This book presents the proceedings of the international conference Particle Systems and Partial Differential Equations I, which took place at the Centre of Mathematics of the University of Minho, Braga, Portugal, from the 5th to the 7th of December, 2012.

The purpose of the conference was to bring together world leaders to discuss their topics of expertise and to present some of their latest research developments in those fields. Among the participants were researchers in probability, partial differential equations and kinetics theory. The aim of the meeting was to present to a varied public the subject of interacting particle systems, its motivation from the viewpoint of physics and its relation with partial differential equations or kinetics theory and to stimulate discussions and possibly new collaborations among researchers with different backgrounds.

The book contains lecture notes written by Francois Golse on the derivation of hydrodynamic equations (compressible and incompressible Euler and Navier-Stokes) from the Boltzmann equation, and several short papers written by some of the participants in the conference. Among the topics covered by the short papers are hydrodynamic limits; fluctuations; phase transitions; motions of shocks and anti shocks in exclusion processes; large number asymptotics for systems with self-consistent coupling; quasi-variational inequalities; unique continuation properties for PDEs and others.

The book will benefit probabilists, analysts and mathematicians who are interested in statistical physics, stochastic processes, partial differential equations and kinetics theory, along with physicists."

This book includes the joint proceedings of the International Conference on Particle Systems and PDEs VI, VII and VIII. Particle Systems and PDEs VI was held in Nice, France, in November/December 2017, Particle Systems and PDEs VII was held in Palermo, Italy, in November 2018, and Particle Systems and PDEs VIII was held in Lisbon, Portugal, in December 2019. Most of the papers are dealing with mathematical problems motivated by different applications in physics, engineering, economics, chemistry and biology. They illustrate methods and topics in the study of particle systems and PDEs and their relation. The book is recommended to probabilists, analysts and to those mathematicians in general, whose work focuses on topics in mathematical physics, stochastic processes and differential equations, as well as to those physicists who work in statistical mechanics and kinetic theory.