PHYSICS OF CRITICAL PHENOMENA
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- Versione italiana
- Academic year
- 2018/2019
- Teacher
- ROBERTO ZIVIERI
- Credits
- 6
- Didactic period
- Primo Semestre
- SSD
- FIS/03
Training objectives
- The course is addressed to students that want to acquire the basic knowledge in the subject of phase transitions and critical phenomena at equilibrium. The aim of the course is:
1) To give a general overview of the theory of phase transitions in the different fields of physics.
2) To underline the concept of critical phenomena and critical phase transitions via the definition of critical points.
3) To give examples of physical systems of general interest and for applications undergoing equilibrium phase transitions.
4) To convey a message of interdisciplinarity to the subject treated that represents a fundamental field of present research in various sectors of physics, but also of chemistry and biology.
To summarize, the course has the scope to give to the student a deep knowledge about the physics of phase transitions by means of a systematic presentation of the formalism and of the models describing them and to give an overview of the main experiments able to confirm the predictions of the models. The final aim is thus to enable the student to recognize the nature and the kind of most of the phase transitions occurring in nature and to investigate independently the main implications. Moreover, basing on the learned knowledges, the student acquires the skills necessary to solve the exercizes suggested durante the lectures e the possible applications useful for the future professional activity. Prerequisites
- To have a full understanding of the topics of the course are needed basic knowledges both of Physics and of Mathematics mainly acquired during the courses of the three-year Degree and partly during the course of the Master's Degree. However, the topics will be recalled during the lectures.
In particular, regarding knowledges of Physics, the student must have clear the following subjects:
1)Basic principles of thermodynamics, statistical mechanics and classical electromagnetism
2) Applications of quantum mechanics with special reference to time-independent perturbation theory and quantization of atomic and molecular levels)
3) Basic knowledges of crystal lattices and more generally of solid state physics.
Regarding the mathematical knowledges, in particular it is necessary that the student must have clear the following subjects:
1)General properties of series and convergence criteria
2)Series expansion of trascendental functions (logarithmic, exponential and trigonometric functions)
3) Derivability of a composed function
4) Eigenvalues and eigenvectors problem. Course programme
- The course can be subdivided into 4 parts:
1) Basics of classical and quantum magnetism including examples of phase transitions in magnetic systems
2) Statistical and thermodinamic approach to phase transitions and classification scheme
3) Most representative models describing systems which undergo phase transitions
4) Exact and approximated analytical methods and numerical methods exploited to solve the models and to establish the behaviour of the described systems in proximity of critical points. In particular, mean-field theories, transfer matrix method, series expansions, renormalization group and quantum field theory, numerical simulation methods.
The total number of hours is 48 corresponding to 6 credits. Hours are subdivided according to the following scheme:
1) 10 hours to give the generalities about equilibrium phase transitions, to classify phase transitions and for introducing the critical exponents
2) 10 hours to discuss the classical and quantum models predicting a critical behaviour
3) 12 hours to describe the most important mean-field theories for the approximated calculation of critical exponents
4) 14 hours to introduce the renormalization group theory and to illustrate some important application in physical systems having critical behaviour
5) 2 hours to present the transfer matrix method and to discuss its application to one-dimensional physical systems exhibiting critical properties like e.g. the one-dimensional Ising model with background in the biological field. Didactic methods
- Lectures will be delivered by the lecturer mostly on the blackboard in English. Slides will be shown to complete the subjects presented at the blackboard. Lecture notes prepared by the teacher are available on-line. There are not hours devoted to exercises but some hints will be given to help students to solve some of the exercises of the reference book.
Learning assessment procedures
- Learning will be checked during the final oral examination to verify for the student: 1) the preparation on the most important subjects of the course; 2) the attitude to correlate the different topics evaluating the level of detail of the concepts; 3) the ability to give examples also outside the subjects treated to check the autonomy of the student.
The final evaluation bases on the student response mode to the general and specific questions (3-4 questions) on the different topics of the course and on the presentation of the subjects.
There will be no written examination during the course. Reference texts
- Books
1) J. M. Yeomans, "Statistical Mechanics of Phase Transitions", Clarendon Press (1992)
2) H. E. Stanley, "Introduction to Phase transitions and Critical Phenomena", Oxford Science Publications (1987).
For an in-depth analysis of some topics
1) K. Huang, "Statistical mechanics", John Wiley & Sons (1987)
2) N. Goldenfeld, "Lectures on Phase Transitions and the Renormalization Group", Levant Books (2005).
Notes
Teacher's course notes on all the course topics.
