## Brazilian Journal of Probability and Statistics

- Braz. J. Probab. Stat.
- Volume 29, Number 2 (2015), 211-281.

### From equilibrium to nonequilibrium statistical mechanics. Phase transitions and the Fourier law

#### Abstract

These are the notes of my lectures at EBP, August 2013. I have added some proofs which being of more technical nature have been omitted in the talks. The notes and the lectures are based on a course on atomistic and continuous descriptions of matter which I gave a few years earlier in Sperlonga, Italy. In the present notes, I have tried to underline the more probabilistic aspects of the theory.

I am afraid I have not been able to reproduce in the written notes the very lively atmosphere of the talks. Many old friends of mine were attending the lectures and helped me a lot with questions, comments and criticism, it was a pleasure for me to speak at EBP and I hope also the audience enjoined all that. In particular, I want to renew my deepest thanks to Stefan Luckhaus who is undoubtedly the best help for a speaker to have in the audience.

These notes are divided into four chapters, like in the lectures I have mostly avoided proofs trying to give qualitatively the main ideas of the theory. The only exception is in the second lecture of these notes where I have given more details on the proof of phase transitions in the canonical Ising model. This partly for completeness and partly because I have been asked by several people for details after the talk.

I am not very good with bibliography so I just quoted some of the papers I am most familiar with, the reader will forgive me, I hope, for the many omissions.

I conclude these preface by renewing my warmest thanks to the people who attended EBP and in particular to Maria Eulalia Vares for inviting me at EBP, for the nice words when she introduced my lectures but especially for the very long friendship (in the past and hopefully in the future).

#### Article information

**Source**

Braz. J. Probab. Stat., Volume 29, Number 2 (2015), 211-281.

**Dates**

First available in Project Euclid: 15 April 2015

**Permanent link to this document**

https://projecteuclid.org/euclid.bjps/1429105589

**Digital Object Identifier**

doi:10.1214/14-BJPS270

**Mathematical Reviews number (MathSciNet)**

MR3336867

**Zentralblatt MATH identifier**

1318.82024

**Keywords**

Phase transitions Ising model Kac potentials free energy functionals stochastic evolutions nonequilibrium stationary measures

#### Citation

Presutti, Errico. From equilibrium to nonequilibrium statistical mechanics. Phase transitions and the Fourier law. Braz. J. Probab. Stat. 29 (2015), no. 2, 211--281. doi:10.1214/14-BJPS270. https://projecteuclid.org/euclid.bjps/1429105589