ADVANCED STATISTICAL MECHANICS

The course aims at the understanding of the thermodynamic properties of macroscopic systems on the basis of statistical-dynamical behavior of their microscopic constituents.

In particular the objectives of the course are:

Critical understanding of the most advanced developments of Modern Physics, both theoretical and experimental, and their interrelations, also across different subjects.

Adequate knowledge of advanced mathematical and numerical tools, currently used in both basic and applied research.

Remarkable acquaintance with the scientific method, understanding of nature, and of the research in Physics.

Ability to identify the essential elements in a phenomenon, in terms of orders of magnitude and approximation level, and being able to perform the required approximations

Ability to use analytical and numerical tools, or science computing, including the development of specific software.

Ability to discuss about advanced physical concepts, both in Italian and in English.

Ability to present one's own research activity or a review topic both to an expert and to an non-expert audience.

Ability to acquire adequate tools for the continuous update of one's knowledge.

Ability to access to specialized literature both in the specific field of one's expertise, and in closely related fields.

Lectures and excercises in the classroom

None

Compulsory

Principles of Thermodynamics. Thermodynamic equilibrium. Thermodynamic Potentials. Kinetic Theory. H theorem of Boltzmann. Maxwell-Boltzmann distribution. Ensemble theory of Gibbs. Classical Statistical Mechanics: Phase space. Liouville's theorem. Principle of a priori equiprobability. Microcanonical ensemble. Virial theorem. Equipartition of energy. Classical ideal gas. Derivation of thermodynamics for almost isolated systems. Gibbs paradox. System in contact with a thermostat. Statistical concept of temperature. Canonical ensemble. Energy fluctuations. Systems with variable number of particles. Chemical potential. Grand-canonical ensemble. Fluctuations in density in open systems. Gibbs paradox and correct counting of microscopic states. Postulates of quantum statistical mechanics. Density matrix. Quantum Liouville equation. Formulation of the quantum theory of Gibbs ensemble. Third Law of Thermodynamics. Ideal gas of Fermi and Bose. Bose-Einstein condensation and superfluid systems. Electromagnetic excitations in a cavity. Thermal excitations in solids. Statistical equilibrium in white dwarf stars. Electron gas in metals. Low-temperature behavior of Bose and Fermi of a weakly imperfect gas. Elementary excitations in helium liquid. Classical interacting systems. Development cluster for a classic real gas. Development of the virial equation of state of a perfect gas. Derivation of Van der Waals forces. Phase transitions and critical phenomena. Critical indices and scale invariance. The Ising model for ferromagnetism and model of the lattice gas. The mean field theory. Renormalization group theory and its applications. Numerical Methods: The Monte Carlo method and molecular dynamics - Some algorithms and applications. Deterministic chaos and the foundations of statistical mechanics - Lyapunov Exponents - Kolmogorov-Sinai entropy. Stochastic processes.

K. Huang : Statistical Mechanics, J. Wiley & Sons (1987)
R.K. Pathria : Statistical Mechanics, Pergamon Press (1996)
E. Ott: Chaos in Dynamical systems, Cambridge University Press (1993)

Intermediate tests during the semester

For the final exam it is requested to prepare a short written dissertation on one of the topics of the programme for a general oral discussion on the main topics presented during the lectures

Thermodynamic potentials

Ensemble equivalence

1st order and 2nd order Phase transitions

Bose-Einstein condensation

Solutions of the Ising model

Universality and Critical exponents