems using the mathematical background covered in the theoretical part of the course. Criteria, rules, and procedures for the exam The exam is aimed at ascertaining knowledge of the topics listed in the official teaching program and the ability to apply the theory and related numerical methods developed to the solution of exercises. The assessments are expressed in thirtieths and the exam is passed if the grade reported is at least 18/30. The exam is made of two oral tests of equal weight. The final mark will be the average of the two individual marks. The first test will cover the theoretical part and will assess the ability of the candidate to navigate through the concepts presented in the first part of the course. The discussion will be based on three broad questions and some simple exercises to assess the capacity of the candidate to apply in concrete cases the theory developed. The second test is an individual numerical project to evaluate the problem-solving ability of the student. The project consists of a numerical implementation based on the theory developed in the course, of a concrete example of complex systems. The description of the project that the candidates will develop, will be described on the last day of the course. The project is to be presented as: (i) a short written report (5-10 pages) or as a Jupyter Notebook describing the different steps the candidate has taken to solve the task; (ii) a discussion where the different computational strategies adopted by the candidate will be ascertained, together with her/his coding skills.

In addition to the message sent by the online system, students with disabilities or Specific Learning Disorders (SLD) are invited to directly inform the professor in charge of the course about the special arrangements for the exam that have been agreed with the Special Needs Unit. The professor has to be informed at least one week before the beginning of the examination session in order to provide students with the most suitable arrangements for each specific type of exam.