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© 2008

Stochastic Ordinary and Stochastic Partial Differential Equations

Transition from Microscopic to Macroscopic Equations

Book

Table of contents

About this book

Introduction

This book provides the first rigorous derivation of mesoscopic and macroscopic equations from a deterministic system of microscopic equations. The microscopic equations are cast in the form of a deterministic (Newtonian) system of coupled nonlinear oscillators for N large particles and infinitely many small particles. The mesoscopic equations are stochastic ordinary differential equations (SODEs) and stochastic partial differential equatuions (SPDEs), and the macroscopic limit is described by a parabolic partial differential equation.

 A detailed analysis of the SODEs and (quasi-linear) SPDEs is presented. Semi-linear (parabolic) SPDEs are represented as first order stochastic transport equations driven by Stratonovich differentials. The time evolution of correlated Brownian motions is shown to be consistent with the depletion phenomena experimentally observed in colloids. A covariance analysis of the random processes and random fields as well as a review section of various approaches to SPDEs are also provided.

An extensive appendix makes the book accessible to both scientists and graduate students who may not be specialized in stochastic analysis.

 Probabilists, mathematical and theoretical physicists as well as mathematical biologists and their graduate students will find this book useful.

 Peter Kotelenez is a professor of mathematics at Case Western Reserve University in Cleveland, Ohio.

Keywords

Kotelenez Macroscopic Microscopic Ordinary Partial Differential Equations Stochastic Variance partial differential equation

Authors and affiliations

  1. 1.Department of MathematicsCase Western Reserve UniversityClevelandUSA

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Reviews

From the reviews:

"This book treats the transition from microscopic to macroscopic equations for particle systems. … Peter Kotelenez … has written a monograph in which he rigorously constructs the theory of correlated Brownian motion in interacting particle systems. … Researchers working on interacting particle systems and probability theory will definitely find this book very useful." (J. Dubbeldam, Kwantitatieve Methoden, Issue R11, 2008)

"This interesting book treats in detail stochastic partial differential equations (SPDEs) describing mass distribution of particles. The book introduces in an essentially self-contained manner the author’s research on the evolution of large particles interacting with an environment consisting, for instance, of small particles. … To help the reader, the book devotes four of the fourteen chapters to the more lengthy proofs of some theorems. … the reviewer recommends this book to any researcher interested in SPDEs." (Carlos M. Mora González, Mathematical Reviews, Issue 2009 h)

"The monography under review presents mathematical models for physical dynamical systems of particle and mass evolution on different levels … . Extremely helpful are the excellent summaries at the beginning of each chapter. … This monography is a highly impressive result of many years of concise scientific studies on SODEs, SPDEs and mathematical physics. I warmly recommend this book for scientific and personal enlightenment to graduate students … as well as to mathematical scientists in mathematical physics, theoretical physics and mathematical biology." (Michael Högele, Zentralblatt MATH, Vol. 1159, 2009)