00695 - Statistical Mechanics

Learning outcomes

At the end of the course, the student will:
- acquire knowledge of the statistical laws that govern the thermodynamic
behavior of the classical macroscopic systems consisting of a huge number
of molecules,
- learn the theory and applications of the Boltzmann and Gibbs
distributions and the foundations of physical kinetics,
- acquire knowledge of the statistical laws that govern the thermodynamic
behavior of macroscopic quantum systems consisting of a huge number of
particles,
- learn the theory and applications of statistics to Fermi-Dirac and Bose-Einstein.

Course contents

I semester



1.    Hamiltonian mechanics

1.1  Hamiltonian formulation of classical mechanics   

1.2  Canonical transformations

1.3  Liouville theorem

1.4  Hamiltonian flow  

1.5  Integrals of motion

1.6  Symmetries, canonical transformations and integrals of motion

1.7  External fields and generalized forces

1.8. Systems of identical and indistinguishable units

1.9  Open systems


2.    Classical statistical mechanics

2.1  Method of the statistical ensemble in classical mechanics

2.2  Phase space probability distribution

2.3  Statistical average, variance and distribution function of a phase function

2.4  Gibbs statistical entropy

2.5  Relation between entropy ed information

2.6  Irreversibility, attainment of equilibrium and entropy increase

2.7  Time dependence of the probability distribution, Liouville equation
        ed kinetic equation

2.8  Master equation, Fokker Planck equation and Langevin equation

2.9  H-theorem

2.10 Irreversibility and phase space cell decomposition

2.11 Boltzmann statistical entropy

2.12 Entropy maximization at equilibrium


3.    Classical equilibrium statistical mechanics

3.1  Probability distribution at equilibrium and integrals of motion

3.2  The ergodic problem

3.3  Statistical Independence and law of  large numbers

3.4  Gaussian distribution central limit theorem


4.   Classical probability distributions

4.1  Structure and partition function and their general properties

4.2  Energy equipartition and virial theorem

4.3  Adiabatic systems and the microcanonical distribution

4.4  Thermodynamic variables in the microcanonical distribution
 
4.5  Applications of the microcanonical distribution
 
4.6  Isothermal systems and the canonical distribution

4.7  Thermodynamic variables in the canonical distribution

4.8  Applications of the canonical distribution
 
4.10 Relation of the microcanonical and canonical distributions

4.11 Open systems and grandcanonic distribution

4.12 Thermodynamic variables in the grandcanonical distribution

4.13 Applications of the grandcanonical distribution



II semester



1.  The Gibbs ensemble

1.1 The Gibbs ensemble in quantum statistical mechanics

1.2 The Gibbs ensemble 

1.3 States and configurations of the Gibbs ensemble 

1.4 The statistical entropy of the Gibbs ensemble 

1.5 The most likely configuration of the Gibbs ensemble

1.6 The union of Gibbs ensembles


2.   Statistical thermodynamics

2.1 Canonical distribution 

2.2  Canonical theory of thermodynamics 

2.3  The free energy in canonical theory

2.4  Thermodynamics in canonical theory

2.5  Grandcanonical distribution

2.6  Grandcanonical theory of thermodynamics

2.7  The grandpotential in grandcanonical theory 

2.8  Thermodynamics in grandcanonical theory 

2.9  The Nernst theorem


3.   Quantum ideal gases

3.1  Quantum ideal gases

3.2  Quantum ideal gas with non conserved number of units

3.3  Quantum ideal gas with conserved number of units 

3.4  The saddle point method

3.5  Non degeneration and classical limit


4.   Ideal gases of quanta

4.1  Ideal gas of quanta

4.2  Ideal gas of quanta: thermodynamics

4.3  Ideal gas of quanta: Planck's energy distribution

4.4  Generalities on solids

4.5  Solids: low temperature regime

4.6  Solids: high temperature regime

4.7  Solids: Debye's interpolation method


5.    Ideal gases of quantum quasi–particles

5.1  Ideal gas of quantum quasi–particles

5.2  Ideal gas of quantum quasi–particles: thermodynamics

5.3  Ideal gas of quantum quasi–particles: non degeneration

5.4  Degenerate ideal Fermi–Dirac gas

5.5  Electron gas in solids

5.6  Degenerate ideal Bose–Einstein gas

5.7  Bose-Einstein degeneration

Readings/Bibliography

1) K. Huang, Statistical Mechanics (2nd Edition),
J. Wiley & sons

2) R. K. Pathria, Statistical Mechanics (2nd edition),
Butterworth-Heinemann

3) B. Touschek & G. Rossi, Meccanica Statistica,
Boringhieri

4) L. D. Landau & E. M. Lifshitz, Statistical Physics, Course of Theoretical Physics, Volume 5 (3rd edition),
Pergamon

Teaching methods

lectures and tutorials

Assessment methods

written and oral examination

Teaching tools

lecture notes in English

Office hours

See the website of Roberto Zucchini