55229 - Complements of Statistical Mechanics

Learning outcomes

The Student is expected to acquire the knowledge of
the probabilistic laws that govern the behavior of
macroscopic systems formed by a very large number of
molecules on the basis of classical and quantum mechanics.
The student is expected to learn the properties of the basic
classical and quantum macroscopic systems, the applications
of the Boltzmann, Gibbs, Fermi-Dirac and Bose-Einstein and
the foundations of kinetic physics.

Course contents

I semester

0.1 Basic notions of thermodynamics

0.2 Thermodynamic states and transformations and thermodynamic equilibrium

0.3 Thermal equilibrium and temperature

0.4 Heat, work, internal energy and the 1st law of thermodynamics

0.5 The 2nd law of thermodynamics, absolute temperature and entropy

0.6 The 3rd law of thermodynamics,

0.7 Relevant thermodynamic systems

0.8 Thermodynamic relations and inequalities


1.1 Hamiltonian formulation of classical mechanics

1.2 Canonical transformations

1.3 Liouville theorem for canonical transformations

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


2.4 Distribution function of a phase function

2.5 Statistical independence and law of large numbers

2.6 Gaussian distribution and central limit theorem

2.7 Statistical entropy

2.8 Relation of entropy and information


3.1 Time dependence of the distribution function

3.2 Liouville equation for the distribution function

3.3 Equilibrium distribution function and integrals of motion

3.4 The ergodic problem


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

5.1 Law of entropy increase in classical statistical mechanics

5.2 Foundation of kinetic physics

5.3 The BBGKY hierarchy

5.4 Boltzmann's transport equation

5.5 Boltzmann's H theorem

5.6 Transport phenomena


6.1 Quantum statistical mechanics

6.2 Quantum systems with many degrees of freedom

6.3 Ensembles, states and configurations

6.4 Statistical entropy of a configuration

6.5 Principle of maximal entropy

6.6 General derivation of Gibbs' statistics


7.1 Ideal gases

7.2 Boltzmann's distribution

7.3 State equation

7.4 Monoatomic ideal gases

7.5 Polyatomic ideal gases, rotational and vibrational degrees of freedom

7.6 Diatomic ideal gases with molecules of different atoms

7.7 Diatomic ideal gases with molecules of identical atoms

7.8 Magnetism of ideal gases


8.1 The Fermi-Dirac distribution

8.2 the Bose-Einstein distribution

8.3 Fermi and Bose gases of elementary particles

8.4 Degenerated Fermi gas

8.5 The specific heat of a degenerated Fermi gas

8.6 Magnetism of a gas of electrons

8.7 Degenerated Bose gas

8.8. Black body radiation


9.1 Solids at low temperatures

9.2 Solids at high temperatures

9.3 Debye's interpolation formula

9.4 Thermal expansion of solids

9.5 Crystal lattices and their vibrations, phonons

9.6 Phonon gas

Readings/Bibliography

Khinchin, A. I., Mathematical Foundations of Statistical Mechanics
(Dover, New York USA).

Landau, L. D., Lifshitz, E. M., Statistical Physics
(Pergamon, Oxford UK).

Huang, K., Statistical Mechanics
(J.Wiley & sons, USA).

Touschek B., Rossi G-C., Meccanica Statistica
(Boringhieri, Italia)

Teaching methods

blackboard lectures

Assessment methods

Written and oral exam

Teaching tools

Slide projectors

Office hours

See the website of Roberto Zucchini