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© 2009

Scaling Limits in Statistical Mechanics and Microstructures in Continuum Mechanics

Book

Part of the Theoretical and Mathematical Physics book series (TMP)

Table of contents

  1. Front Matter
    Pages i-xxix
  2. Statistical Mechanics of Ising systems

  3. Mesoscopic Theory, non-local functionals

  4. Phase transitions in systems with Kac potentials

  5. Back Matter
    Pages 434-469

About this book

Introduction

Collective behavior in systems with many components, blow-ups with emergence of microstructures are proofs of the double, continuum and atomistic, nature of macroscopic systems, an issue which has always intrigued scientists and philosophers. Modern technologies have made the question more actual and concrete with recent, remarkable progresses also from a mathematical point of view. The book focuses on the links connecting statistical and continuum mechanics and, starting from elementary introductions to both theories, it leads to actual research themes. Mathematical techniques and methods from probability, calculus of variations and PDE are discussed at length.

Keywords

Gamma convergence Ising models Kac potentials LMP model Potential Statistical mechanics generalized Pirogov-Sinai theory non local free energy functionals partial differential equation phase transitions in the continuum probabilistic methods

Authors and affiliations

  1. 1.Dipartimento di MatematicaUniversitàdi Roma Tor VergataRomaItaly

About the authors

Errico Presutti is professor of classical mechanics at the Department of Mathematics of the University of Roma Tor Vergata.

Bibliographic information

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Reviews

From the reviews:

"The author takes an effort to make accessible a variety of problems and mathematical techniques involved in describing how a continuum description emerges from mesoscopic and microscopic models. Technically that refers to large scale phenomena in statistical mechanics and their mathematical formulations in terms of thermodynamic and hydrodynamic limits. The book is intended to both (mathematical) physics and mathematical communities … ." (Piotr Garbaczewski, Zentralblatt MATH, Vol. 1156, 2009)

“This book is a beautiful overview of the state of the art. … The link between notions and theories developed in continuum mechanics and PDEs and statistical mechanics is illustrated here. The book is an effort to build bridges between the two areas and, since it is meant for both communities it is written with the presumption that reader may not be expert in all topics. … The various topics can be easily singled out to be used for courses or lectures … .” (Nicoletta Cancrini, Mathematical Reviews, Issue 2011 a)