Noise in spatially extended systems.

*(English)*Zbl 0938.60002
Institute for Nonlinear Science. New York, NY: Springer. xiii, 307 p. (1999).

The last two decades are marked out by growing interest to the problems of self-organization and ordering in the presence of fluctuations. It has been elaborated on behalf of many examples in the last decade that noise, surprisingly, is able to create order in nonlinear systems far from equilibrium. It means that an increasing intensity of noise applied to a nonlinear system may exhibit a more ordered response of the system than smaller or vanishing values of this intensity.

The book sheds the light to this counterintuitive role of external fluctuations in the behavior of spatially extended nonlinear systems and provides a state-of-the-art review of this topic. The book appears to be a welcome addition to the fast growing body of articles in journals devoted to the ordering role of noise in nonequilibrium coupled systems. It presents a detailed overview of recent progress in understanding of noise-induced effects in distributed systems under local spatial coupling.

The notion of noise-induced phenomena is introduced as a broad concept that includes phase transitions, phase separation, pattern formation and front dynamics in the presence of external fluctuations. The variety of fluctuational phenomena, their physical origin and possible applications, discussed in the Introduction will attract interest of a wide readership in the field of physics, chemistry, biology and applied mathematics. Chapter 2 deals with fundamental tools and describes clearly and concisely the basic ideas of methods to study stochastic systems with spatial coupling. A short description of theoretical concepts and numerical methods are combined with references which provide deeper information about this point. Other mathematical tools are explained in detail in a number of appendices. The style, level and content of the main part of the book makes it useful for graduate students who have interest in applied stochastic processes. Only knowledge of Langevin and Fokker-Planck equations are required for an understanding of the phenomena described in the book. On the other hand, the book discusses contemporary research of such nontrivial phenomena as pure noise-induced ordering transitions, noise-induced phase separation, pattern formation and front propagation in fluctuating media, and will be of interest for researchers, who are specialized in the study of noise-induced effects. The book is concluded by a clearly written summary, manifesting the main results of the phenomena in coupled and distributed systems, which are an extension of a behavior, found earlier in purely temporal systems. An outlook of open questions is presented.

In conclusion, this competently written book will be an invaluable reference for anyone who enters the world of noise-induced effects or needs some up-to-date information about this subject. The reader will remark throughoutly that the authors are highly qualified specialists which are able to write a stimulating monograph on a topic to which development the authors themselves have contributed a lot.

The book sheds the light to this counterintuitive role of external fluctuations in the behavior of spatially extended nonlinear systems and provides a state-of-the-art review of this topic. The book appears to be a welcome addition to the fast growing body of articles in journals devoted to the ordering role of noise in nonequilibrium coupled systems. It presents a detailed overview of recent progress in understanding of noise-induced effects in distributed systems under local spatial coupling.

The notion of noise-induced phenomena is introduced as a broad concept that includes phase transitions, phase separation, pattern formation and front dynamics in the presence of external fluctuations. The variety of fluctuational phenomena, their physical origin and possible applications, discussed in the Introduction will attract interest of a wide readership in the field of physics, chemistry, biology and applied mathematics. Chapter 2 deals with fundamental tools and describes clearly and concisely the basic ideas of methods to study stochastic systems with spatial coupling. A short description of theoretical concepts and numerical methods are combined with references which provide deeper information about this point. Other mathematical tools are explained in detail in a number of appendices. The style, level and content of the main part of the book makes it useful for graduate students who have interest in applied stochastic processes. Only knowledge of Langevin and Fokker-Planck equations are required for an understanding of the phenomena described in the book. On the other hand, the book discusses contemporary research of such nontrivial phenomena as pure noise-induced ordering transitions, noise-induced phase separation, pattern formation and front propagation in fluctuating media, and will be of interest for researchers, who are specialized in the study of noise-induced effects. The book is concluded by a clearly written summary, manifesting the main results of the phenomena in coupled and distributed systems, which are an extension of a behavior, found earlier in purely temporal systems. An outlook of open questions is presented.

In conclusion, this competently written book will be an invaluable reference for anyone who enters the world of noise-induced effects or needs some up-to-date information about this subject. The reader will remark throughoutly that the authors are highly qualified specialists which are able to write a stimulating monograph on a topic to which development the authors themselves have contributed a lot.

Reviewer: Lutz Schimansky-Geier (Berlin)

##### MSC:

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

82-02 | Research exposition (monographs, survey articles) pertaining to statistical mechanics |

60K40 | Other physical applications of random processes |

82C26 | Dynamic and nonequilibrium phase transitions (general) in statistical mechanics |