STATISTICAL PHYSICS II
- Overview
- Assessment methods
- Learning objectives
- Contents
- Bibliography
- Teaching methods
- Contacts/Info
Basic knowledge of equilibrium statistical mechanics, as the one gained through the Statistical Physics I course.
The examination is oral and it is essentially divided into two parts:
First of all, students are asked to choose (in agreement with the instructor) a topic among those covered in the course. In the first part of the exam, they will be asked to discuss it in sufficient depth both at a qualitative and quantitative level.
This part of the exam is aimed at verifying:
- the knowledge of the specific topic.
- the understanding of its most important features
- the capability of discussing it from a technical viewpoint.
In the second part of the exam – in order to asses if the students have acquired a sufficient knowledge of critical phenomena -- some aspect of the remaining topics will be discussed at a slightly more qualitative level, without the need of dwelling too deeply in the technical aspects.
To successfully pass the exam students should demonstrate knowledge at the required level of all the topics presented in the course. The deeper the knowledge the better the evaluation. Full mark with laude is assigned only to students that accomplish completely the learning outcomes discussed above, showing a solid qualitative and quantitative comprehension of the course topics and the ability to re-elaborate them.
The theory of critical phenomena and the renormalization group are one of the great successes of modern theoretical physics. Its applications have spanned a wide range of different fields in the natural and even social sciences. The concept of universality and universal behaviour justifies the adoption of simplified minimal models in the study of many physical phenomena, and the theory of critical and collective phenomena is essential for our understanding of complex systems.
The course illustrates the theoretical approach to the study of equilibrium critical phenomena through the study of simple microscopic models, statistical field theory and renormalization group methods.
We will also discuss numerical methods for the study of critical phenomena.
On completion of the course the student should
• Have developed a refined understanding of the concepts of phase transitions, collective phenomena, universality and scale invariance.
• Show familiarity with the basic technical aspects of statistical field theory and the renormalization group
• Be able to study numerically the critical behaviour of simple equilibrium models.
Introduction to phase transitions (10h):
Example of common universality classes, the phase diagram of simple liquids. The Ising model: interpretation, mean field solution. Exact solution in one dimension. Landau Teory. Landau and Landau Peierls arguments and the lower critical dimension.
Scaling ad correlations (4h):
Gaussian model, correlation functions, fluctuation dissipation theorem, upper critical dimension.
Monte Carlo methods (8 h):
Detailed balance. Metropolis and heat bath algorithms. Finite size scaling analysis and numerical implementation.
Renormalization Group (RG) (20 h):
General formulation in real space. Widom’s scaling and critical exponents. Scaling laws. Universality.
Implementation of real space RG: one dimensional Ising, Migdal-Kadanoff approximation.
Momentum shell RG: General idea and application to scalar Ginzburg-Landau theory. The Gaussian fixed point. Epsilon expansion: linear order and qualitative walk-through of order squared corrections.
Continuous Symmetries (6 h):
n-vector model. Spontaneous symmetry breaking and Goldstone Bosons. Classical Mermin-Wagner theorem and quasi-long-range-theorem. Kosterlitz-Thouless transition in the XY model: the role of vortices.
The main reference text for this course is:
H. Nishimori & G. Ortiz, Elements of Phase Transitions and Critical Phenomena, (2011, Oxford University Press)
For further reading, other suggested textbooks are:
J. J. Binney, N. J. Dowrick,
A. J. Fisher and M. E. J. Newman, The Theory of Critical Phenomena, (2002, Oxford University Press).
N. Goldenfeld, Lectures on phase transitions and the renormalization group (Taylor & Francis)
Detailed lecture notes will also made available.
The course is essentially based on lectures, during which the teacher
presents the contents of the course in full detail, including mathematical
derivations.
This course is NOT active in 2021/22 and runs every two years, alternating with Physics of Complex Systems.
Therefore Statistical Physics II will be active and available in 2022/23.
For questions/discussion/comments, students are invited to contact the
teacher via email at the following address: francesco.ginelli@uninsubria.it
You are also welcomed to directly visit me in my office, but to be sure to
find me is always better to contact me by email beforehand.