ID:
509492
Durata (ore):
112
CFU:
12
SSD:
FISICA TEORICA, MODELLI E METODI MATEMATICI
Anno:
2024
Dati Generali
Periodo di attività
Annualità Singola (30/09/2024 - 13/06/2025)
Syllabus
Obiettivi Formativi
Module #1: Theoretical Physics
This module presents, at a basic level, the approach of theoretical physics to the quantitative description of phenomena, especially in interdisciplinary settings. The focus of the course is on concepts that have general applicability, and particularly on those that are commonly used in machine learning research, such as scaling, stability, chaos, conservation laws, variational principles, temperature, entropy.
The successful student will be able to
• Use dimensional analysis and scaling arguments to obtain constraints for quantitative laws in physical and non-physical phenomena.
• Characterize discrete-state dynamical systems in terms of their fixed points, cycles, connected components.
• Compute the stability of fixed points in continuous-state dynamical systems in one dimension.
• Formulate and solve simple optimization problems via Euler-Lagrange equations.
• Distinguish between microscopic and macroscopic degrees of freedom in a thermodynamic system.
• Compute physical quantities from thermodynamic potentials, or, in very simple examples, from the corresponding microscopic models.
Module #2: Quantum Physics
The objective of this module is providing the student with the basic knowledge of quantum physics. At the end of the module, the student is expected
• to be familiar with the main concepts of quantum physics, such as observable, quantum state, measurement, distinguishability, quantum superposition, quantum entanglement.
• To be familiar with the basic concepts of quantum information theory, and their relevance for quantum communication and quantum computation.
• To be able to understand and describe the behavior of simple quantum systems (e.g., a two-qubits state), including their evolution.
This module presents, at a basic level, the approach of theoretical physics to the quantitative description of phenomena, especially in interdisciplinary settings. The focus of the course is on concepts that have general applicability, and particularly on those that are commonly used in machine learning research, such as scaling, stability, chaos, conservation laws, variational principles, temperature, entropy.
The successful student will be able to
• Use dimensional analysis and scaling arguments to obtain constraints for quantitative laws in physical and non-physical phenomena.
• Characterize discrete-state dynamical systems in terms of their fixed points, cycles, connected components.
• Compute the stability of fixed points in continuous-state dynamical systems in one dimension.
• Formulate and solve simple optimization problems via Euler-Lagrange equations.
• Distinguish between microscopic and macroscopic degrees of freedom in a thermodynamic system.
• Compute physical quantities from thermodynamic potentials, or, in very simple examples, from the corresponding microscopic models.
Module #2: Quantum Physics
The objective of this module is providing the student with the basic knowledge of quantum physics. At the end of the module, the student is expected
• to be familiar with the main concepts of quantum physics, such as observable, quantum state, measurement, distinguishability, quantum superposition, quantum entanglement.
• To be familiar with the basic concepts of quantum information theory, and their relevance for quantum communication and quantum computation.
• To be able to understand and describe the behavior of simple quantum systems (e.g., a two-qubits state), including their evolution.
Prerequisiti
Basic concepts of calculus and linear algebra: multivariable functions, derivatives, integrals, Taylor series, vectors, eigenvalues and eigenvectors of a linear operator. Basic concepts of physics: scalar and vectorial quantities, fundamental concepts of Newtonian mechanics, principle of energy conservation.
Metodi didattici
Module #1: Theoretical Physics
This module will consist of lectures (at the blackboard), a few pair/group activities in class, and around 16 hours of dedicated training sessions, where students will work on assignments individually. Additional exercise sheets with solved exercises will be provided.
Module #2: Quantum Physics
In this module lectures will be implemented by using Power-Point slides and notes given at the blackboard. The module also includes a training session (some 16 hours) in which the teacher will show how to solve problems and exercises connected with the lectures’ topics. Attendance and interaction with the teacher during the entire course are strongly recommended.
This module will consist of lectures (at the blackboard), a few pair/group activities in class, and around 16 hours of dedicated training sessions, where students will work on assignments individually. Additional exercise sheets with solved exercises will be provided.
Module #2: Quantum Physics
In this module lectures will be implemented by using Power-Point slides and notes given at the blackboard. The module also includes a training session (some 16 hours) in which the teacher will show how to solve problems and exercises connected with the lectures’ topics. Attendance and interaction with the teacher during the entire course are strongly recommended.
Verifica Apprendimento
The exam consists in two tests, one for each module. Tests contain multiple-choice questions to establish the student knowledge of the concepts presented in the course and the capability to solve simple exercises. The duration of the test is up to 120 minutes. The written exam can be integrated with an oral exam at the discretion of the teacher.
Testi
Module #1: Theoretical Physics
• Ain A. Sonin, "The physical basis of dimensional analysis" [available online at https://web.mit.edu/2.25/www/pdf/DA_unified.pdf]
• M. Cencini, F. Cecconi, A. Vulpiani, "Chaos: from simple models to complex systems", World Scientific Pub, 2009
• P. Hamill, "A student's guide to Lagrangians and Hamiltonians", Cambridge University Press, 2013
• R. Shankar, "Fundamentals of physics I: mechanics, relativity, and thermodynamics", Yale University Press (Expanded edition), 2019
Note: Some of these books are quite advanced: only excerpts from them will be used. Additional reading material will be provided during the course.
Module #2: Quantum Physics –
B. Schumacher and M. Westmoreland – Quantum Processes Systems, and Information 2010 Cambirdge University Press (ISBN 978-0-521-87534-9)
• Ain A. Sonin, "The physical basis of dimensional analysis" [available online at https://web.mit.edu/2.25/www/pdf/DA_unified.pdf]
• M. Cencini, F. Cecconi, A. Vulpiani, "Chaos: from simple models to complex systems", World Scientific Pub, 2009
• P. Hamill, "A student's guide to Lagrangians and Hamiltonians", Cambridge University Press, 2013
• R. Shankar, "Fundamentals of physics I: mechanics, relativity, and thermodynamics", Yale University Press (Expanded edition), 2019
Note: Some of these books are quite advanced: only excerpts from them will be used. Additional reading material will be provided during the course.
Module #2: Quantum Physics –
B. Schumacher and M. Westmoreland – Quantum Processes Systems, and Information 2010 Cambirdge University Press (ISBN 978-0-521-87534-9)
Contenuti
Module #1: Theoretical Physics
• INTRODUCTION. A map of theoretical physics. The role of theoretical physics for AI.
• DIMENSIONAL ANALYSIS AND SCALING ARGUMENTS
Dimensions and units. Dimensional analysis for checking results and for guessing results. Scaling arguments. Power laws, scale invariance, characteristic scales. Scaling arguments to predict the existence of bounds and to validate models.
• DYNAMICAL SYSTEMS
Dynamical systems with continuous and discrete states and time. Orbits, cycles, fixed points, transient/recurrent states. Reversibility. Connected components and conserved quantities. Cobweb plot for iterated 1-dimensional maps. Linearization around a fixed point; stability analysis. Bifurcation diagrams. The logistic map. First-order ordinary differential equations as dynamical systems.
• VARIATIONAL PRINCIPLES AND LAGRANGIAN MECHANICS
Fermat's principle of least time and Snell's law. Difference between stationarity and optimality. Elements of functional derivatives. The action and the Lagrangian. The Euler-Lagrange equations. The Lagrangian of classical mechanics. Particle in a potential. Euler-Lagrange formulation of the shortest path between two points.
• THERMODYNAMICS
Thermodynamic systems, microscopic and macroscopic degrees of freedom. Temperature. Quantity of heat exchanged and its conservation. The equation of state of the ideal gas; Boltzmann constant. Thermodynamic equilibrium. Thermodynamic transformations, quasi-static transformations. Work and the first law of thermodynamics. Elementary kinetic theory of the ideal gas. Microscopic interpretation of temperature from kinetic theory. Probability distribution of velocities, fluctuations. The second principle of thermodynamics. Entropy and its microscopic origin. Thermodynamic potentials.
Module #2: Quantum Physics
This module deals with the basic knowledge of quantum systems and quantum information theory. Topics include: bits and quanta, qubits, definition of states and observables; measurement of a quantum system; evolution of a quantum system; entanglement; description of pure and mixed states; main concepts connected with the application of quantum physics in communications and computations; discrete and continuous degrees of freedom
• INTRODUCTION. A map of theoretical physics. The role of theoretical physics for AI.
• DIMENSIONAL ANALYSIS AND SCALING ARGUMENTS
Dimensions and units. Dimensional analysis for checking results and for guessing results. Scaling arguments. Power laws, scale invariance, characteristic scales. Scaling arguments to predict the existence of bounds and to validate models.
• DYNAMICAL SYSTEMS
Dynamical systems with continuous and discrete states and time. Orbits, cycles, fixed points, transient/recurrent states. Reversibility. Connected components and conserved quantities. Cobweb plot for iterated 1-dimensional maps. Linearization around a fixed point; stability analysis. Bifurcation diagrams. The logistic map. First-order ordinary differential equations as dynamical systems.
• VARIATIONAL PRINCIPLES AND LAGRANGIAN MECHANICS
Fermat's principle of least time and Snell's law. Difference between stationarity and optimality. Elements of functional derivatives. The action and the Lagrangian. The Euler-Lagrange equations. The Lagrangian of classical mechanics. Particle in a potential. Euler-Lagrange formulation of the shortest path between two points.
• THERMODYNAMICS
Thermodynamic systems, microscopic and macroscopic degrees of freedom. Temperature. Quantity of heat exchanged and its conservation. The equation of state of the ideal gas; Boltzmann constant. Thermodynamic equilibrium. Thermodynamic transformations, quasi-static transformations. Work and the first law of thermodynamics. Elementary kinetic theory of the ideal gas. Microscopic interpretation of temperature from kinetic theory. Probability distribution of velocities, fluctuations. The second principle of thermodynamics. Entropy and its microscopic origin. Thermodynamic potentials.
Module #2: Quantum Physics
This module deals with the basic knowledge of quantum systems and quantum information theory. Topics include: bits and quanta, qubits, definition of states and observables; measurement of a quantum system; evolution of a quantum system; entanglement; description of pure and mixed states; main concepts connected with the application of quantum physics in communications and computations; discrete and continuous degrees of freedom
Lingua Insegnamento
INGLESE
Corsi
Corsi
ARTIFICIAL INTELLIGENCE
Laurea
3 anni
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