# 99512 - STATISTICAL PHYSICS FOR CLIMATE SCIENCE

• Moduli: Elisa Ercolessi (Modulo 1) Marco Lenci (Modulo 2)
• Campus: Bologna
• Corso: Second cycle degree programme (LM) in Science of Climate (cod. 5895)

## Learning outcomes

At the end of the course, the student will have a basic knowledge of theoretical concepts and methods of statistical physics, including: the probabilistic laws that rule the microscopic description for modeling the behaviour of thermodynamic and complex systems; description of systems at equilibrium; an approach to dynamics and non equilibrium physics. The student will be able to describe the main theoretical concepts and tools in order to use them to solve -analytically or with the aid of numerical simulations- simple but paradigmatic models, with applications to different branches of physics and in particular to problems of climate science.

## Course contents

• Elements of Probability for Applications (24 h, prof. Marco Lenci)

Mathematical foundations of probability: probability spaces, events; conditional probability, independence; Bayes’ Theorem.

Random variables: general theory; discrete and continuous random variables; moments; important examples and applications; joint distribution.

Limit theorems: law of large numbers, characteristic function, Central Limit Theorem; moment-generating function.

Elements of stochastic processes: stationarity; i.i.d. random variables; Markov chains.

• Statistical Models for Physics (24 h, prof. Elisa Ercolessi)

Thermodynamics and its microscopic interpretation: work, heat, entropy; the laws of thermodynamics;

Kinetic theory of gases.

Introduction to (classical) statistical mechanics: the state of a system of many particles; microcanocical ensemble.

The canonical and grandcanonical ensemble: partition function, free energy, grand-potential and other tjhermodynamic potentials; averages and fluctuations; the notion of entropy.

Applications: the equipartition theorem; ideal gas; the virial expansion and the real gas.

S. Ross, ntroduction to Probability Models (Academic Press)

K. Huang, Statistical Mechanics (John Wiley & Sons).

Further reading suggestions and other didactic materials will be made available in the Virtuale platform.

## Teaching methods

The course is divided into 2 modules of 24 hours each.

Classes will consists in front lectures on theory, applications and exercises.

## Assessment methods

The exam will consist of:

- a written part, with exercises on probability and simple applications to statistical physics;

- an oral exam, that will consist of (at least) two questions on the written exam and on concepts introduce in the classes.

Students should demonstrate to be familiar and have a good understanding of the different subjects.

They will be asked to present an introduction to the main general topics and to argue in order to discussspecific applications, making connections among the different parts of the syllabus.

The organization of the presentation and a rigorous scientific language will be also considered for the formulation of the final grade.

The “cum laude” honor is granted to students who demonstrate a personal and critical rethinking of the subject.