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Stochastic simulation methods in physics

01SPNPF

A.A. 2021/22

Course Language

Inglese

Course degree

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
Teachers
Teacher Status SSD h.Les h.Ex h.Lab h.Tut Years teaching
Pagnani Andrea   Professore Ordinario FIS/03 60 0 0 0 5
Teaching assistant
Espandi

Context
SSD CFU Activities Area context
FIS/02
FIS/03
FIS/03
MAT/06
3
2
1
2
B - Caratterizzanti
B - Caratterizzanti
F - Altre attivitÓ (art. 10)
F - Altre attivitÓ (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
2020/21
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.
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.
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.
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: It˘ 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.
Frontal lectures and computer lab sessions.
Frontal lectures and computer lab sessions. Both frontal and lab lectures will be also recorded online.
- 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).
- 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).
ModalitÓ di esame: Prova orale obbligatoria; Progetto individuale;
Exam: Compulsory oral exam; Individual project;
... 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 disabilitÓ 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'UnitÓ Special Needs, al fine di permettere al/la docente la declinazione pi¨ idonea in riferimento alla specifica tipologia di esame.
Exam: Compulsory oral exam; Individual project;
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.
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.
ModalitÓ di esame: Prova orale obbligatoria; Elaborato progettuale individuale;
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.
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.
ModalitÓ di esame: Prova orale obbligatoria; Elaborato progettuale individuale;
The two tests (theoretical part and individual project) will be either discussed in remote or in presence.
Exam: Compulsory oral exam; Individual project;
The two tests (theoretical part and individual project) will be either discussed in remote or in presence.
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