D1 - Knowledge and understanding: The course will provide an overview of the most important aspects of phase transitions and critical phenomena in thermodynamic systems and their description using modern statistical mechanics techniques. D2 – Ability to apply knowledge and understanding: By the end of the course, students will be able to independently use the thermodynamic formalism and basic theories to describe phase transitions and critical phenomena. D3 - Judgement autonomy: Upon completion of the course, students will have developed the ability to evaluate the suitability of statistical mechanics theories for describing phase transitions. D4 – Communication skills: Students will achieve sufficient proficiency in using technical language. Furthermore, they will have developed the ability to describe the relevant aspects and limitations of theories and approximations. D5 - Learning ability: By the end of the course, students will be able to access relevant literature and autonomously extend their knowledge.

Fundamental concepts of thermodynamics and statistical physics.

RECAP OF THERMODYNAMICS. Thermodynamic potentials. Legendre transforms. Thermodynamics of phase transitions. Classification of phase transitions. Existence of the thermodynamic limit. CRITICAL PHENOMENA. Van der Waals equation for the gas-liquid critical point. Ising model for the paramagnet-ferromagnet transition. Spontaneous symmetry breaking. Critical exponents. CLASSICAL THEORIES. Mean-field approximation. Landau theory for second-order phase transitions. Functional formulation of the partition function for the Ising model. CRITICAL PHENOMENA AND MODERN THEORIES: THE RENORMALIZATION GROUP. Universality and scaling theory. Scaling laws and relations among the critical exponents. Block variable transformations and general properties of the renormalization group transformation. Wilsonian renormalization. Calculation of critical exponents.

N. Goldenfeld, Lectures on Phase transitions and the Renormalization Group. H.E. Stanley, Introduction to Phase Transitions and Critical Phenomena. S.-K. Ma, Modern Theory of Critical Phenomena.

RECAP OF THERMODYNAMICS. Thermodynamic potentials. Legendre transforms. Thermodynamics of phase transitions. Classification of phase transitions. Existence of the thermodynamic limit. CRITICAL PHENOMENA. Van der Waals equation for the gas-liquid critical point. Ising model for the paramagnet-ferromagnet transition. Spontaneous symmetry breaking. Critical exponents. CLASSICAL THEORIES. Mean-field approximation. Landau theory for second-order phase transitions. Functional formulation of the partition function for the Ising model. CRITICAL PHENOMENA AND MODERN THEORIES: THE RENORMALIZATION GROUP. Universality and scaling theory. Scaling laws and relations among the critical exponents. Block variable transformations and general properties of the renormalization group transformation. Wilsonian renormalization. Calculation of critical exponents.

Lectures will have a theoretical aspect. The student's active participation will be stimulated by simple exercises.

Oral exam (two questions) on the course syllabus. The final grade will reflect the student's grasp of the physical concepts and theoretical tools presented in the course.