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    Silvio BACCARI

    Insegnamento di MODELING OF COMPLEX SYSTEMS

    Corso di laurea magistrale in PHYSICS

    SSD: FIS/02

    CFU: 6,00

    ORE PER UNITÀ DIDATTICA: 48,00

    Periodo di Erogazione: Primo Semestre

    Italiano

    Lingua di insegnamento

    INGLESE

    Contenuti

    Random numbers. Random geometrical models. Non equilibrium systems. Branching processes. Monte Carlo methods. Cluster algorithms. Basics of molecular dynamics.

    Testi di riferimento

    R. Livi and P. Politi,
    Nonequilibrium statistical physics,
    Cambridge University Press, Cambridge 2017,
    L Bottcher and H.J. Herrmann, Computational Statistical Physics, Cambridge University Press, Cambridge 2021

    Harris, T. E. (1963). The Theory of Branching Processes. Springer, Berlin

    A. Corral and F. Font-Clos, Criticality and self-organization in branching processes: application to natural hazards, arXiv: 1207.2589

    Kaufman, L. and Rousseeuw, P. J. (1990). Finding Groups in Data, anIntroduction to Cluster Analysis. New York: Wiley.

    S. Landau, I. Chis Ster, Cluster Analysis: Overview, International Encyclopedia of Education (Third Edition), Elsevier, 2010

    Obiettivi formativi

    The course aims to provide basic theoretical and numerical methods to develop models for complex systems.

    Prerequisiti

    Statistical ensembles, basic probability theory,
    critical phenomena

    Metodologie didattiche

    The course is structured in 40 hours of frontal lectures and
    12 hours of exercises and assisted study.
    Attendance is not compulsory but strongly recommended.

    Metodi di valutazione

    Oral exam

    Programma del corso

    Random Numbers. Definition of Random Numbers. Congruential RNG (multiplicative). Lagged Fibonacci RNG (additive). Available Libraries. How Good is a RNG? Non-Uniform Distributions
    Random-Geometrical Models. Percolation. The Sol-Gel Transition. The Percolation Model. The Burning Method. The Percolation Threshold. The Order Parameter. The Cluster Size Distribution. Correlation Length. Finite Size Effects. Finite Size Scaling. Size Dependence of the Order Parameter. The Shortest Path. Bootstrap Percolation. Fractals. Self-Similarity. Fractal Dimension: Mathematical Definition. The Box Counting Method. The Sandbox Method. The Ensemble Method. The Correlation-Function Method. Fractal Dimension in Percolation. Volatile Fractals. Random Walks. Self-Avoiding Walks.
    Non-Equilibrium Systems. Directed Percolation and Gillespie Algorithms. Cellular Automata. Classification of Cellular Automata. Time Evolution. Classes of Automata. The Game of Life. Q2R. Irreversible Growth. Random Deposition. Dielectric Breakdown Model (DBM). Random Fuse Model and Fracture. Simulated Annealing. Traveling Salesman.
    Epidemic Branching processes: Point-processes. The Galton-Watson process. Generating Function. Probability of extinction. Non-equilibrium phase transition. Size distribution of the population. Self-organized branching processes. Epidemic models for virus spreading. The time-dependent reproduction number.
    Monte-Carlo Methods. Computation of Integrals. Integration Errors. Higher Dimensional Integrals. Markov Chains. M(RT)2 Algorithm. Glauber Dynamics (Heat Bath Dynamics). Binary Mixtures and Kawasaki Dynamics. Creutz Algorithm. Boundary Conditions. Application to Interfaces. Self-Affine Scaling. Next-Nearest Neighbors. Shape of a Drop.
    Phase Transitions. Temporal Correlations. Decorrelated Configurations. Finite-Size Scaling. Binder Cumulant. First-Order Transitions.
    Cluster Algorithms. Potts Model. The Kasteleyn and Fortuin Theorem. Coniglio-Klein Clusters. Swendsen-Wang Algorithm.
    Overview of Cluster Analysis. Proximity Measures. Hierarchical clustering. Optimization clustering. Model Based clustering. Number of clusters.
    Renormalization Group. Real Space Renormalization. Renormalization and Free Energy. Majority Rule. Decimation of the One-dimensional Ising Model. Generalization. Monte Carlo Renormalization Group.
    Boltzmann Machine. Hopfield Network. Neuronal networks. Boltzmann Machine Learning.
    Basic Molecular Dynamics. Introduction. Equations of Motion. Contact Time. Verlet Method. Leapfrog Method.

    English

    Teaching language

    English

    Contents

    Random numbers. Random geometrical models. Non equilibrium systems. Branching processes. Monte Carlo methods. Cluster algorithms. Basics of molecular dynamics.

    Textbook and course materials

    R. Livi and P. Politi,
    Nonequilibrium statistical physics,
    Cambridge University Press, Cambridge 2017,

    L Bottcher and H.J. Herrmann, Computational Statistical Physics, Cambridge University Press, Cambridge 2021

    Harris, T. E. (1963). The Theory of Branching Processes. Springer, Berlin

    A. Corral and F. Font-Clos, Criticality and self-organization in branching processes: application to natural hazards, arXiv: 1207.2589

    Kaufman, L. and Rousseeuw, P. J. (1990). Finding Groups in Data, anIntroduction to Cluster Analysis. New York: Wiley.

    S. Landau, I. Chis Ster, Cluster Analysis: Overview, International Encyclopedia of Education (Third Edition), Elsevier, 2010

    Course objectives

    The course aims to provide basic theoretical and numerical methods to develop models for complex systems.

    Prerequisites

    Statistical ensembles, basic probability theory,
    critical phenomena

    Teaching methods

    The course is structured in 40 hours of frontal lectures and
    12 hours of exercises and assisted study.
    Attendance is not compulsory but strongly recommended.

    Evaluation methods

    Oral exam

    Course Syllabus

    Random Numbers. Definition of Random Numbers. Congruential RNG (multiplicative). Lagged Fibonacci RNG (additive). Available Libraries. How Good is a RNG? Non-Uniform Distributions
    Random-Geometrical Models. Percolation. The Sol-Gel Transition. The Percolation Model. The Burning Method. The Percolation Threshold. The Order Parameter. The Cluster Size Distribution. Correlation Length. Finite Size Effects. Finite Size Scaling. Size Dependence of the Order Parameter. The Shortest Path. Bootstrap Percolation. Fractals. Self-Similarity. Fractal Dimension: Mathematical Definition. The Box Counting Method. The Sandbox Method. The Ensemble Method. The Correlation-Function Method. Fractal Dimension in Percolation. Volatile Fractals. Random Walks. Self-Avoiding Walks.
    Non-Equilibrium Systems. Directed Percolation and Gillespie Algorithms. Cellular Automata. Classification of Cellular Automata. Time Evolution. Classes of Automata. The Game of Life. Q2R. Irreversible Growth. Random Deposition. Dielectric Breakdown Model (DBM). Random Fuse Model and Fracture. Simulated Annealing. Traveling Salesman.
    Epidemic Branching processes: Point-processes. The Galton-Watson process. Generating Function. Probability of extinction. Non-equilibrium phase transition. Size distribution of the population. Self-organized branching processes. Epidemic models for virus spreading. The time-dependent reproduction number.
    Monte-Carlo Methods. Computation of Integrals. Integration Errors. Higher Dimensional Integrals. Markov Chains. M(RT)2 Algorithm. Glauber Dynamics (Heat Bath Dynamics). Binary Mixtures and Kawasaki Dynamics. Creutz Algorithm. Boundary Conditions. Application to Interfaces. Self-Affine Scaling. Next-Nearest Neighbors. Shape of a Drop.
    Phase Transitions. Temporal Correlations. Decorrelated Configurations. Finite-Size Scaling. Binder Cumulant. First-Order Transitions.
    Cluster Algorithms. Potts Model. The Kasteleyn and Fortuin Theorem. Coniglio-Klein Clusters. Swendsen-Wang Algorithm.
    Overview of Cluster Analysis. Proximity Measures. Hierarchical clustering. Optimization clustering. Model Based clustering. Number of clusters.
    Renormalization Group. Real Space Renormalization. Renormalization and Free Energy. Majority Rule. Decimation of the One-dimensional Ising Model. Generalization. Monte Carlo Renormalization Group.
    Boltzmann Machine. Hopfield Network. Neuronal networks. Boltzmann Machine Learning.
    Basic Molecular Dynamics. Introduction. Equations of Motion. Contact Time. Verlet Method. Leapfrog Method.

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