Math Courses

Probability and Statistical Mechanics

by Prof. Errico Presutti (GSSI), Dr Lu Xu (GSSI), Dr Marielle Simon (GSSI), Dr Serena Cenatiempo (GSSI), Stefano Olla (GSSI)

Europe/Rome
Description

Lecturers
Stefano Olla, stefano.olla@gssi.it
Marielle Simon, marielle.simon@gssi.it
Errico Presutti, errico.presutti@gmail.com
Serena Cenatiempo, serena.cenatiempo@gssi.it
Lu Xu, lu.xu@gssi.it


Timetable and workload
Lectures: 60 hours
Final project assignment: 24 hours

Course description and outcomes
This is a probability course that will constitute an introduction to the main concepts of Statistical Mechanics.  One of the purposes of statistical mechanics is to explain thermodynamics in terms of the “microscopic” dynamics. The general consensus is that thermodynamics describe the “macroscopic” behavior of some quantities (energy, pressure, . . . ) under the influence of external forces or heat bath. Macroscopic means that laws of thermodynamics emerge for “large” systems and in a large time scale. We present in these lectures a precise mathematical approach where thermodynamics laws (Carnot cycles, Kelvin and Clausius principles, etc., and the corresponding first and second principle of thermodynamics) are obtained through space-time scaling limits. In this approach the problem of thermodynamics is reduced to a purely mathematical problem, very difficult indeed.
We will avoid making a general theory, and in the first part of the course we will work with the simple system of one dimensional hard core classical dynamics and some variants of this. We will study the stationary measures and scaling limits.

In the second part of the course we will study phase transitions in equilibrium for some lattice models, like the Ising model.

In the third part we will give an introduction to the statistical mechanics of quantum systems.

The classes will be complemented by TA sessions, aimed to recall some of the mathematical background.


Examination and grading
Each student, after having delivered a written report on the final project, will be evaluated according to the ECTS grading scale.

 

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