The present volume contains expanded and substantially reworked records of invitedlecturesdeliveredduringthe38thKarpaczWinterSchoolofTheoretical Physics on "Dynamical Semigroups: Dissipation, Chaos, Quanta", which took placeinLadek , Zdr' oj,(Poland)intheperiod6-15February2002. Themainpurposeoftheschoolwastocreateaplatformfortheconfrontation ofviewpointsandresearchmethodologiesrepresentedbytwogroupsofexperts actually working in the very same area of theoretical physics. This situation is quite distinct in non-equilibrium statistical physics of open systems, where classicalandquantumaspectsareaddressedseparatelybymeansofverydi?erent andevenincompatibleformaltools. TheschooltopicsselectionbytheLecturersreads:dissipativedynamicsand chaoticbehaviour,modelsofenvironment-systemcouplingandmodelsofth- mostats;non-equilibriumstatisticalmechanicsandfarfromequilibriumphen- ena;quantumopensystems,decoherenceandlinkstoquantumchaos;quantum andclassicalapplicationsofMarkovsemigroupsandthevalidityofMarkovian approximations. Theorganizingprincipleforthewholeendeavourwastheissueofthedyn- ics of open systems and more speci?cally -15February2002. Themainpurposeoftheschoolwastocreateaplatformfortheconfrontation ofviewpointsandresearchmethodologiesrepresentedbytwogroupsofexperts actually working in the very same area of theoretical physics. This situation is quite distinct in non-equilibrium statistical physics of open systems, where classicalandquantumaspectsareaddressedseparatelybymeansofverydi?erent andevenincompatibleformaltools. TheschooltopicsselectionbytheLecturersreads:dissipativedynamicsand chaoticbehaviour,modelsofenvironment-systemcouplingandmodelsofth- mostats;non-equilibriumstatisticalmechanicsandfarfromequilibriumphen- ena;quantumopensystems,decoherenceandlinkstoquantumchaos;quantum andclassicalapplicationsofMarkovsemigroupsandthevalidityofMarkovian approximations. Theorganizingprincipleforthewholeendeavourwastheissueofthedyn- ics of open systems and more speci?cally - thedynamics of dissipation. Since this research area is extremely broad and varied, no single book can cover all importantdevelopments. Therefore,linkswithdynamicalchaoswerechosento representasupplementaryconstraint. Theprogrammeoftheschoolandits?naloutcomeintheformofthepresent volumehasbeenshapedwiththehelpofthescienti?ccommitteecomprising:R. Alicki,Ph. Blanchard,J. R. Dorfman,G. Gallavotti,P. Gaspard,I. Guarneri, ? F. Haake, M. Ku's, A. Lasota, B. Zegarlinski ' and K. Zyczkowski. Some of the committeememberstookchargeoflecturingtoo. Weconveyourthankstoall ofthem. Wewouldliketoexpresswordsofgratitudetomembersofthelocalorgan- ingcommittee,W. Ceg laandP. Lugiewicz, fortheirhelp. Specialthanksmust beextendedtoMrsAnnaJadczykforherhelpatvariousstagesoftheschool organizationandthecompetenteditorialassistance. Theschoolwas?nanciallysupportedbytheUniversityofWroc law,Univ- sityofZielonaG' ora,PolishMinistryofEducation,PolishAcademyofSciences, FoundationfortheKarpaczWinterSchoolofTheoreticalPhysicsandthe- nationfromtheDrWilhelmHeinrichHeraeusundElseHeraeusStiftung. Wrocla wandZielonaG' ora,Poland PiotrGarbaczewski June2002 RobertOlkiewicz TableofContents Introduction...1 ChapterI NonequilibriumDynamics SomeRecentAdvancesinClassicalStatisticalMechanics E. G. D. Cohen...7 DeterministicThermostatsandFluctuationRelations L. Rondoni...35 WhatIstheMicroscopicResponseofaSystem DrivenFarFromEquilibrium? C. Jarzynski...63 Non-equilibriumStatisticalMechanics ofClassicalandQuantumSystems D. Kusnezov,E. Lutz,K. Aoki...8 3 ChapterII DynamicsofRelaxationandChaoticBehaviour DynamicalTheoryofRelaxation inClassicalandQuantumSystems P. Gaspard...111 RelaxationandNoiseinChaoticSystems S. Fishman,S. Rahav...165 FractalStructuresinthePhaseSpace ofSimpleChaoticSystemswithTransport J. R. Dorfman...193 ChapterIII DynamicalSemigroups MarkovSemigroupsandTheirApplications R. Rudnicki,K. Pich'or,M. Tyran-Kaminska ' ...215 VIII TableofContents InvitationtoQuantumDynamicalSemigroups R. Alicki...239 FiniteDissipativeQuantumSystems M. Fannes...265 CompletePositivityinDissipativeQuantumDynamics F. Benatti,R. Floreanini,R. Romano...283 QuantumStochasticDynamicalSemigroup W. A. Majewski ...305 ChapterIV Driving,DissipationandControlinQuantumSystems DrivenChaoticMesoscopicSystems, DissipationandDecoherence D. Cohen...317 QuantumStateControlinCavityQED T. WellensandA. Buchleitner...351 SolvingSchrodinger'sEquationforanOpenSystem andItsEnvironment W. T. Strunz...377 ChapterV DynamicsofLargeSystems ThermodynamicBehaviorofLargeDynamicalSystems -Quantum1dConductorandClassicalMultibakerMap- S. Tasaki...395 CoherentandDissipativeTransport inAperiodicSolids:AnOverview J. Bellissard...

Based on a one-year course taught by the author to graduates at the University of Missouri, this book provides a student-friendly account of some of the standard topics encountered in an introductory course of ordinary differential equations. In a second semester, these ideas can be expanded by introducing more advanced concepts and applications. A central theme in the book is the use of Implicit Function Theorem, while the latter sections of the book introduce the basic ideas of perturbation theory as applications of this Theorem. The book also contains material differing from standard treatments, for example, the Fiber Contraction Principle is used to prove the smoothness of functions that are obtained as fixed points of contractions. The ideas introduced in this section can be extended to infinite dimensions.

This work systematically investigates a large number of oscillatory network configurations that are able to describe many real systems such as electric power grids, lasers or even the heart muscle, to name but a few. The book is conceived as an introduction to the field for graduate students in physics and applied mathematics as well as being a compendium for researchers from any field of application interested in quantitative models.

Unlike the conventional research for the general theory of stability, this mono graph deals with problems on stability and stabilization of dynamic systems with respect not to all but just to a given part of the variables characterizing these systems. Such problems are often referred to as the problems of partial stability (stabilization). They naturally arise in applications either from the requirement of proper performance of a system or in assessing system capa bility. In addition, a lot of actual (or desired) phenomena can be formulated in terms of these problems and be analyzed with these problems taken as the basis. The following multiaspect phenomena and problems can be indicated: * "Lotka-Volterra ecological principle of extinction;" * focusing and acceleration of particles in electromagnetic fields; * "drift" of the gyroscope axis; * stabilization of a spacecraft by specially arranged relative motion of rotors connected to it. Also very effective is the approach to the problem of stability (stabilization) with respect to all the variables based on preliminary analysis of partial sta bility (stabilization). A. M. Lyapunov, the founder of the modern theory of stability, was the first to formulate the problem of partial stability. Later, works by V. V. Rumyan tsev drew the attention of many mathematicians and mechanicians around the world to this problem, which resulted in its being intensively worked out. The method of Lyapunov functions became the key investigative method which turned out to be very effective in analyzing both theoretic and applied problems.

This monograph combines the knowledge of both the field of nonlinear dynamics and non-smooth mechanics, presenting a framework for a class of non-smooth mechanical systems using techniques from both fields. The book reviews recent developments, and opens the field to the nonlinear dynamics community. This book addresses researchers and graduate students in engineering and mathematics interested in the modelling, simulation and dynamics of non-smooth systems and nonlinear dynamics.

Complexity science has been a source of new insight in physical and social systems and has demonstrated that unpredictability and surprise are fundamental aspects of the world around us. This book is the outcome of a discussion meeting of leading scholars and critical thinkers with expertise in complex systems sciences and leaders from a variety of organizations, sponsored by the Prigogine Center at The University of Texas at Austin and the Plexus Institute, to explore strategies for understanding uncertainty and surprise. Besides contributions to the conference, it includes a key digest by the editors as well as a commentary by the late nobel laureate Ilya Prigogine, "Surprises in half of a century." The book is intended for researchers and scientists in complexity science, as well as for a broad interdisciplinary audience of both practitioners and scholars. It will well serve those interested in the research issues and in the application of complexity science to physical and social systems.

By now, most academics have heard something about the new science of complexity. In a manner reminiscent of Einstein and the last hundred years of physics, complexity science has captured the public imagination. (R) One can go to Amazon. com and purchase books on complexification (Casti 1994), emergence (Holland 1998), small worlds (Barabasi 2003), the web of life (Capra 1996), fuzzy thinking (Kosko 1993), global c- plexity (Urry 2003) and the business of long-tails (Anderson 2006). Even television has incorporated the topics of complexity science. Crime shows (R) (R) such as 24 or CSI typically feature investigators using the latest advances in computational modeling to "simulate scenarios" or "data mine" all p- sible suspects-all of which is done before the crime takes place. The (R) World Wide Web is another example. A simple search on Google. Com using the phrase "complexity science" gets close to a million hits! C- plexity science is ubiquitous. What most scholars do not realize, however, is the remarkable role sociologists are playing in this new science. C- sider the following examples. 0. 1 Sociologists in Complexity Science The first example comes from the new science of networks (Barabasi 2003). By now, most readers are familiar with the phenomena known as six-degrees of separation-the idea that, because most large networks are comprised of a significant number of non-random weak-ties, the nodes (e. g. , people, companies, etc.

This book is drawn from across many active fields of mathematics and physics. It has connections to atmospheric dynamics, spherical codes, graph theory, constrained optimization problems, Markov Chains, and Monte Carlo methods. It addresses how to access interesting, original, and publishable research in statistical modeling of large-scale flows and several related fields. The authors explicitly reach around the major branches of mathematics and physics, showing how the use of a few straightforward approaches can create a cornucopia of intriguing questions and the tools to answer them.

This IMA Volume in Mathematics and its Applications FRACTALS IN MULTIMEDIA is a result of a very successful three-day minisymposium on the same title. The event was an integral part of the IMA annual program on Mathemat ics in Multimedia, 2000-2001. We would like to thank Michael F. Barnsley (Department of Mathematics and Statistics, University of Melbourne), Di etmar Saupe (Institut fUr Informatik, UniversiUit Leipzig), and Edward R. Vrscay (Department of Applied Mathematics, University of Waterloo) for their excellent work as organizers of the meeting and for editing the proceedings. We take this opportunity to thank the National Science Foundation for their support of the IMA. Series Editors Douglas N. Arnold, Director of the IMA Fadil Santosa, Deputy Director of the IMA v PREFACE This volume grew out of a meeting on Fractals in Multimedia held at the IMA in January 2001. The meeting was an exciting and intense one, focused on fractal image compression, analysis, and synthesis, iterated function systems and fractals in education. The central concerns of the meeting were to establish within these areas where we are now and to develop a vision for the future."

In this text the authors consider the Korteweg-de Vries (KdV) equation (ut = - uxxx ] 6uux) with periodic boundary conditions. Derived to describe long surface waves in a narrow and shallow channel, this equation in fact models waves in homogeneous, weakly nonlinear and weakly dispersive media in general.

Viewing the KdV equation as an infinite dimensional, and in fact integrable Hamiltonian system, we first construct action-angle coordinates which turn out to be globally defined. They make evident that all solutions of the periodic KdV equation are periodic, quasi-periodic or almost-periodic in time. Also, their construction leads to some new results along the way.

Subsequently, these coordinates allow us to apply a general KAM theorem for a class of integrable Hamiltonian pde's, proving that large families of periodic and quasi-periodic solutions persist under sufficiently small Hamiltonian perturbations.

The pertinent nondegeneracy conditions are verified by calculating the first few Birkhoff normal form terms -- an essentially elementary calculation.

Besides turbulence there is hardly any other scientific topic which has been considered as a prominent scientific challenge for such a long time. The special interest in turbulence is not only based on it being a difficult scientific problem but also on its meaning in the technical world and our daily life. This carefully edited book comprises recent basic research as well as research related to the applications of turbulence. Therefore, both leading engineers and physicists working in the field of turbulence were invited to the iTi Conference on Turbulence held in Bad Zwischenahn, Gemany 25th - 28th of September 2005. Discussed topics include, for example, scaling laws and intermittency, thermal convection, boundary layers at large Reynolds numbers, isotropic turbulence, stochastic processes, passive and active scalars, coherent structures, numerical simulations, and related subjects.

Ergodic theory is hard to study because it is based on measure theory, which is a technically difficult subject to master for ordinary students, especially for physics majors. Many of the examples are introduced from a different perspective than in other books and theoretical ideas can be gradually absorbed while doing computer experiments. Theoretically less prepared students can appreciate the deep theorems by doing various simulations. The computer experiments are simple but they have close ties with theoretical implications. Even the researchers in the field can benefit by checking their conjectures, which might have been regarded as unrealistic to be programmed easily, against numerical output using some of the ideas in the book. One last remark: The last chapter explains the relation between entropy and data compression, which belongs to information theory and not to ergodic theory. It will help students to gain an understanding of the digital technology that has shaped the modern information society.

Since the publication of our first book [80], there has been a real resiu-gence of interest in the study of almost automorphic functions and their applications ([16, 17, 28, 29, 30, 31, 32, 40, 41, 42, 46, 51, 58, 74, 75, 77, 78, 79]). New methods (method of invariant s- spaces, uniform spectrum), and new concepts (almost periodicity and almost automorphy in fuzzy settings) have been introduced in the literature. The range of applications include at present linear and nonlinear evolution equations, integro-differential and functional-differential equations, dynamical systems, etc...It has become imperative to take a bearing of the main steps of the the ory. That is the main purpose of this monograph. It is intended to inform the reader and pave the road to more research in the field. It is not a self contained book. In fact, [80] remains the basic reference and fimdamental source of information on these topics. Chapter 1 is an introductory one. However, it contains also some recent contributions to the theory of almost automorphic functions in abstract spaces. VIII Preface Chapter 2 is devoted to the existence of almost automorphic solutions to some Unear and nonUnear evolution equations. It con tains many new results. Chapter 3 introduces to almost periodicity in fuzzy settings with applications to differential equations in fuzzy settings. It is based on a work by B. Bede and S. G. Gal [40].

The present volume is an introduction to nonlinear waves and soliton theory in the special environment of compact spaces such as closed curves and surfaces and other domain contours.

The first part of the book introduces the mathematical concept required for treating the manifolds considered. An introduction to the theory of motion of curves and surfaces is given. The second and third parts discuss the modeling of various physical solitons on compact systems.

Multibody Mechanics and Visualization is designed to appeal to computer-savvy students who will acquire significant skills in mathematical and physical modelling of mechanical systems in the process of producing attractive computer simulations and animations. The emphasis here is on general skills with all-round applicability rather than the ability to solve "cooked-up problems. The approachable style and clear presentation of this text will help you grasp the essentials of: modeling the kinematics and dynamics of arbitrary multibody mechanisms; formulating a mathematical description of general motions of such mechanisms; implementing the description in a computer-graphics application for the animation/visualization of the movement. Multibody Mechanics and Visualization plays down the prediction of dynamics by formal analysis of differential equations while preparing its students to perform such analyses with greater understanding later. The text relies on the following principles for effective tuition: an inductive approach to learning - discerning general patterns from particular observations; repetition and review of important principles to reinforce your learning through numerous examples; obvious visual guidance that shows you at a glance which information you need for different levels of understanding; computer tools, visual representations and elements of active learning integrated into the text to suit the way you want to learn. Supported in the text in parallel with the theoretical presentation is the simulation and animation application Mambo. In contrast with existing commercially available educational software tools, Mambo requires detailed input from you in order to define the specific geometry of a mechanism as well as the differential equations governing its behavior while allowing you to visualize the results of your efforts. The Mambo toolbox enables you to provide these specifications for mechanisms that would pose insurmountable algebraic challenges to manual calculation. With these tools, you will be able to see the implications of decisions made throughout the modeling process, to check your mathematical analyses, and to enjoy the fruit of your labor Mambo can be freely downloaded from the author's website and runs under any version of MS Windows(r). The toolbox is compatible with the Maple software environment and the Matlab(r) extended symbolic toolbox."

This book explains why complex systems research is important in understanding the structure, function and dynamics of complex natural and social phenomena. It illuminates how complex collective behavior emerges from the parts of a system, due to the interaction between the system and its environment. Readers will learn the basic concepts and methods of complex system research. The book is not highly technical mathematically, but teaches and uses the basic mathematical notions of dynamical system theory, making the book useful for students of science majors and graduate courses.

The notion of a ?xed point plays a crucial role in numerous branches of mat- maticsand its applications. Informationabout the existence of such pointsis often the crucial argument in solving a problem. In particular, topological methods of ?xed point theory have been an increasing focus of interest over the last century. These topological methods of ?xed point theory are divided, roughly speaking, into two types. The ?rst type includes such as the Banach Contraction Principle where the assumptions on the space can be very mild but a small change of the map can remove the ?xed point. The second type, on the other hand, such as the Brouwer and Lefschetz Fixed Point Theorems, give the existence of a ?xed point not only for a given map but also for any its deformations. This book is an exposition of a part of the topological ?xed and periodic point theory, of this second type, based on the notions of Lefschetz and Nielsen numbers. Since both notions are homotopyinvariants, the deformationis used as an essential method, and the assertions of theorems typically state the existence of ?xed or periodic points for every map of the whole homotopy class, we refer to them as homotopy methods of the topological ?xed and periodic point theory.

The main purpose of the book is to acquaint mathematicians, physicists and engineers with classical mechanics as a whole, in both its traditional and its contemporary aspects. As such, it describes the fundamental principles, problems, and methods of classical mechanics, with the emphasis firmly laid on the working apparatus, rather than the physical foundations or applications. Chapters cover the n-body problem, symmetry groups of mechanical systems and the corresponding conservation laws, the problem of the integrability of the equations of motion, the theory of oscillations and perturbation theory.

Thank heavens for Jens Wittenburg, of the University of Karlsruhe in Germany. Anyone who 's been laboring for years over equation after equation will want to give him a great big hug. It is common practice to develop equations for each system separately and to consider the labor necessary for deriving all of these as inevitable. Not so, says the author. Here, he takes it upon himself to describe in detail a formalism which substantially simplifies these tasks.

Besides turbulence, there is hardly any other scientific topic which has been considered a prominent scientific challenge for such a long time. The special interest in turbulence is not only based on it being a difficult scientific problem but also on its meaning in the technical world and our daily life. This carefully edited book comprises recent basic research as well as research related to the applications of turbulence. Therefore, both leading engineers and physicists working in the field of turbulence were invited to the iTi Conference on Turbulence held in Bad Zwischenahn, Gemany 21st - 24th of September 2003. Topics discussed include, for example, scaling laws and intermittency, thermal convection, boundary layers at large Reynolds numbers, isotropic turbulence, stochastic processes, passive and active scalars, coherent structures, numerical simulations, and related subjects.

The seminal 1970 Moscow thesis of Grigoriy A. Margulis, published for the first time. Entitled "On Some Aspects of the Theory of Anosov Systems," it uses ergodic theoretic techniques to study the distribution of periodic orbits of Anosov flows. The thesis introduces the "Margulis measure" and uses it to obtain a precise asymptotic formula for counting periodic orbits. This has an immediate application to counting closed geodesics on negatively curved manifolds. The thesis also contains asymptotic formulas for the number of lattice points on universal coverings of compact manifolds of negative curvature.

The thesis is complemented by a survey by Richard Sharp, discussing more recent developments in the theory of periodic orbits for hyperbolic flows, including the results obtained in the light of Dolgopyat's breakthroughs on bounding transfer operators and rates of mixing.

This text is well-designed with respect to the exposition from the preliminary to the more advanced and the applications interwoven throughout. It provides the essential foundations for the theory as well as the basic facts relating to almost periodicity. In six structured and self-contained chapters, the author unifies the treatment of various classes of almost periodic functions, while uniquely addressing oscillations and waves in the almost periodic case.

This is the first text to present the latest results in almost periodic oscillations and waves. The presentation level and inclusion of several clearly presented proofs make this work ideal for graduate students in engineering and science. The concept of almost periodicity is widely applicable to continuuum mechanics, electromagnetic theory, plasma physics, dynamical systems, and astronomy, which makes the book a useful tool for mathematicians and physicists.

The fourteen chapters of this book cover the central ideas and concepts of chaos and fractals as well as many related topics including: the Mandelbrot set, Julia sets, cellular automata, L-systems, percolation and strange attractors. This new edition has been thoroughly revised throughout. The appendices of the original edition were taken out since more recent publications cover this material in more depth. Instead of the focussed computer programs in BASIC, the authors provide 10 interactive JAVA-applets for this second edition.

Nature is inherently noisy and nonlinear. It is noisy in the sense that all macroscopic systems are subject to the fluctuations of their environments and also to internal fluctuations. It is nonlinear in the sense that the restoring force on a system displaced from equilibrium does not usually vary linearly with the size of the displacement. To calculate the properties of stochastic (noisy) nonlinear systems is in general extremely difficult, although considerable progress has been made in the past. The three volumes that make up Noise in Nonlinear Dynamical Systems comprise a collection of specially written authoritative reviews on all aspects of the subject, representative of all the major practitioners in the field. The first volume deals with the basic theory of stochastic nonlinear systems. It includes an historical overview of the origins of the field, chapters covering some developed theoretical techniques for the study of coloured noise, and the first English-language translation of the landmark 1933 paper by Pontriagin, Andronov and Vitt.

This work of applied mathematics focuses on the functional study of the nonlinear boundary value problems relating to water flow in porous media, a topic which has not up to now been explored in book form. The author shows that abstract theory may be sometimes easier and richer in consequences for applications than standard classical approaches are. The volume deals with diffusion type models, emphasizing the mathematical treatment of their nonlinear aspects.