Methods of Contemporary Mathematical Statistical Physics

ISBN-13: 9783540927952 / Angielski / Miękka / 2009 / 364 str.

Marek Biskup; Anton Bovier; W. Th F. Den Hollander

Opis

The Lecture Notes collect seven mini-courses presented at the 5th Prague Summer School on Mathematical Statistical Physics that took placeduringtwoweeksofSeptember2006.Aswithprecedingschools, it was aimed at PhD students and young postdocs. The central theme of the volume is what could be called mathematics of phase transitions in diverse contexts. Even though all courses were meant to introduce the reader to recent progress of a particular topic of modern statis- cal physics, attention has been paid to providing a solid grounding by carefully developing various basic tools. One of the techniques that led, more than two decades ago, to a seriesofimportantoutcomesinthetheoryofphasetransitionsoflattice models was re?ection positivity. Recently it resurfaced and was used to obtain interesting new results in various settings. The lectures of Marek Biskup include a thorough introduction to re?ection positivity as well as a review of its recent applications. In addition, it contains a crash course on lattice spin models that is useful as a background for other lectures of the collection. Also the following two contributions concern equilibrium statistical physics.ThelecturesofDmitriIo?earedevotedtoastochasticgeom- ricreformulationofclassicalaswellasquantumIsingmodels.Auni?ed approachtotheFortuin-Kasteleynandrandomcurrentrepresentations in terms of path integrals is presented. Statistical mechanics of directed polymers interacting with o- dimensionalspatiale?ectsisatopicwithvariousapplicationsinphysics and biophysics. The lectures of Fabio Toninelli are devoted to a th- ough discussion of the localization/delocalization transition in these models."

Spis treści

Reflection Positivity and Phase Transitions in Lattice Spin Models.- Stochastic Geometry of Classical and Quantum Ising Models.- Localization Transition in Disordered Pinning Models.- Metastability.- Three Lectures on Metastability Under Stochastic Dynamics.- A Selection of Nonequilibrium Issues.- Facilitated Spin Models: Recent and New Results.

Blurb

This volume presents a collection of courses introducing the reader to the recent progress with attention being paid to laying solid grounds and developing various basic tools.
An introductory chapter on lattice spin models is useful as a background for other lectures of the collection.
The topics include new results on phase transitions for gradient lattice models (with introduction to the techniques of the reflection positivity), stochastic geometry reformulation of classical and quantum Ising models, the localization/delocalization transition for directed polymers.
A general rigorous framework for theory of metastability is presented and particular applications in the context of Glauber and Kawasaki dynamics of lattice models are discussed.
A pedagogical account of several recently discussed topics in nonequilibrium statistical mechanics with an emphasis on general principles is followed by a discussion of kinetically constrained spin models that are reflecting important peculiar features of glassy dynamics.