This book is devoted to the study of evolution of nonequilibrium systems. Such a system usually consists of regions with different dominant scales, which coexist in the space-time where the system lives. In the case of high nonuniformity in special direction, one can see patterns separated by clearly distinguishable boundaries or interfaces. The author considers several examples of nonequilibrium systems. One of the examples describes the invasion of the solid phase into the liquidphase during the crystallization process. Another example is the transition from oxidized to reduced states in certain chemical reactions. An easily understandable example of the transition in the temporal direction is a sound beat, and the author describes typical patterns associated with thisphenomenon. The main goal of the book is to present a mathematical approach to the study of highly nonuniform systems and to illustrate it with examples from physics and chemistry. The two main theories discussed are the theory of singular perturbations and the theory of dissipative systems. A set of carefully selected examples of physical and chemical systems nicely illustrates the general methods described in the book.

Stemming from the IHP trimester "Stochastic Dynamics Out of Equilibrium", this collection of contributions focuses on aspects of nonequilibrium dynamics and its ongoing developments. It is common practice in statistical mechanics to use models of large interacting assemblies governed by stochastic dynamics. In this context "equilibrium" is understood as stochastically (time) reversible dynamics with respect to a prescribed Gibbs measure. Nonequilibrium dynamics correspond on the other hand to irreversible evolutions, where fluxes appear in physical systems, and steady-state measures are unknown. The trimester, held at the Institut Henri Poincaré (IHP) in Paris from April to July 2017, comprised various events relating to three domains (i) transport in non-equilibrium statistical mechanics; (ii) the design of more efficient simulation methods; (iii) life sciences. It brought together physicists, mathematicians from many domains, computer scientists, as well as researchers working at the interface between biology, physics and mathematics. The present volume is indispensable reading for researchers and Ph.D. students working in such areas.

A comprehensive and unified introduction to describing and understanding complex interacting systems.

The NATO Advanced Study Institute on "Scale Invariance, Interfaces and Non Equilibrium Dynamics" was held at the Isaac Newton Institute for Mathematical Sciences in Cambridge, UK from 20-30 June 1994. The topics discussed at the Institute were all concerned with the origin and nature of complex structures found far from equilibrium. Examples ranged from reaction diffusion systems and hydrodynamics through to surface growth due to deposition. A common theme was that of scale invariance due to the self-similarity of the underly ing structures. The topics that were covered can be broadly classified as pattern for mation (theoretical, computational and experimental aspects), the non-equilibrium dynamics of the growth of interfaces and other manifolds, coarsening phenomena, generic scale invariance in driven systems and the concept of self-organized critical ity. The main feature of the Institute was the four one-hour-Iong lectures given each day by invited speakers. In addition to thirty-seven of these lectures, two contributed lectures were also given. The many questions that were asked after the lectures attested to the excitement and interest that the lecturers succeeded in generating amongst the students. In addition to the discussions initiated by lectures, an im portant component of the meeting were the poster sessions, where participants were able to present their own work, which took place on three of the afternoons. The list of titles given at the end of these proceedings gives some idea of the range and scope of these posters.

The past decade has seen unprecedented developments in the understanding of relativistic fluid dynamics in and out of equilibrium, with connections to astrophysics, cosmology, string theory, quantum information, nuclear physics and condensed matter physics. Romatschke and Romatschke offer a powerful new framework for fluid dynamics, exploring its connections to kinetic theory, gauge/gravity duality and thermal quantum field theory. Numerical algorithms to solve the equations of motion of relativistic dissipative fluid dynamics as well as applications to various systems are discussed. In particular, the book contains a comprehensive review of the theory background necessary to apply fluid dynamics to simulate relativistic nuclear collisions, including comparisons of fluid simulation results to experimental data for relativistic lead-lead, proton-lead and proton-proton collisions at the Large Hadron Collider (LHC). The book is an excellent resource for students and researchers working in nuclear physics, astrophysics, cosmology, quantum many-body systems and string theory.

Optimal growth theory studies the problem of efficient resource allocation over time, a fundamental concern of economic research. Since the 1970s, the techniques of nonlinear dynamical systems have become a vital tool in optimal growth theory, illuminating dynamics and demonstrating the possibility of endogenous economic fluctuations. Kazuo Nishimura's seminal contributions on business cycles, chaotic equilibria and indeterminacy have been central to this development, transforming our understanding of economic growth, cycles, and the relationship between them. The subjects of Kazuo's analysis remain of fundamental importance to modern economic theory. This book collects his major contributions in a single volume. Kazuo Nishimura has been recognized for his contributions to economic theory on many occasions, being elected fellow of the Econometric Society and serving as an editor of several major journals. Chapter “Introduction” is available open access under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License via link.springer.com.

Topological defects formed at symmetry-breaking phase transitions play an important role in many different fields of physics. They appear in many condensed-matter systems at low temperature; examples include vortices in superfluid helium-4, a rich variety of defects in helium-3, quantized mag netic flux tubes in type-II superconductors, and disclination lines and other defects in liquid crystals. In cosmology, unified gauge theories of particle interactions suggest a sequence of phase transitions in the very early uni verse some of which may lead to defect formation. In astrophysics, defects play an important role in the dynamics of neutron stars. In 1997 the European Science Foundation started the scientific network "Topological defects" headed by Tom Kibble. This network has provided us with a unique opportunity of establishing a collaboration between the representatives of these very different branches of modern physics. The NATO-ASI (Advanced Study Institute), held in Les Houches in February 1999 thanks to the support of the Scientific Division of NATO, the European Science Foundation and the CNRS, represents a key event of this ESF network. It brought together participants from widely different fields, with diverse expertise and vocabulary, fostering the exchange of ideas. The lectures given by particle physicists, cosmologists and condensed matter physicists are the result of the fruitful collaborations established since 1997 between groups in several European countries and in the U.S.A.