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

The physics of non-equilibrium many-body systems is a rapidly expanding area of theoretical physics. Traditionally employed in laser physics and superconducting kinetics, these techniques have more recently found applications in the dynamics of cold atomic gases, mesoscopic and nano-mechanical systems, and quantum computation. This book provides a detailed presentation of modern non-equilibrium field-theoretical methods, applied to examples ranging from biophysics to the kinetics of superfluids and superconductors. A highly pedagogical and self-contained approach is adopted within the text, making it ideal as a reference for graduate students and researchers in condensed matter physics. In this Second Edition, the text has been substantially updated to include recent developments in the field such as driven-dissipative quantum systems, kinetics of fermions with Berry curvature, and Floquet kinetics of periodically driven systems, among many other important new topics. Problems have been added throughout, structured as compact guided research projects that encourage independent exploration.

This book provides a challenging and stimulating introduction to the contemporary topics of complexity and criticality, and explores their common basis of scale invariance, a central unifying theme of the book. Criticality refers to the behaviour of extended systems at a phase transition where scale invariance prevails. The many constituent microscopic parts bring about macroscopic phenomena that cannot be understood by considering a single part alone. The phenomenology of phase transitions is introduced by considering percolation, a simple model with a purely geometrical phase transition, thus enabling the reader to become intuitively familiar with concepts such as scale invariance and renormalisation. The Ising model is then introduced, which captures a thermodynamic phase transition from a disordered to an ordered system as the temperature is lowered in zero external field. By emphasising analogies between percolation and the Ising model, the reader's intuition of phase transitions is developed so that the underlying theoretical formalism may be appreciated fully. These equilibrium systems undergo a phase transition only if an external agent finely tunes certain external parameters to particular values. Besides fractals and phase transitions, there are many examples in Nature of the emergence of such complex behaviour in slowly driven non-equilibrium systems: earthquakes in seismic systems, avalanches in granular media and rainfall in the atmosphere. A class of non-equilibrium systems, not constrained by having to tune external parameters to obtain critical behaviour, is addressed in the framework of simple models, revealing that the repeated application of simple rules may spontaneously give rise to emergent complex behaviour not encoded in the rules themselves. The common basis of complexity and criticality is identified and applied to a range of non-equilibrium systems. Finally, the reader is invited to speculate whether self-organisation in non-equilibrium systems might be a unifying concept for disparate fields such as statistical mechanics, geophysics and atmospheric physics. Visit http://www.complexityandcriticality.com for animations for the models in the book (available for Windows and Linux), solutions to exercises, as well as a list with corrections. Contents:Percolation:Percolating Phase TransitionPercolation in One DimensionPercolation on the Bethe LatticePercolation in Two DimensionsGeometric Properties of ClustersScaling Ansatz, Scaling Functions and Scaling RelationsFinite-Size ScalingUniversalityReal-Space Renormalisation GroupIsing Model:Review of Thermodynamics and Statistical MechanicsSymmetry BreakingFerromagnetic Phase TransitionIsing Model in One DimensionMean-Field Ising ModelIsing Model in Two DimensionsLandau Theory of Continuous Phase TransitionsScaling Ansatz, Scaling Functions and Scaling RelationsUniversalityReal-Space Renormalisation GroupSelf-Organised Criticality:Non-equilibrium steady state systemBTW Model in One DimensionMean-Field Theory of the BTW ModelBranching ProcessScaling Ansatz, Scaling Functions and Scaling RelationsBTW Model in Two DimensionsA Rice Pile Experiment and the Oslo ModelEarthquakes and the OFC ModelRainfallSelf-Organised Criticality as a Unifying Principle Readership: Students at all levels, researchers and instructors looking for an introduction to the ideas of complexity and criticality.

Markedly apart from elementary particle physics, another current has been building up and cont i nuous ly growi ng within contemporary phys i cs for severa 1 decades, and even expanding into many other disciplines, especially chemistry, biology and, quite recently, economics. Several reasons account for this: presumably the most impor tant one lies in the fact that, whatever the specific problem, model or material concerned, the same basic mathematical features are always involved. In this way, a general phenomenology has emerged which, unlike thermodynamics, is no longer depen dent upon the details or specifics: what largely prevails is the nonlinear charac ter of the underlying dynamics. Perhaps we are witnessing the emergence of a "non linear physics"--In a way similar to the birth of "quantum physics" in the twen ties - a physics which deals with the general behaviour of systems, whatever they are or may be. Over the past fifteen years, chemical systems evolving sufficiently far from equilibrium have proved to be particularly well fitted to experimental research on nonlinear behaviour: oscillation, multistability, birhythmicity, chaotic evolution, spatial self-organization and hysteresis are displayed by chemical reactions whose number is growing each year. In this volume are collected the lectures, communica tions and posters (abstracts) presented at an international meeting entitled: "Non-Equilibrium Dynamics in Chemical Systems", held in Bordeaux (France), Septem ber 3 rd-lth, 1984.

Non-equilibrium systems cover a tremendously wide range of different systems, in experiments and in simulations as well as in the real world. The fact that these systems are not in thermodynamic equilibrium is in many cases responsible for unique effects and behavior. A fundamental understanding of non-equilibrium systems is crucial to gain insights into such behavior and exploit it in a manifold of use cases. A recent growth in attention to non-equilibrium systems is a consequence. Especially deep insights into the nature of certain non-equilibrium systems can be gained through the study of phase behavior in these very systems. To do so, this thesis utilizes computer simulations of different systems: Discrete and continuous active matter systems on the one hand and skyrmion lattices on the other hand. The active matter systems being discussed in this work consist of active particles. These particles are not only subject to Brownian motion but they are in addition “actively” performing directed motion, which drives the systems out of equilibrium. Active lattice gas models are studied as a discrete realization of such particles with comparably low computational effort. At sufficiently high activity they undergo a motility induced phase separation (MIPS) that closely resembles the gas-liquid transition known from equilibrium. However, a determination of critical points and exponents for different model realizations performed in this work showed some model dependent deviations from 2D Ising universality class. This raises the question, whether the concept of universality holds for active matter and non-equilibrium systems at all. The critical behavior of active Brownian particles (ABPs) around MIPS has already been studied before and showed even stronger deviations. In this work additional focus is put on quenches of ABPs from homogeneous phase right into the phase separated regime and to the critical point. Following the quench, the systems far-from-steady-state dynamics, structure growth and aging can be studied. Results obtained in this work appear to be similar to those observed during phase separation in the 2D Ising model. However, for the active lattice models, there are deviations in the case of quenches inside the coexistence regions. Skyrmion lattices consist of densely packed skyrmions. These topologically stabilized whirls of magnetization can be described as quasiparticles. By modelling them as soft disks similar to ABPs, the underlying interaction potential of experimental skyrmions was determined with the help of computer simulations in this work. Furthermore, different experimental skyrmion lattices were characterized according to their phase state and degree of hexagonal order with the help of a newly developed parameter. Thereby the onset of a hexatic phase was found.

This book describes two main classes of non-equilibrium phase-transitions: static and dynamics of transitions into an absorbing state, and dynamical scaling in far-from-equilibrium relaxation behavior and ageing.