Bachelor in Physics and knowledge of basic statistical mechanics.
What do a magnet, a Bose-Einstein condensate and a flock of birds have in common? All these systems exhibit a collective behavior and have large-scale physical properties that cannot be understood in terms of a simple extrapolation of the properties of a few particles. Conversely, systems comprising many interacting subunits often present entirely new properties, that scientists refer to as emergent.
Statistical Physics A, is the first part of a two-part introductory course on emergent phenomena in equilibrium and non-equilibrium systems. The course provides an introduction to phase transitions and critical phenomena at equilibrium. The second (elective) part of this course is given in Statistical Physics B and is focused on emergent phenomena in non-equilibrium systems (i.e. collective motion in animals and other biological systems etc.)
The course consists of 10 lectures and 5 tutorials (exercise classes), during which the students will practice the concepts and the techniques learned during the lectures by solving problems.
Introduction to phase transitions and critical phenomena in statistical mechanics.
The one-dimensional Ising model: exact solution via transfer matrix method.
The two-dimensional Ising model: domain walls and Peierls’ argument.
Mean field theory.
Fluctuations theory and field-theoretical approach to critical phenomena.
Universality, scaling and critical dimensions.
Real-space renormalization group.
Momentum-shell renormalization group and ε-expansion.
The aim of the course is to develop a strong foundation in advanced statistical mechanics with an emphasis on emergent phenomena. Furthermore, the course aims to provide the students with a toolbox of mathematical techniques that can be readily used in theoretical and experimental research projects.
Specifically, at the end of the course, successful students will have learned how to:
Solve 1-dimensional spin models and calculate quantities of physical interests such as the average magnetization, magnetic susceptibility, energy, entropy etc.
Set up a mean-field theory for the Ising model and analogous models and use it to calculate the critical exponents.
Recognize the scale-invariance and how it connects with second order phase transitions.
Construct a field theory for magnetism by perfuming the continuum limit of the mean-field Ising model and use it to calculate the spin-spin correlation function and other relevant quantities.
Apply Wilson’s renormalization group, in real and Fourier space, to go beyond mean-field and accurately describe the critical behavior of ferromagnets.
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Mode of instruction
Final exam with open questions and homework assignments for bonus points.
Nigel Goldenfeld, Lectures of phase transitions and the renormalization group (Perseus Books, 1992).
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Lecturer: Dr. Luca Giomi
At the end of the course, students will have been trained how to:
Decompose a complex concept into simpler steps, each of which can be rationalized at an intuitive level.
Think across individual academic disciplines and use analogies with other fields (e.g. computer graphics) to build up physical intuition.
Not to stop after the first hints of success, but aim at greater goals.