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...83 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...