Mathematics

Activity type

Formazione modellistico-applicativa

SSD

MAT/07

Year

1st year

Semester

2nd semester

CFU

3

Hours distribution

24 classroom hours

Lecturers

GIADA BASILE
GIADA BASILE

Activity type

Formazione modellistico-applicativa

SSD

MAT/06

Year

1st year

Semester

2nd semester

CFU

3

Hours distribution

24 classroom hours

Lecturers

GIADA BASILE
GIADA BASILE

General targets:
acquire basic knowledge on a rigorous approach to statistical equilibrium mechanics.

Applying knowledge and understanding:
knowledge of statistical ensembles, Gibbs measures and thermodynamic functionals; understanding of phase transitions for paradigmatic lattice particle models.

Making judgements:
ability to describe mechanical and thermodynamic behavior of large systems of particles.



Communication skills:
ability to identify the main points of the theory, to be able to illustrate the most interesting elements by using appropriate examples, and to discuss the mathematic details for simple models.


Learning skills:
the acquired knowledge will allow to face advanced studies, i.e. at PhD level, related to equilibrium and non-equilibrium statistical mechanics, and to use the basic tools of statistical mechanics in other contexts.

Module: Module I - Statistical methods
Knowledge of basic argument and toold of equlibrium statistical mechanics


Module: Module II - Physical mathematics methods
Knowledge of basic argument and toold of equlibrium statistical mechanics

Module: Module I - Statistical methods
Probability, functional analysis, measure theory, thermodynamics


Module: Module II - Physical mathematics methods
Probability, functional analysis, measure theory, thermodynamics

Module: Module I - Statistical methods
-Hamiltonian dynamics of large systems. Statistical ensembles: micro-canonical, canonical, grand canonical. Entropy and thermodynamic functions.
-Statistical mechanics of lattice models. Gibbs measures, partition function and its thermodynamic limit. DLR equations. Gibbs variational principle.
-Phase transition. Uniqueness of the Gibbs state in the infinite volume limit: Dobrushin's criterion. One-dimensional systems and transfer matrix. Ferromagnetism and Curie's temperature. Mean-field models: the Curie-Weiss model. Phase transition of the Ising model.


Module: Module II - Physical mathematics methods
-Hamiltonian dynamics of large systems. Statistical ensembles: micro-canonical, canonical, grand canonical. Entropy and thermodynamic functions.
-Statistical mechanics of lattice models. Gibbs measures, partition function and its thermodynamic limit. DLR equations. Gibbs variational principle.
-Phase transition. Uniqueness of the Gibbs state in the infinite volume limit: Dobrushin's criterion. One-dimensional systems and transfer matrix. Ferromagnetism and Curie's temperature. Mean-field models: the Curie-Weiss model. Phase transition of the Ising model.

Module: Module I - Statistical methods
N/D
Module: Module II - Physical mathematics methods
N/D

Module: Module I - Statistical methods
N/D
Module: Module II - Physical mathematics methods
N/D

Module: Module I - Statistical methods
-


Module: Module II - Physical mathematics methods
-

Module: Module I - Statistical methods
N/D
Module: Module II - Physical mathematics methods
N/D

Module: Module I - Statistical methods
Discussion on the topics of the course.


Module: Module II - Physical mathematics methods
Discussion on the topics of the course.

Module: Module I - Statistical methods
Free argument.
The entropy of ideal gas.
Infinite volume Gibbs measures and their properties.
The Curie-Weiss model.
The 1-dim Ising model.
The Dobrushin criterion.


Module: Module II - Physical mathematics methods
Free argument.
The entropy of ideal gas.
Infinite volume Gibbs measures and their properties.
The Curie-Weiss model.
The 1-dim Ising model.
The Dobrushin criterion.

Module: Module I - Statistical methods
N/D
Module: Module II - Physical mathematics methods
N/D