Physics

GENERAL OBJECTIVES: The course aims at presenting to the students the probability thoery and its applications in theoretical and experimental physics. Starting from the fundamental concepts of the theory, the course leads to the knowledge of probability theorems, Bayesian inference techniques, experimental data analysis techniques, general modeling of Markov processes, algorithms and stochastic numerical methods. For the acquisition of this knowledge various examples and applications will be studied. SPECIFIC OBJECTIVES: A - Knowledge and understanding OF 1) To know the assiomatic theory and the fundamental theorems in probability theory (large numbers, central limit, large deviations, generating functions per moments and cumulants) in general cases (correlated random variables, random variables with diverging variance). OF 2) To know baesyan inference and how to critically apply it to specific problems, on the base of their a priori knowledge. OF3) To know the theory beyond data analysis techniques. OF 4) To understand Markov chains, in abstract and in specific cases such as the random walker, chain reactions e recurrent events. OF 5) To know markovian stochatsic processes in continuous time: master equation, Fokker-Planck equation. Hints about the Langevin equation. OF 6) To understand connections between probability theory and statistical mechanics. B - Application skills OF 7) To deduce probabilities in bayesian theory. OF 8) To know how to compute probability distributions in the central limit, in the large deviations regime , even for a finite number of variables. OF 9) To apply fundamental probability theory to compute statistical properties of specific systems. OF 10) To be able to tackle data analysis for correlated and anomalous data. OF 11) To apply teh knowledge of Markov chains to algorithmic problems, such as the Monte Carlo method for large dimensional integrals and its application to the equilibrium dynamics of statistical mechanical problems, or the page ranking algorithm for websites in the WWW. C - Autonomy of judgment OF 12) To be able to assess relevant knowledge and variables in a probability problem. OF 13) To integrate acquired knowledge in order to apply it to data analysis, sttaistical mechanical problems and to general problems requiring the usage of probability techniques application (as COVID-19 epidemic spreading, evaluation of swab tests or vaccines). D - Communication skills OF 14) To know how to orally present a demonstration procedure or an application assessing the most relevant and clarifying steps and their meaning. E - Ability to learn OF 15) To be able to critically read handbooks or scientific information articles. OF 16) To be able to eavluate the consequences of a given probablistic approach and compare different methods. OF 17) To be able to identify the correct hypothesis and to disergard inconsistent hypothesis in the estimate procedure for phenomenological parameters and, more generaly, in statistical inference.