الصفحة 1
الصفحة 1
Noise-Induced Transitions : Theory and Applications in Physics, Chemistry, and Biology
This classic text, an often-requested reprint, develops and explains the foundations of noise-induced processes. At its core is a self-contained, textbook-style presentation of the elements of probability theory, of the theory of Markovian diffusion processes and of the theory of stochastic differential equations, on which the modeling of fluctuating natural and artificial environments is based. Following an introduction to the mathematical tools, the occurrence and the properties of noise-induced transitions are then analyzed for rapidly fluctuating environments describable by the white-noise idealization. Subsequently, more realistic and general types of colored noises are considered. Appropriate practical methods for dealing with these situations are developed. The latter part of the book contains applications and experimental studies illustrating the many facets of noise-induced transitions. The following applications are considered in Noise-Induced Transitions: population dynamics, electrical circuits, chemical and photochemical reactions, non-linear optics, and hydrodynamical systems.
Noise-Induced Phenomena in Slow-Fast Dynamical Systems : A Sample-Paths Approach
Stochastic differential equations play an increasingly important role in modeling the dynamics of a large variety of systems in the natural sciences, and in technological applications. This book presents a new constructive approach to the quantitative description of solutions to systems of stochastic differential equations evolving on well-separated timescales. The method, which combines techniques from stochastic analysis and singular perturbation theory, allows the domains of concentration for typical sample paths to be determined, and provides precise estimates on the transition probabilities between these domains. In addition to the detailed presentation of the set-up and mathematical results, applications to problems in physics, biology, and climatology are discussed. The emphasis lies on noise-induced phenomena such as stochastic resonance, hysteresis, excitability, and the reduction of bifurcation delay.
Complex Nonlinearity : Chaos, Phase Transitions, Topology Change and Path Integrals
The book starts with a textbook-like expose on nonlinear dynamics, attractors and chaos, both temporal and spatio-temporal, including modern techniques of chaos–control. Chapter 2 turns to the edge of chaos, in the form of phase transitions (equilibrium and non-equilibrium, oscillatory, fractal and noise-induced), as well as the related field of synergetics. While the natural stage for linear dynamics comprises of flat, Euclidean geometry (with the corresponding calculation tools from linear algebra and analysis), the natural stage for nonlinear dynamics is curved, Riemannian geometry (with the corresponding tools from nonlinear, tensor algebra and analysis). The extreme nonlinearity – chaos – corresponds to the topology change of this curved geometrical stage, usually called configuration manifold. Chapter 3 elaborates on geometry and topology change in relation with complex nonlinearity and chaos. Chapter 4 develops general nonlinear dynamics, continuous and discrete, deterministic and stochastic, in the unique form of path integrals and their action-amplitude formalism.
