International Teaching | STATISTICAL MECHANICS

cod. 0522600016

STATISTICAL MECHANICS

	0522600016
	DEPARTMENT OF PHYSICS "E. R. CAIANIELLO"
	EQF7
	PHYSICS
	2024/2025

PERCORSO SHARED STUDY PATHWAY

	YEAR OF COURSE 1
	YEAR OF DIDACTIC SYSTEM 2021
	SPRING SEMESTER

TEACHING UNITS

SSD	CFU	HOURS	ACTIVITY	TYPE OF ACTIVITY
FIS/02	4	32	LESSONS	SUPPLEMENTARY COMPULSORY SUBJECTS
FIS/02	2	24	EXERCISES	SUPPLEMENTARY COMPULSORY SUBJECTS

	Objectives
	THE COURSE AIMS TO EXTEND THE BASIC KNOWLEDGE OF STATISTICAL MECHANICS ACQUIRED IN THE THREE-YEARS DEGREE TO ADVANCED PROBLEMS AS INTERACTING SYSTEMS, PHASE TRANSITIONS AND NON-EQUILIBRIUM SYSTEMS. KNOWLEDGE AND COMPREHENSION: THE COURSE FURNISHES THE NECESSARY TOOLS TO THE STUDENT TO DEEPLY UNDERSTAND IMPORTANT AND SOPHISTICATED CONCEPTS AS SYMMETRY BREAKING, SCALE INVARIANCE, SELF SIMILARITY, TIME REVERSAL INVARIANCE AND OTHERS. APPLYING KNOWLEDGE AND UNDERSTANDING: THE AIM OF THE COURSE IS TO PRESENT TO THE STUDENT SOLUTIONS TO COMPLEX STATISTICAL MECHANICAL PROBLEMS, SUCH AS THOSE OBTAINED IN THE STUDY OF PHASE TRANSITIONS BY THE SOPHISTICATED TECHNIQUE OF THE RENORMALIZATION GROUP, THUS PROVIDING HIM WITH THE ABILITY TO SPOT THE SOURCES OF COMPLEXITY AND TO AUTONOMOUSLY DEAL WITH DIFFICULT PROBLEMS IN AN ORIGINAL WAY.

THE COURSE AIMS TO EXTEND THE BASIC KNOWLEDGE OF STATISTICAL MECHANICS ACQUIRED IN THE THREE-YEARS DEGREE TO ADVANCED PROBLEMS AS INTERACTING SYSTEMS, PHASE TRANSITIONS AND NON-EQUILIBRIUM SYSTEMS.

KNOWLEDGE AND COMPREHENSION:
THE COURSE FURNISHES THE NECESSARY TOOLS TO THE STUDENT TO DEEPLY UNDERSTAND IMPORTANT AND SOPHISTICATED CONCEPTS AS SYMMETRY BREAKING, SCALE INVARIANCE, SELF SIMILARITY, TIME REVERSAL INVARIANCE AND OTHERS.

APPLYING KNOWLEDGE AND UNDERSTANDING:
THE AIM OF THE COURSE IS TO PRESENT TO THE STUDENT SOLUTIONS TO COMPLEX STATISTICAL MECHANICAL PROBLEMS, SUCH AS THOSE OBTAINED IN THE STUDY OF PHASE TRANSITIONS BY THE SOPHISTICATED TECHNIQUE OF THE RENORMALIZATION GROUP, THUS PROVIDING HIM WITH THE ABILITY TO SPOT THE SOURCES OF COMPLEXITY AND TO AUTONOMOUSLY DEAL WITH DIFFICULT PROBLEMS IN AN ORIGINAL WAY.

	Prerequisites
	GENERALITIES OF EQUILIBRIUM STATISTICAL MECHANICS. ENSEMBLE THEORIES ANS IDENTIFICATION OF THERMODYNAMIC QUANTITIES.NON-INTERACTING CLASSICAL AND QUANTUM SYSTEMS (CLASSICAL PERFECT GAS, BOSON GAS, PARAMAGNET ETC...)

	Contents
	INTERACTING SYSTEMS. (36 HOURS, 5 OF WHICH ARE EXERCISES) PHASE TRANSITIONS. CONDENSATION TRANSITION. URNS MODELS. BOSE-EINSTEIN CONDENSATION. PERCOLATION. PERCOLATION IN ONE DIMENSION. PERCOLATION ON THE BETHE LATTICE. GAS-LIQUID TRANSITION. VAN DER WAALS THEORY. FERRO-PARAMAGNETIC TRANSITION. CRITICAL EXPONENTS. ISING MODEL. OTHER SYSTEMS DESCRIBED BY THE ISING MODEL: GAS-LATTICE, BINARY MIXTURES, ANTIFERROMAGNETS ETC... SPONTANEOUS SYMMETRY BREAKING. FLUCTUATION-DISSIPATION THEOREM (STATIC). MEAN-FIELD THEORIES (LANDAU, WEISS, VAN DER WAALS). CORRELATION FUNCTIONS. ORNSTEIN-ZERNIKE THEORY. LIMIT OF VALIDITY OF MEAN FIELD: GINZBURG CRITERION. EXACTLY SOLUBLE MODELS: 1D ISING MODEL, SPHERICAL MODEL. SCALING THEORY. UNIVERSALITY. BLOCK TRANSFORMATIONS AND RENORMALIZATION GROUP IN REAL SPACE AND A LA WILSON.DISORDERED SYSTEMS. QUENCHED AND ANNEALED AVERAGES. DYNAMICS. (20 HOURS, 3 OF WHICH ARE EXERCISES) HYDRODYNAMIC APPROACH. LANGEVIN EQUATION. FLUCTUATION-DISSIPATION THEOREM (TIME DEPENDENT). MASTER EQUATION. DINAMICS O A PARAMAGNET AND OF A FERROMAGNET IN THE MEAN-FIELD APPROXIMATION. FOKKER-PLANCK EQUATION. BROWNIAN OSCILLATOR. DYNAMICS OF THE SPHERICAL MODEL. BOLTZMANN EQUATION. H-THEOREM. SPECTRAL ANALYSIS AND THE WIENER-KINTCHIN THEOREM. FLUCTUATION THEOREMS.

Contents

INTERACTING SYSTEMS. (36 HOURS, 5 OF WHICH ARE EXERCISES)

PHASE TRANSITIONS. CONDENSATION TRANSITION. URNS MODELS. BOSE-EINSTEIN CONDENSATION. PERCOLATION. PERCOLATION IN ONE DIMENSION. PERCOLATION ON THE BETHE LATTICE. GAS-LIQUID TRANSITION. VAN DER WAALS THEORY. FERRO-PARAMAGNETIC TRANSITION. CRITICAL EXPONENTS. ISING MODEL. OTHER SYSTEMS DESCRIBED BY THE ISING MODEL: GAS-LATTICE, BINARY MIXTURES, ANTIFERROMAGNETS ETC... SPONTANEOUS SYMMETRY BREAKING. FLUCTUATION-DISSIPATION THEOREM (STATIC). MEAN-FIELD THEORIES (LANDAU, WEISS, VAN DER WAALS). CORRELATION FUNCTIONS. ORNSTEIN-ZERNIKE THEORY. LIMIT OF VALIDITY OF MEAN FIELD: GINZBURG CRITERION. EXACTLY SOLUBLE MODELS: 1D ISING MODEL, SPHERICAL MODEL. SCALING THEORY. UNIVERSALITY. BLOCK TRANSFORMATIONS AND RENORMALIZATION GROUP IN REAL SPACE AND A LA WILSON.DISORDERED SYSTEMS. QUENCHED AND ANNEALED AVERAGES.

DYNAMICS. (20 HOURS, 3 OF WHICH ARE EXERCISES)

HYDRODYNAMIC APPROACH. LANGEVIN EQUATION. FLUCTUATION-DISSIPATION THEOREM (TIME DEPENDENT). MASTER EQUATION. DINAMICS O A PARAMAGNET AND OF A FERROMAGNET IN THE MEAN-FIELD APPROXIMATION. FOKKER-PLANCK EQUATION. BROWNIAN OSCILLATOR. DYNAMICS OF THE SPHERICAL MODEL. BOLTZMANN EQUATION. H-THEOREM. SPECTRAL ANALYSIS AND THE WIENER-KINTCHIN THEOREM. FLUCTUATION THEOREMS.

	Teaching Methods
	THEORETICAL LESSONS WITH EXERCISES ON THE BLACKBOARD SOLVED BY THE TEACHER. A TWO-HOURS LESSON DEDICATED TO THE STUDY OF STATISTICAL MODELS WITH THE COMPUTER.

	Verification of learning
	ORAL EXAM AT THE END OF THE COURSE, WHOSE DURATION WILL BE ONE HOUR AT LEAST, WITH QUESTIONS CONCERNING ALL THE MAIN ARGUMENTS OF THE COURSE. IN PARTICULAR, THE STUDENT WILL BE CALLED TO DEMONSTRATE HOW DEEP IS HIS LEVEL OF UNDERSTANDING OF THE VARIOUS CONCEPTS AND PROBLEMS. THE STUDENT IS ASKED TO BE ABLE TO MAKE CONNECTIONS BETWEEN DIFFERENT PARTS OF THE PROGRAM AND TO SOLVE EXTEMPORANEOUS PROBLEMS. THESE ASPECTS ARE PARTICULARLY CONSIDERED FOR THE FINAL SCORE, BESIDES THE USUAL VERIFICATION OF THE KNOWLEDGE OF THE MATTER AND THE ABILITY TO PRESENT IT. THE MINIMUM MARK (18) IS EARNED BY A STUDENT WHO STUDIED MOST OF THE TOPICS OF THE COURSE BUT WHOSE COMPREHENSION IS NOT PARTICULARLY DEEPENED AND WHOSE CAPACITY TO CONNECT DIFFERENT PARTS IS NOT WELL DEVELOPED. THE MAXIMUM MARK (30) IS GIVEN TO STUDENTS WITH A TOTAL AND DEEP CONTROL OF THE ARGUMENTS OF THE COURSE. THE LAUDE IS GIVEN WHEN THESE CAPACITY IS OWNED AT AN OUTSTANDING LEVEL. .

Verification of learning

ORAL EXAM AT THE END OF THE COURSE, WHOSE DURATION WILL BE ONE HOUR AT LEAST, WITH QUESTIONS CONCERNING ALL THE MAIN ARGUMENTS OF THE COURSE. IN PARTICULAR, THE STUDENT WILL BE CALLED TO DEMONSTRATE HOW DEEP IS HIS LEVEL OF UNDERSTANDING OF THE VARIOUS CONCEPTS AND PROBLEMS. THE STUDENT IS ASKED TO BE ABLE TO MAKE CONNECTIONS BETWEEN DIFFERENT PARTS OF THE PROGRAM AND TO SOLVE EXTEMPORANEOUS PROBLEMS. THESE ASPECTS ARE PARTICULARLY CONSIDERED FOR THE FINAL SCORE, BESIDES THE USUAL VERIFICATION OF THE KNOWLEDGE OF THE MATTER AND THE ABILITY TO PRESENT IT. THE MINIMUM MARK (18) IS EARNED BY A STUDENT WHO STUDIED MOST OF THE TOPICS OF THE COURSE BUT WHOSE COMPREHENSION IS NOT PARTICULARLY DEEPENED AND WHOSE CAPACITY TO CONNECT DIFFERENT PARTS IS NOT WELL DEVELOPED. THE MAXIMUM MARK (30) IS GIVEN TO STUDENTS WITH A TOTAL AND DEEP CONTROL OF THE ARGUMENTS OF THE COURSE. THE LAUDE IS GIVEN WHEN THESE CAPACITY IS OWNED AT AN OUTSTANDING LEVEL. .

	Texts
	NOTES BY THE TEACHER.THEY ARE DISTRIBUTED IN THE CLASS OR SHARED ON THE INTERNET WITH THE STUDENTS AT THE BEGINNING OF THE COURSE. OTHER TEXTBOOKS FOR CONSULTATION AND DEEPENING SOME SUBJECTS: 1) R.K.PATHRIA: "STATISTICAL MECHANICS" 2) K.HUANG: "STATISTICAL MECHANICS" 3) L.PELITI: "APPUNTI DI MECCANICA STATISTICA"

	More Information
	STUDENTS ARE ENCOURAGED TO KEEP IN CONTACT WITH THE TEACHER AT ANY TIME.

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International Teaching | STATISTICAL MECHANICS

STATISTICAL MECHANICS

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International Teaching | STATISTICAL MECHANICS