Noise In Spatially Extended Systems

Intended for graduates and researchers in physics, chemistry, biology, and applied mathematics, this book provides an up-to-date introduction to current research in fluctuations in spatially extended systems. It covers the theory of stochastic partial differential equations and gives an overview of the effects of external noise on dynamical systems with spatial degrees of freedom. Starting with a general introduction to noise-induced phenomena in dynamical systems, the text moves on to an extensive discussion of analytical and numerical tools needed to gain information from stochastic partial differential equations. It then turns to particular problems described by stochastic PDEs, covering a wide part of the rich phenomenology of spatially extended systems, such as nonequilibrium phase transitions, domain growth, pattern formation, and front propagation. The only prerequisite is a minimal background knowledge of the Langevin and Fokker-Planck equations.

This book is about the dynamics of coupled map lattices (CML) and of related spatially extended systems. It will be useful to post-graduate students and researchers seeking an overview of the state-of-the-art and of open problems in this area of nonlinear dynamics. The special feature of this book is that it describes the (mathematical) theory of CML and some related systems and their phenomenology, with some examples of CML modeling of concrete systems (from physics and biology). More precisely, the book deals with statistical properties of (weakly) coupled chaotic maps, geometric aspects of (chaotic) CML, monotonic spatially extended systems, and dynamical models of specific biological systems.

The shared platform of the articles collected in this volume is used to advocate a dynamical systems approach to cognition. It is argued that recent developments in cognitive science towards an account of embodiment, together with the general approach of complexity theory and dynamics, have a major impact on behavioral and cognitive science. The book points out that there are two domains that follow naturally from the stance of embodiment: first, coordination dynamics is an established empirical paradigm that is best able to aid the approach; second, the obvious goal-directedness of intelligent action (i.e., intentionality) is nicely addressed in the framework of the dynamical synergetic approach.

The shared platform of the articles collected in this volume is used to advocate a dynamical systems approach to cognition. It is argued that recent developments in cognitive science towards an account of embodiment, together with the general approach of complexity theory and dynamics, have a major impact on behavioral and cognitive science. The book points out that there are two domains that follow naturally from the stance of embodiment: first, coordination dynamics is an established empirical paradigm that is best able to aid the approach; second, the obvious goal-directedness of intelligent action (i.e., intentionality) is nicely addressed in the framework of the dynamical synergetic approach.

​​Since the parameters in dynamical systems of biological interest are inherently positive and bounded, bounded noises are a natural way to model the realistic stochastic fluctuations of a biological system that are caused by its interaction with the external world. Bounded Noises in Physics, Biology, and Engineering is the first contributed volume devoted to the modeling of bounded noises in theoretical and applied statistical mechanics, quantitative biology, and mathematical physics. It gives an overview of the current state-of-the-art and is intended to stimulate further research. The volume is organized in four parts. The first part presents the main kinds of bounded noises and their applications in theoretical physics. The theory of bounded stochastic processes is intimately linked to its applications to mathematical and statistical physics, and it would be difficult and unnatural to separate the theory from its physical applications. The second is devoted to framing bounded noises in the theory of random dynamical systems and random bifurcations, while the third is devoted to applications of bounded stochastic processes in biology, one of the major areas of potential applications of this subject. The final part concerns the application of bounded stochastic processes in mechanical and structural engineering, the area where the renewed interest for non-Gaussian bounded noises started. Pure mathematicians working on stochastic calculus will find here a rich source of problems that are challenging from the point of view of contemporary nonlinear analysis. Bounded Noises in Physics, Biology, and Engineering is intended for scientists working on stochastic processes with an interest in both fundamental issues and applications. It will appeal to a broad range of applied mathematicians, mathematical biologists, physicists, engineers, and researchers in other fields interested in complexity theory. It is accessible to anyone with a working knowledge of stochastic modeling, from advanced undergraduates to senior researchers.

The theory of stochastic processes originally grew out of efforts to describe Brownian motion quantitatively. Today it provides a huge arsenal of methods suitable for analyzing the influence of noise on a wide range of systems. The credit for acquiring all the deep insights and powerful methods is due ma- ly to a handful of physicists and mathematicians: Einstein, Smoluchowski, Langevin, Wiener, Stratonovich, etc. Hence it is no surprise that until - cently the bulk of basic and applied stochastic research was devoted to purely mathematical and physical questions. However, in the last decade we have witnessed an enormous growth of results achieved in other sciences - especially chemistry and biology - based on applying methods of stochastic processes. One reason for this stochastics boom may be that the realization that noise plays a constructive rather than the expected deteriorating role has spread to communities beyond physics. Besides their aesthetic appeal these noise-induced, noise-supported or noise-enhanced effects sometimes offer an explanation for so far open pr- lems (information transmission in the nervous system and information p- cessing in the brain, processes at the cell level, enzymatic reactions, etc.). They may also pave the way to novel technological applications (noise-- hanced reaction rates, noise-induced transport and separation on the na- scale, etc.). Key words to be mentioned in this context are stochastic r- onance, Brownian motors or ratchets, and noise-supported phenomena in excitable systems.

This volume contains a selection of the lectures given at the Fifth International Workshop on Instabilities and Nonequilibrium Structures, held in Santiago, Chile, in December 1993. The following general subjects are covered: instabilities and pattern formation, stochastic effects in nonlinear systems, nonequilibrium statistical mechanics and granular matter. Review articles on transitions between spatio-temporal patterns and nonlinear wave equations are also included. Audience: This book should appeal to physicists and mathematicians working in the areas of nonequilibrium systems, dynamical systems, pattern formation and partial differential equations. Chemists and biologists interested in self-organization and statistical mechanics should also be interested, as well as engineers working in fluid mechanics and materials science.