Scientists have always attempted to explain the world in terms of a few unifying principles. In the fifth century B.C. Democritus boldly claimed that reality is simply a collection of indivisible and eternal parts or atoms. Over the centuries his doctrine has remained a landmark, and much progress in physics is due to its distinction between subjective perception and objective reality. This book discusses theory reduction in physics, which states that the whole is nothing more than the sum of its parts: the properties of things are directly determined by their constituent parts. Reductionism deals with the relation between different theories that address different levels of reality, and uses extrapolations to apply that relation in different sciences. Reality shows a complex structure of connections, and the dream of a unified interpretation of all phenomena in several simple laws continues to attract anyone with genuine philosophical and scientific interests. If the most radical reductionist point of view is correct, the relationship between disciplines is strictly inclusive: chemistry becomes physics, biology becomes chemistry, and so on. Eventually, only one science, indeed just a single theory, would survive, with all others merging in the Theory of Everything. Is the current coexistence of different sciences a mere historical venture which will end when the Theory of Everything has been established? Can there be a unified description of nature?
Rather than an analysis of full reductionism, this book focuses on aspects of theory reduction in physics and stimulates reflection on related questions: is there any evidence of actual reduction? Are the examples used in the philosophy of science too simplistic? What has been endangered by the search for (the) ultimate truth? Has the dream of reductionist reason created any monsters? Is big science one such monster? What is the point of embedding science Y within science X, if predictions cannot be made on that basis?

The present volume, published at the occasion of his 100th birthday anniversary, is a collection of articles that reviews the impact of Kolomogorov's work in the physical sciences and provides an introduction to the modern developments that have been triggered in this way to encompass recent applications in biology, chemistry, information sciences and finance.

This book offers an informal, easy-to-understand account of topics in modern physics and mathematics. The focus is, in particular, on statistical mechanics, soft matter, probability, chaos, complexity, and models, as well as their interplay. The book features 28 key entries and it is carefully structured so as to allow readers to pursue different paths that reflect their interests and priorities, thereby avoiding an excessively systematic presentation that might stifle interest. While the majority of the entries concern specific topics and arguments, some relate to important protagonists of science, highlighting and explaining their contributions. Advanced mathematics is avoided, and formulas are introduced in only a few cases. The book is a user-friendly tool that nevertheless avoids scientific compromise. It is of interest to all who seek a better grasp of the world that surrounds us and of the ideas that have changed our perceptions.

While statistical mechanics describe the equilibrium state of systems with many degrees of freedom, and dynamical systems explain the irregular evolution of systems with few degrees of freedom, new tools are needed to study the evolution of systems with many degrees of freedom. This book presents the basic aspects of chaotic systems, with emphasis on systems composed by huge numbers of particles. Firstly, the basic concepts of chaotic dynamics are introduced, moving on to explore the role of ergodicity and chaos for the validity of statistical laws, and ending with problems characterized by the presence of more than one significant scale. Also discussed is the relevance of many degrees of freedom, coarse graining procedure, and instability mechanisms in justifying a statistical description of macroscopic bodies. Introducing the tools to characterize the non asymptotic behaviors of chaotic systems, this text will interest researchers and graduate students in statistical mechanics and chaos.

Chaos: from simple models to complex systems aims to guide science and engineering students through chaos and nonlinear dynamics from classical examples to the most recent fields of research. The first part, intended for undergraduate and graduate students, is a gentle and self-contained introduction to the concepts and main tools for the characterization of deterministic chaotic systems, with emphasis to statistical approaches. The second part can be used as a reference by researchers as it focuses on more advanced topics including the characterization of chaos with tools of information theory and applications encompassing fluid and celestial mechanics, chemistry and biology.The book is novel in devoting attention to a few topics often overlooked in introductory textbooks and which are usually found only in advanced surveys such as: information and algorithmic complexity theory applied to chaos and generalization of Lyapunov exponents to account for spatiotemporal and non-infinitesimal perturbations.The selection of topics, numerous illustrations, exercises and proposals for computer experiments make the book ideal for both introductory and advanced courses.

Chaos: from simple models to complex systems aims to guide science and engineering students through chaos and nonlinear dynamics from classical examples to the most recent fields of research. The first part, intended for undergraduate and graduate students, is a gentle and self-contained introduction to the concepts and main tools for the characterization of deterministic chaotic systems, with emphasis to statistical approaches. The second part can be used as a reference by researchers as it focuses on more advanced topics including the characterization of chaos with tools of information theory and applications encompassing fluid and celestial mechanics, chemistry and biology.The book is novel in devoting attention to a few topics often overlooked in introductory textbooks and which are usually found only in advanced surveys such as: information and algorithmic complexity theory applied to chaos and generalization of Lyapunov exponents to account for spatiotemporal and non-infinitesimal perturbations.The selection of topics, numerous illustrations, exercises and proposals for computer experiments make the book ideal for both introductory and advanced courses.

Scientists have always attempted to explain the world in terms of a few unifying principles. In the fifth century B.C. Democritus boldly claimed that reality is simply a collection of indivisible and eternal parts or atoms. Over the centuries his doctrine has remained a landmark, and much progress in physics is due to its distinction between subjective perception and objective reality. This book discusses theory reduction in physics, which states that the whole is nothing more than the sum of its parts: the properties of things are directly determined by their constituent parts. Reductionism deals with the relation between different theories that address different levels of reality, and uses extrapolations to apply that relation in different sciences. Reality shows a complex structure of connections, and the dream of a unified interpretation of all phenomena in several simple laws continues to attract anyone with genuine philosophical and scientific interests. If the most radical reductionist point of view is correct, the relationship between disciplines is strictly inclusive: chemistry becomes physics, biology becomes chemistry, and so on. Eventually, only one science, indeed just a single theory, would survive, with all others merging in the Theory of Everything. Is the current coexistence of different sciences a mere historical venture which will end when the Theory of Everything has been established? Can there be a unified description of nature? Rather than an analysis of full reductionism, this book focuses on aspects of theory reduction in physics and stimulates reflection on related questions: is there any evidence of actual reduction? Are the examples used in the philosophy of science too simplistic? What has been endangered by the search for (the) ultimate truth? Has the dream of reductionist reason created any monsters? Is big science one such monster? What is the point of embedding science Y within science X, if predictions cannot be made on that basis?

This book reviews the basic ideas of the Law of Large Numbers with its consequences to the deterministic world and the issue of ergodicity. Applications of Large Deviations and their outcomes to Physics are surveyed. The book covers topics encompassing ergodicity and its breaking and the modern applications of Large deviations to equilibrium and non-equilibrium statistical physics, disordered and chaotic systems, and turbulence.

Complexity is a puzzling and important concept in contemporary research in many disciplines. This book addresses the problem of defining complexity by carefully analysing in what sense complexity means measure in such areas as the theory of dynamical systems, condensed matter physics, ecology, immunology and the theory of neural networks. The information content of complexity is studied and similarities and differences in the various concepts of complexity are highlighted, sometimes provocatively. The book could open the way to finding a paradigm of complexity, and should become a standard reference for a wide audience of researchers in the physical and biological sciences.

At the present moment, after the success of the renormalization group in providing a conceptual framework for studying second-order phase tran sitions, we have a nearly satisfactory understanding of the statistical me chanics of classical systems with a non-random Hamiltonian. The situation is completely different if we consider the theory of systems with a random Hamiltonian or of chaotic dynamical systems. The two fields are connected; in fact, in the latter the effects of deterministic chaos can be modelled by an appropriate stochastic process. Although many interesting results have been obtained in recent years and much progress has been made, we still lack a satisfactory understanding of the extremely wide variety of phenomena which are present in these fields. The study of disordered or chaotic systems is the new frontier where new ideas and techniques are being developed. More interesting and deep results are expected to come in future years. The properties of random matrices and their products form a basic tool, whose importance cannot be underestimated. They playa role as important as Fourier transforms for differential equations. This book is extremely interesting as far as it presents a unified approach for the main results which have been obtained in the study of random ma trices. It will become a reference book for people working in the subject. The book is written by physicists, uses the language of physics and I am sure that many physicists will read it with great pleasure."

The present volume, published at the occasion of his 100th birthday anniversary, is a collection of articles that reviews the impact of Kolomogorov's work in the physical sciences and provides an introduction to the modern developments that have been triggered in this way to encompass recent applications in biology, chemistry, information sciences and finance.

La meccanica statistica (MS) nell'insegnamente universitario e' spesso confinata in una posizione itermedia tra le tre grandi aree della fisica teorica, la fisica della materia e la fisica matematica. In genere vengono discussi gli aspetti "pratici," di supporto alla fisica della materia, che pur importanti non esauriscono la rilevanza concettule della meccanica statistica. Esistono molti ottimi libri (Huang, Landau-Lifsits, Chandler, Peliti etc) che trattano in modo dettagliato gli aspetti tecnici della meccanica statistica. Lo scopo del nostro libro non e' quello di presentare metodi (esatti ed approssimati) per determinare le proprieta termodinamiche a partire dalle interazioni microscopiche, quanto discutere alcuni aspetti concettuali della meccanica statistica che sono spesso poco trattati. In particolare: 1- Il ruolo dell'ipotesi ergodica 2- L'importanza dei tanti gradi di liberta per le leggi statistiche 3- L'interpretazione degli ensemble in termini di probabilita; 4- L'irreversibilita macroscopica 5- L'utilizzo della meccanica statistica per provare l'ipotesi atomistica e la determinazione delle scale (spaziali ed egernetiche) del mondo microscopico."

This book offers an informal, easy-to-understand account of topics in modern physics and mathematics. The focus is, in particular, on statistical mechanics, soft matter, probability, chaos, complexity, and models, as well as their interplay. The book features 28 key entries and it is carefully structured so as to allow readers to pursue different paths that reflect their interests and priorities, thereby avoiding an excessively systematic presentation that might stifle interest. While the majority of the entries concern specific topics and arguments, some relate to important protagonists of science, highlighting and explaining their contributions. Advanced mathematics is avoided, and formulas are introduced in only a few cases. The book is a user-friendly tool that nevertheless avoids scientific compromise. It is of interest to all who seek a better grasp of the world that surrounds us and of the ideas that have changed our perceptions.

This book, first published in 1998, treats turbulence from the point of view of dynamical systems. The exposition centres around a number of important simplified models for turbulent behaviour in systems ranging from fluid motion (classical turbulence) to chemical reactions and interfaces in disordered systems.The modern theory of fractals and multifractals now plays a major role in turbulence research, and turbulent states are being studied as important dynamical states of matter occurring also in systems outside the realm of hydrodynamics, i.e. chemical reactions or front propagation. The presentation relies heavily on simplified models of turbulent behaviour, notably shell models, coupled map lattices, amplitude equations and interface models, and the focus is primarily on fundamental concepts such as the differences between large and small systems, the nature of correlations and the origin of fractals and of scaling behaviour. This book will be of interest to graduate students and researchers interested in turbulence, from physics and applied mathematics backgrounds.

Questo testo, che nasce dall'esperienza didattica degli autori, si propone di introdurre gli aspetti fondamentali della teoria della probabilita e dei processi stocastici, guardando con particolare attenzione alle connessioni con la meccanica statistica, il caos, le applicazioni modellistiche ed i metodi numerici. La prima parte e costituita da un'introduzione generale alla probabilita con particolare enfasi sulla probabilita condizionata, le densita marginali e i teoremi limite. Nella seconda parte, prendendo spunto dal moto Browniano, sono presentati i concetti fondamentali dei processi stocastici (catene di Markov, equazione di Fokker- Planck). La terza parte e una selezione di argomenti avanzati che possono essere trattati in corsi della laurea specialistica."

This book, first published in 1998, treats turbulence from the point of view of dynamical systems. The exposition centres around a number of important simplified models for turbulent behaviour in systems ranging from fluid motion (classical turbulence) to chemical reactions and interfaces in disordered systems.The modern theory of fractals and multifractals now plays a major role in turbulence research, and turbulent states are being studied as important dynamical states of matter occurring also in systems outside the realm of hydrodynamics, i.e. chemical reactions or front propagation. The presentation relies heavily on simplified models of turbulent behaviour, notably shell models, coupled map lattices, amplitude equations and interface models, and the focus is primarily on fundamental concepts such as the differences between large and small systems, the nature of correlations and the origin of fractals and of scaling behaviour. This book will be of interest to graduate students and researchers interested in turbulence, from physics and applied mathematics backgrounds.