# Fisica | STATISTICAL PHYSICS

## Fisica STATISTICAL PHYSICS

 0512600007 DIPARTIMENTO DI FISICA "E.R. CAIANIELLO" EQF6 PHYSICS 2018/2019

 OBBLIGATORIO YEAR OF COURSE 3 YEAR OF DIDACTIC SYSTEM 2016 ANNUALE
SSD CFU HOURS ACTIVITY TYPE OF ACTIVITY FIS/02 5 40 LESSONS SUPPLEMENTARY COMPULSORY SUBJECTS FIS/02 1 12 EXERCISES SUPPLEMENTARY COMPULSORY SUBJECTS
 GAETANO BUSIELLO
Objectives
TWO DISTINCTS COURSES ARE REFERRED TO HERE, “ANALYTICAL MECHANICS AND STATISTICAL MECHANICS” (12 CFU, SECOND YEAR), OFFERED FOR THE FIRST TIME IN THE ACADEMIC YEAR 2018-19, AND “STATISTICAL PHYSICS” (6 CFU, THIRD YEAR). REFERENCES TO EACH OF THE TWO COURSES ARE POINTED OUT IN THE FOLLOWING.

THE GENERAL AIMS OF THE COURSE OF “ANALYTICAL MECHANICS AND STATISTICAL MECHANICS” ARE:

A)PROVIDE AN INTRODUCTION TO ADVANCED FORMULATIONS OF CLASSICAL MECHANICS.
B)PROVIDE THE BASIC KNOWLEDGE OF THE FUNDAMENTAL STATISTICAL LAWS WHICH GOVERN THE BEHAVIOUR OF MANY-PARTICLE SYSTEMS.

THE LATTER ALSO REPRESENTS THE GENERAL AIM OF THE COURSE OF “STATISTICAL PHYSICS”.

KNOWLEDGE AND UNDERSTANDING:
THE COURSE OF “ANALYTICAL MECHANICS AND STATISTICAL MECHANICS” PROVIDES A DETAILED KNOWLEDGE, AT A THEORETICAL LEVEL AND IN A WAY SUITABLE TO APPLICATIONS, OF THE FOLLOWING TOPICS:
A)LAGRANGIAN AND HAMILTONIAN FORMALISMS AND THEORY OF HAMILTON-JACOBI;
B)PRINCIPLES AND METHODS OF STATISTICAL PHYSICS, WITH A SPECIAL ATTENTION TO THE THEORY OF THE STATISTICAL ENSEMBLES AND ITS APPLICATIONS.
IN BOTH FRAMEWORKS, ATTENTION WILL BE DEVOTED TO THE ANALYTICAL METHODS WHICH ALLOW THE SOLUTION OF SIMPLE MODELS, ALSO IN VIEW OF THE QUANTITATIVE DESCRIPTION OF SOME FUNDAMENTAL PHYSICAL PHENOMENA.

CONCERNING THE COURSE OF “STATISTICAL PHYSICS”, THE SAME KIND OF APPROACH APPLIES, WITH REFERENCE TO TOPICS B, EXTENDED WITH THE INCLUSION OF THE BASIC PRINCLIPLES OF THE QUANTUM STATISTICAL MECHANICS.

APPLYING KNOWLEDGE AND UNDERSTANDING:
THE COURSE IS EXPECTED TO LEAD TO A FULL COMPREHENSION OF THE THEORETICAL CONCEPTS UNDERLYING ANALYTICAL AND STATISTICAL MECHANICS, MAKING STUDENTS ABLE TO:
A)PRESENT LAGRANGE, HAMILTON AND HAMILTON-JACOBI METHODS AND APPLY THEM TO THE SOLUTION OF COMPLEX PROBLEMS CONCERNING STATICS AND DYNAMICS OF SYSTEMS OF PARTICLES AND RIGID BODIES IN THE PRESENCE OF CONSTRAINTS, WITH A SPECIAL ATTENTION TO THE ROLE PLAYED BY CONSERVATIONS LAWS.
B)PRESENT THE GENERAL THEORY OF THE STATISTICAL ENSEMBLES, WITH APPLICATION TO THE SOLUTION OF EXERCISES REQUIRING A SUITABLE CHOICE OF THE ENSEMBLE, WITH THE RELATED LINK BETWEEN THE MICROSCOPICAL APPROACH AND THE MACROSCOPIC LAWS OF THERMODYNAMICS.

AS FAR AS THE COURSE OF “STATISTICAL PHYSICS” IS CONCERNED, THE STUDENT IS EXPECTED TO ACQUIRE THE ABILITY OF SUITABLY PRESENTING THE THEORETICAL CONCEPTS UNDERLYING STATISTICAL MECHANICS, ALSO WITH REFERENCE TO QUANTUM SYSTEMS, AND APPLYING THE ACQUIRED KNOWLEDGE TO THE SOLUTION OF SIMPLE EXERCISES, ACCORDING TO WHAT SPECIFIED ABOVE AT POINT B.
Prerequisites
AS FAR AS THE COURSE OF "ANALYTICAL MECHANICS AND STATISTICAL MECHANICS" IS CONCERNED, THE MINIMAL BACKGROUND INCLUDES THE KNOWLEDGE OF THE VECTOR CALCULUS AND THE NEWTONIAN MECHANICS (KINEMATICS, DYNAMICS OF A SINGLE PARTICLE, DYNAMICS OF A SYSTEM OF PARTICLES AND OF RIGID BODIES). CONCERNING MATHEMATICAL ASPECTS, STUDENTS ARE EXPECTED TO KNOW THE LINEAR ALGEBRA AND THE THEORY OF THE FUNCTIONS OF A SINGLE VARIABLE, TOGETHER WITH SOME ELEMENTS OF THE THEORY OF THE FUNCTIONS OF SEVERAL VARIABLES (PARTIAL DERIVATIVES AND DIFFERENTIALS).

THE BASIC KNOWLEDGE REQUIRED FOR THE COURSE OF "STATISTICAL PHYSICS" COVERS THE CONTENT OF THE FIRST AND SECOND YEAR COURSES OF GENERAL PHYSICS AND MATHEMATICAL ANALYSIS, WITH A SPECIAL ATTENTION TO THE THEORY OF THE FUNCTIONS OF SEVERAL VARIABLES, THE THERMODYNAMICS AND THE ANALYTICAL MECHANICS.
Contents
ARGUMENTS OF ANALYTICAL MECHANICS

1) LAGRANGIAN DYNAMICS (LECT. H. 18, EXERC. H. 5):
NEWTON’S LAWS. CONSTRAINTS, DEGREES OF FREEDOM AND GENERALIZED COORDINATES. D’ALEMBERT’S PRINCIPLE AND PRINCIPLE OF VIRTUAL WORKS. LAGRANGE’S EQUATIONS OF MOTION WITH APPLICATIONS. GENERALIZED POTENTIALS: MOTION OF A CHARGED PARTICLE IN AN ELECTROMAGNETIC FIELD. CYCLIC COORDINATES. SYMMETRY PROPERTIES AND CONSERVATIONS LAWS: NOETHER’S THEOREM. HAMILTON’S PRINCIPLE AND ITS EQUIVALENCE WITH LAGRANGE’S EQUATIONS. APPLICATIONS OF THE CALCULUS OF VARIATIONS (BRACHISTOCRONE, GEODETICS, MINIMAL SURFACE OF REVOLUTION). CALCULUS OF VARIATIONS WITH CONSTRAINTS. SUSPENDED CHAIN PROBLEM. SYSTEMS WITH NON-HOLONOMIC CONTRAINTS. LAGRANGE’S MULTIPLIERS.

2) COUPLED OSCILLATION (LECT. H. 6, EXERC. H. 3):
STABILITY OF THE EQUILIBRIUM POSITIONS. EXPANSION OF THE INTERACTION POTENTIAL AND LAGRANGIAN. EIGENVALUE PROBLEM. SIMULTANEOUS DIAGONALIZATION OF THE KINETIC AND THE POTENTIAL ENERGY AS QUADRATIC FORMS. CHARACTERISTIC FREQUENCIES AND NORMAL MODES OF OSCILLATION. LINEAR TRIATOMIC MOLECULE.

3) MOTION UNDER CENTRAL FORCES (LECT. H. 4, EXERC. H. 1):
TWO-BODY PROBLEM AND REDUCTION TO A ONE-BODY PROBLEM IN THE PRESENCE OF A CENTRAL FORCE. EQUATIONS OF MOTION AND SOLUTION VIA CONSERVATION LAWS. EFFECTIVE POTENTIAL AND ORBIT CLASSIFICATION. ORBIT DIFFERENTIAL EQUATION AND INTEGRABLE POTENTIALS SHOWING A POWER-LAW DEPENDENCE ON THE DISTANCE. KEPLER’S PROBLEM: FORCES GOING AS THE INVERSE OF THE SQUARE OF THE DISTANCE.

4) HAMILTONIAN DYNAMICS (LECT. H. 13, EXERC. H. 3):
LEGENDRE’S TRANSFORMATION AND HAMILTON’S EQUATIONS OF MOTIONS. CYCLIC COORDINATES AND ROUTH’S METHOD. THEOREMS ON CONSERVATION LAWS AND PHYSICAL INTERPRETATION OF THE HAMILTONIAN FUNCTION. POISSON BRACKETS, JACOBI IDENTITY AND JACOBI-POISSON’S THEOREM. QUALITATIVE DESCRIPTION OF THE DERIVATION FROM THE HAMILTONIAN FORMALISM OF THE HEISENBERG FORMULATION OF QUANTUM MECHANICS. DERIVATION OF HAMILTON’S EQUATIONS FROM A VARIATIONAL PRINCIPLE. CANONICAL TRANSFORMATIONS. GENERATING FUNCTION AND CONDITIONS OF CANONICITY. SYMPLECTIC APPROACH TO CANONICAL TRANSFORMATIONS.

5) HAMILTON-JACOBI’S METHOD (LECT. H. 6, EXERC. H. 1):
H-J EQUATION AND HAMILTON’S PRINCIPAL FUNCTION. REDUCED H-J EQUATION FOR TIME INDEPENDENT HAMILTONIANS AND HAMILTON’S CHARACTERISTIC FUNCTION USED AS GENERATING FUNCTION. SOLUTION OF THE H-J EQUATION IN THE PRESENCE OF CYCLIC COORDINATES AND METHOD OF SEPARATION OF THE VARIABLES, WITH APPLICATION TO RELEVANT PROBLEMS (HARMONIC OSCILLATOR, MOTION IN THE PRESENCE OF CENTRAL FORCES, ETC.). ACTION-ANGLE VARIABLES

ARGUMENTS OF STATISTICAL MECHANICS

6) BASIC ELEMENTS OF THEORY OF PROBABILITY AND CENTRAL LIMIT THEOREM (2 H)

7) THERMODYNAMICS (8 H):
FIRST AND SECOND PRINCIPLE. ENERGY AND ENTROPY REPRESENTATIONS. STATE EQUATIONS. LEGENDRE TRANSFORMATION AND THERMODYNAMIC POTENTIALS.

8) INTRODUCTION TO STATISTICAL MECHANICS (4 H):
PHASE SPACE. STATISTICAL AND TEMPORAL AVERAGES. ERGODICITY. LIOUVILLE THEOREM

9) FIXED ENERGY SYSTEMS (8 H):
MICROCANONICAL ENSEMBLE. MICROSCOPIC INTERPRETATINO OF THE ENTROPY. TWO-LEVEL SYSTEMS. FREE PARTICLE GAS. GIBBS PARADOX. THEOREM OF THE ENERGY EQUIPARTITION.

10) FIXED TEMPERATURE SYSTEMS (14 H):
CANONICAL ENSEMBLE. THE PARTITION FUNCTION AND ITS CONNECTION WITH HELMHOLTZ FREE ENERGY. APPLICATIONS: TWO-LEVEL SYSTEMS, IDEAL CLASSICAL GAS, INDEPENDENT HARMONIC OSCILLATOR SYSTEMS. CLASSICAL PARAMAGNET.

11) SYSTEMS WITH FIXED TEMPERATURE AND CHEMICAL POTENTIAL (6 H):
GRAND-CANONICAL ENSEMBLE. GRAN-PARTITION FUNCTION AND ITS CONNECTION WITH THE GRAND-CANONICAL POTENTIAL.

ARGUMENTS 1-11 DEFINE THE CONTENT OF THE COURSE OF "ANALYTICAL MECHANICS AND STATISTICAL MECHANICS".

ARGUMENTS 6-12 DEFINE THE CONTENT OF THE COURSE OF "STATISTICAL PHYSICS".
Teaching Methods
ABOUT TWO THIRDS OF THE COURSE OF “ANALYTICAL MECHANICS AND STATISTICAL MECHANICS” ARE DEVOTED TO THEORETICAL LECTURES FOCUSING ON THE PRESENTATION OF THE LAGRANGIAN AND HAMILTONIAN FORMULATIONS OF CLASSICAL MECHANICS, AS WELL AS OF THE PRINCIPLES AND METHODS OF CLASSICAL STATISTICAL MECHANICS.

SIMILARLY, ABOUT TWO THIRDS OF THE COURSE OF “STATISTICAL PHYSICS” ARE DEVOTED TO THEORETICAL LECTURES FOCUSING ON THE PRESENTATION OF THE PRINCIPLES AND METHODS OF CLASSICAL STATISTICAL MECHANICS, AS WELL AS OF THE BASIC NOTIONS OF QUANTUM STATISTICAL MECHANICS.

THE REMAINING PARTS OF THE TWO COURSES CONSISTS OF LECTURES FOCUSED ON THE APPLICATION OF SUITABLE METHODS OF SOLUTION TO EXERCISES AND PROBLEMS RELATED TO THE TOPICS THEORETICALLY INVESTIGATED.
Verification of learning
AS FAR AS THE COURSE OF “ANALYTICAL MECHANICS AND STATISTICAL MECHANICS” IS CONCERNED, THE ASSESSMENT OF THE LEVEL OF STUDENT'S LEARNING IS MADE THROUGH A FINAL EXAMINATION CONSISTING, FOR EACH OF THE TWO PARTS OF THE COURSE, OF A WRITTEN TEST FOLLOWED BY AN ORAL DISCUSSION.

THE WRITTEN TEST IN ANALYTICAL MECHANICS CONSISTS IN DETERMINING THE SOLUTION OF TWO EXERCISES ON LAGRANGIAN AND HAMILTONIAN DYNAMICS, AIMING TO EVALUATE STUDENT'S ABILITY OF SUITABLY OPERATING IN BOTH CONTEXTS. THE MAXIMUM SCORES FOR EACH OF THE EXERCISES ARE 18/30 AND 12/30, RESPECTIVELY. IF THE TOTAL SCORE IS NOT LOWER THAN 18/30, STUDENTS ARE ALLOWED TO TAKE THE ORAL EXAMINATION.

THE WRITTEN TEST IN STATISTICAL MECHANICS CONSISTS IN THE SOLUTION OF SEVERAL IN-ITINERE TESTS, EACH REQUIRING THE SOLUTION OF TWO OR THREE EXERCISES, FOCUSING ON DISTINCT PORTIONS OF THE COURSE PROGRAMME. IF THE TOTAL SCORE, AVERAGED OVER ALL THE TESTS, IS NOT LOWER THAN 18/30, STUDENTS ARE ALLOWED TO TAKE THE ORAL EXAMINATION. BELOW THIS THRESHOLD VALUE, STUDENTS ARE REQUIRED TO PASS A SINGLE, GLOBAL WRITTEN TEST FOCUSING ON ALL THE TOPICS THEORETICALLY ANALYZED IN THE COURSE.

THE FINAL GRADE COMES FROM THE AVERAGE OF THE GRADES OBTAINED IN THE FOUR PARTS OF THE EXAMINATION, WEIGHTED BY THE NUMBER OF CFU ASSOCIATED TO THE TWO PARTS OF THE COURSE (7 FOR THE ANALYTICAL MECHANICS, 5 FOR THE STATISTICAL MECHANICS). A BONUS UP TO A MAXIMUM OF 3 POINTS CAN BE ADDED, DEPENDING ON HOW CLEARLY THE ARGUMENTS ARE SET OUT IN THE ORAL EXAMINATIONS.

THE MINIMUM GRADE OF 18/30 IS OBTAINED WHEN THE STUDENT EXHIBITS A SUFFICIENT KNOWLEDGE OF THE PROCEDURE OF ELIMINATION OF THE CONSTRAINTS LEADING TO THE LAGRANGIAN FORMALISM AND TO THE SUBSEQUENT INTRODUCTION, VIA LEGENDRE TRANSFORMATION, OF THE HAMILTONIAN FORMALISM. A SUFFICIENT KNOWLEDGE OF THE GENERAL SCHEME USED IN STATISTICAL PHYSICS, BASED ON THE INTRODUCTION OF THE STATISTICAL ENSEMBLES, IS ALSO REQUIRED, TOGETHER WITH ITS APPLICATION TO THE STUDY OF SIMPLE CLASSICAL SYSTEMS.

THE MAXIMUM GRADE CUM LAUDE REQUIRES A DEEP AND EXTENDED KNOWLEDGE OF THE THEORY AND THE APPLICATIONS OF ALL THE ARGUMENTS OF THE COURSE, TOGETHER WITH THE ABILITY OF APPLYING THE RELATED PRINCIPLES AND METHODS ALSO WITHIN CONTEXTS GOING BEYOND THE ONES INVESTIGATED IN THE COURSE.
Texts
ANALYTICAL MECHANICS:
- H. GOLDSTEIN, C. P. POOLE, J. L. SAFKO, “MECCANICA CLASSICA”, ZANICHELLI
- F.R. GANTMACHER, “LEZIONI DI MECCANICA ANALITICA”, EDITORI RIUNITI
- L. LANDAU, E. LIFSHITZ, “MECCANICA“, EDITORI RIUNITI

STATISTICAL MECHANICS:
- K. HUANG, “MECCANICA STATISTICA” , ZANICHELLI
- L.D.LANDAU, E.M.LIFSITS, L.P.PITAEVSKIJ, “FISICA STATISTICA I“ ED. RIUNITI
- L.E.REICHL, “A MODERN COURSE IN STATISTICAL PHYSICS” EDWARD ARNOLD PUBLISHERS LTD
- LECTURE NOTES