Scaling Limits of Interacting Particle Systems Grundlehren Der Mathematischen Wissenschaften Springer

The idea of writing up a book on the hydrodynamic behavior of interacting particle systems was born after a series of lectures Claude Kipnis gave at the University of Paris 7 in the spring of 1988. At this time Claude wrote some notes in French that covered Chapters 1 and 4, parts of Chapters 2, 5 and Appendix 1 of this book. His intention was to prepare a text that was as self-contained as possible. lt would include, for instance, all tools from Markov process theory ( cf. Appendix 1, Chaps. 2 and 4) necessary to enable mathematicians and mathematical physicists with some knowledge of probability, at the Ievel of Chung (1974), to understand the techniques of the theory of hydrodynamic Iimits of interacting particle systems. In the fall of 1991 Claude invited me to complete his notes with him and transform them into a book that would present to a large audience the latest developments of the theory in a simple and accessible form. To concentrate on the main ideas and to avoid unnecessary technical difficulties, we decided to consider systems evolving in finite lattice spaces and for which the equilibrium states are product measures. To illustrate the techniques we chose two well-known particle systems, the generalized exclusion processes and the zero-range processes. We also conceived the book in such a manner that most chapters can be read independently of the others. Here are some comments that might help readers find their way. "This book presents in a progressive way the techniques used in the proof of the hydrodynamic behavior of interacting particle systems. It starts with introductory material on independent particles and goes all the way to nongradient systems, covering the entropy and the relative entropy methods, asymmetric processes from which hyperbolic equations emerge, the equilibrium fluctuations and the large deviations theory for short-range stochastic dynamics. It reviews, in appendices, some tools of Markov process theory and derives estimates on the spectral gap of reversible, conservative generators. The book is self-contained and can be read by graduate students in mathematics or mathematical physics with standard probability background. It can be used as a support for a graduate on stochastic processes."--THE PUBLISHER Front Matter....Pages I-XVI Introduction....Pages 1-6 An Introductory Example: Independent Random Walks....Pages 7-20 Some Interacting Particle Systems....Pages 21-40 Weak Formulations of Local Equilibrium....Pages 41-46 Hydrodynamic Equation of Symmetric Simple Exclusion Processes....Pages 47-66 An Example of Reversible Gradient System: Symmetric Zero Range Processes....Pages 67-114 The Relative Entropy Method....Pages 115-139 Hydrodynamic Limit of Reversible Nongradient Systems....Pages 141-189 Hydrodynamic Limit of Asymmetric Attractive Processes....Pages 191-229 Conservation of Local Equilibrium for Attractive Systems....Pages 231-256 Large Deviations from the Hydrodynamic Limit....Pages 257-285 Equilibrium Fluctuations of Reversible Dynamics....Pages 287-310 Back Matter....Pages 311-445

This book has been long awaited in the "interacting particle systems" community. Begun by Claude Kipnis before his untimely death, it was completed by Claudio Landim, his most brilliant student and collaborator. It presents the techniques used in the proof of the hydrodynamic behavior of interacting particle systems.