B010508 - STATISTICAL PHYSICS AND THEORY OF INFORMATION

Italian

Scientific programming. Discrete dynamical systems. Chaos. Stochastic processes. Statistical ensembles. Markov processes. Extended Systems and mean field. Dynamical Phase transitions. Review of thermodynamics. Equilibrium statistical mechanics. Ising model. The Monte-Carlo method. Disordered systems. Stochastic optimization. Network theory. Epidemic dynamics. Sociophysics. Neural networks. Theory of evolution. The theory of repeated games.

Luca Peliti, Appunti di meccanica statistica, Bollati Boringhieri 2003, ISBN-13: 9788833957128.

Steven Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, Westview Press 1994, ISBN-13: 9780738204536

Werner Krauth, Statistical Mechanics: Algorithms and Computations, Oxford University Press 2006, ISBN-13: 978-0198515364

Crispin W. Gardiner, Handbook of Stochastic Methods, Springer 2009, ISBN-13: 9783540707127

Notes, computer code and recordings on fisico.complexworld.net

-Provide the foundations of the theory of discrete dynamical systems (maps)
-Furnish the concept of of chaos theory.
-Motivate the choice of using a stochastic approach.
-Provide the basic elements of stochastic processes.
-Provide a basic idea of equilibrium statistical mechanics.
-Present the techniques of numerical simulation of dynamical processes, stochastic and statistical mechanics.
-Give the concept of global optimization.
-Provide examples of complex problems and how one can use statistical mechanics to treat them.
-Make the point on current research topics in statistical mechanics in connection with computing (networks, disordered systems and combinatorial problems, evolution, game theory and genetic algorithms, social patterns,
and psychological models of epidemics).

-Ability to write and run a program for a scientific simulation and data processing.
-Elements of the theory of discrete dynamical systems.
-Elements of the theory of Markov processes.
-Knowledge of the fundamental aspects of statistical mechanics of equilibrium.
-Theory and practice of Monte-Carlo simulations.
-Elements of the theory of stochastic optimization, in particular applied to combinatorial problems.
-Knowledge of some aspects of the current applications of statistical mechanics.

-Basic knowledge of physics and thermodynamics.
-Differential calculus in several variables, differential equations.
-Knowledge of the basic programming and use of an operating system.

Frontal lessons

Written test and oral examination or presentation of a work consisting of: problem definition, modeling, working code, data processing.

Introduction. Statistical physics and information theory. Deterministic equations of motion from stochastic approach. The analytical and numerical methods of statistical physics. Applications
the global optimization problem. Statistical physics today. Examples of application of science and information systems.

Scientific programming. Program structure, input / oputput, cycle calculation and printing results. Unix philosophy and file processing. Presentation of results and production of scientific articles. Fortran 90 and C / C for scientific computing. The gnuplot program for
data visualization and production of scientific images. Filtering data with awk and perl.

Discrete dynamical systems (maps). Sections, stroboscopic vision. The maps of the interval. Examples. Graphic illustration (COWEB). Fixed points. Stable, trajectories, transient basin of attraction. Limit cycles and fixed points of iterated map. Dependence on parameters, the logistic map. Bifurcations.

Transition to chaos. Doubling period. Intermittently. Bernoulli map and the map tent, symbolic dynamics. Lyapunov exponents. Histograms and probability distributions. Time series analysis and reconstruction of the attractor. The importance of temporal resolution.

Stochastic processes. The random walk. The generation of random numbers on computers. Implementing a stochastic process. Observable. Control parameters and order parameters. Stochastic and deterministic trajectories.

Statistical ensembles. The statistical probability distribution. Binomial distributions, Poisson and Gauss. Average values and fluctuations. Moments. Central limit theorem.

Markov. Evolution of the probability distribution. The Markov approximation. Classification of states. Ergodicity. Uniqueness of the asymptotic state. Relaxation to the asymptotic state and correlations.

Extended systems. Cellular automata. Directed percolation. The Domany-Kinzel model. System status. Mean field, and phase transitions. Reduced probability distributions. The mean field approach.

Dynamical phase transitions. Numerical examples. The correlation length. Observables and critical exponents. Mean field analysis and bifurcations. Universality.

Review of thermodynamics. Systems and thermodynamic variables. Principle of maximum entropy. Mechanical and thermal equilibrium. First law of thermodynamics. Equations of state. Thermodynamic potentials. Magnetic systems and ideal gas.

Equilibrium statistical mechanics. Entropy and information. The principle of maximum entropy. Microcanical and canonical distribution (Boltzmann). The partition function. Free energy and observables. Properties of the partition function.

The Ising model. Analysis of the behavior of a single isolated spin. The Ising model on a graph is completely connected. Phase transitions. Analysis of mean field models and on trees.

The Monte-Carlo method. Implementation and phenomenology of the Ising model on the lattice. Correlation length. Renormalization.

Disordered systems. The 1D Ising model in random field. Constraints and fluctuations. K-sat and traveling salesman.

Stochastic optimization. Simulating annealing. Application to the ordered and disordered Ising model and the salesman.

Network theory. Adjacency matrix; regular lattices, graphs, trees. Dynamics on trees and graphs. The effect Smallworld. Social networks.

Epidemic dynamics. Epidemic models. Models SIS, SIR, Seir, etc.. Connection with percolation and directed percolation. The minimum threshold for infectivity. Epidemics on scale-free networks.

Sociofisica Models opinion-forming. Models majority and minority.

Neural networks. The neuron. The Hopfield model. Neural networks. The formal neuron McCullogh
and Pitts. Attractor neural network. Learning and generalization in neural networks.

Theory of evolution. Mutation, fitness selection. Coevolution. Competition. Genetic algorithms.