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Quantum Thermodynamics ... Emergence Of Thermodynamic Behavior Within Composite .

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内容提示: Lecture Notes in PhysicsEditorial BoardR. Beig, Wien, AustriaW. Beiglb¨ ock, Heidelberg, GermanyW. Domcke, Garching, GermanyB.-G. Englert, SingaporeU. Frisch, Nice, FranceP. H¨ anggi, Augsburg, GermanyG. Hasinger, Garching, GermanyK. Hepp, Z¨ urich, SwitzerlandW. Hillebrandt, Garching, GermanyD. Imboden, Z¨ urich, SwitzerlandR. L. Jaffe, Cambridge, MA, USAR. Lipowsky, Golm, GermanyH. v. L¨ ohneysen, Karlsruhe, GermanyI. Ojima, Kyoto, JapanD. Sornette, Nice, France, and Los Angeles, CA, USAS. Theisen, ...

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Lecture Notes in PhysicsEditorial BoardR. Beig, Wien, AustriaW. Beiglb¨ ock, Heidelberg, GermanyW. Domcke, Garching, GermanyB.-G. Englert, SingaporeU. Frisch, Nice, FranceP. H¨ anggi, Augsburg, GermanyG. Hasinger, Garching, GermanyK. Hepp, Z¨ urich, SwitzerlandW. Hillebrandt, Garching, GermanyD. Imboden, Z¨ urich, SwitzerlandR. L. Jaffe, Cambridge, MA, USAR. Lipowsky, Golm, GermanyH. v. L¨ ohneysen, Karlsruhe, GermanyI. Ojima, Kyoto, JapanD. Sornette, Nice, France, and Los Angeles, CA, USAS. Theisen, Golm, GermanyW. Weise, Garching, GermanyJ. Wess, M¨ unchen, GermanyJ. Zittartz, K¨ oln, Germany The Editorial Policy for MonographsThe series Lecture Notes in Physics reports new developments in physical research andteaching - quickly, informally, and at a high level. 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Gemmer M. Michel G. MahlerQuantumThermodynamicsEmergence ofThermodynamic BehaviorWithin Composite Quantum Systems123 AuthorsJ. GemmerUniversit¨ at Osnabr¨ uckFB PhysikBarbarastr. 749069 Osnabr¨ uck, GermanyM. MichelG. MahlerUniversit¨ at StuttgartPfaffenwaldring 5770550 Stuttgart, GermanyJ. Gemmer M. Michel G. Mahler , Quantum Thermodynamics, Lect. Notes Phys. 657(Springer, Berlin Heidelberg 2005), DOI 10.1007/b98082Library ofCongress Control Number: 2004110894ISSN 0075-8450ISBN 3-540-22911-6 Springer Berlin Heidelberg New YorkThis work is subject to copyright. All rights are reserved, whether the whole or part ofthematerial is concerned, specifically the rights of translation, reprinting, reuse of illustra-tions, recitation, broadcasting, reproduction on microfilm or in any other way, andstorage in data banks. 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Mahler (Eds.),Quantum Thermodynamics PrefaceThis monograph views thermodynamics as an incomplete description of manyfreedom quantum systems. Left unaccounted for may be an environment withwhich the system of interest interacts; closed systems can be described incom-pletely by focussing on any subsystem with fewer particles and declaring theremainder as the environment. Any interaction with the environment bringsthe open system to a mixed quantum state, even if the closed compound stateis pure. Moreover, observables (and sometimes even the density operator) ofan open system may relax to equilibrium values, while the closed compoundstate keeps evolving unitarily ` a la Schr¨ odinger forever.The view thus taken can hardly be controversial for our generation ofphysicists. And yet, the authors offer surprises. Approach to equilibrium,with equilibrium characterized by maximum ignorance about the open sys-tem of interest, does not require excessively many particles: some dozens suf-fice! Moreover, the precise way of partitioning which might reflect subjectivechoices is immaterial for the salient features of equilibrium and equilibration.And what is nicest, quantum effects are at work in bringing about univer-sal thermodynamic behavior of modest size open systems. Von Neumann’sconcept of entropy thus appears as being much more widely useful than some-times feared, way beyond truely macroscopic systems in equilibrium.The authors have written numerous papers on their quantum view ofthermodynamics, and the present monograph is a most welcome coherentreview.Essen,June 2004Fritz Haake AcknowledgementsThe authors thank Dipl. Phys. Peter Borowski (MPI Dresden) for the first nu-merical simulations to test our theoretical considerations and for contributingseveral figures as well as some text material, and Dipl. Phys. Michael Hart-mann (DLR Stuttgart) for contributing some sections. Furthermore, we thankCand. Phys. Markus Henrich for helping us to design some diagrams and,both M. Henrich and Cand. Phys. Christos Kostoglou (Institut f¨ ur theoretis-che Physik, Universit¨ at Stuttgart) for supplying us with numerical data. Wehave profited a lot from fruitful discussions with Dipl. Phys. Harry Schmidt,Dipl. Phys. Marcus Stollsteimer and Dipl. Phys. Friedemann Tonner (In-stitut f¨ ur theoretische Physik, Universit¨ at Stuttgart) and Prof. Dr. KlausB¨ arwinkel, Prof. Dr. Heinz-J¨ urgen Schmidt and Prof. Dr. J¨ urgen Schnack(Fachbereich Physik, Universit¨ at Osnabr¨ uck). We benefitted much from con-versations with Prof. Dr. Wolfram Brenig and Dipl. Phys. Fabian Heidrich-Meisner (Technische Universit¨ at Braunschweig) as well as Dr. Alexander Otteand Dr. Heinrich Michel (Stuttgart). It is a pleasure to thank Springer-Verlag,especially Dr. Christian Caron, for continuous encouragement and excellentcooperation. This cooperation has garanteed a rapid and smooth progress ofthe project. Financial support by the “Deutsche Forschungsgesellschaft” andthe “Landesstiftung Baden-W¨ urttemberg” is gratefully acknowledged. Lastbut not least, we would like to thank Bj¨ orn Butscher, Kirsi Weber and Hen-drik Weimer for helping us with typesetting the manuscript, proof-readingand preparing some of the figures. ContentsPart I Background1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .32Basics of Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.1Introductory Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.2Operator Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.2.1Transition Operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.2.2Pauli Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.2.3State Representation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.2.4Purity and von Neumann Entropy . . . . . . . . . . . . . . . . . .2.2.5Bipartite Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.2.6Multi-Partite Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.3Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.4Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.5Time-Dependent Perturbation Theory. . . . . . . . . . . . . . . . . . . . .2.5.1Interaction Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.5.2Series Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .77889910121415161819203Basics of Thermodynamics and Statistics . . . . . . . . . . . . . . . . .3.1Phenomenological Thermodynamics. . . . . . . . . . . . . . . . . . . . . . .3.1.1Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.1.2Fundamental Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.1.3Gibbsian Fundamental Form . . . . . . . . . . . . . . . . . . . . . . .3.1.4Thermodynamic Potentials . . . . . . . . . . . . . . . . . . . . . . . .3.2Linear Irreversible Thermodynamics . . . . . . . . . . . . . . . . . . . . . .3.3Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.3.1Boltzmann’s Principle, A Priori Postulate . . . . . . . . . . .3.3.2Microcanonical Ensemble . . . . . . . . . . . . . . . . . . . . . . . . . .3.3.3Statistical Entropy, Maximum Principle . . . . . . . . . . . . .21212123262628303132344Brief Review of Pertinent Concepts. . . . . . . . . . . . . . . . . . . . . . .4.1Boltzmann’s Equation and H-Theorem . . . . . . . . . . . . . . . . . . . .4.2Ergodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.3Ensemble Approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .37384243 XContents4.44.54.64.74.8Macroscopic Cell Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .The Problem of Adiabatic State Change . . . . . . . . . . . . . . . . . . .Shannon Entropy, Jaynes’ Principle . . . . . . . . . . . . . . . . . . . . . . .Time-Averaged Density Matrix Approach . . . . . . . . . . . . . . . . .Open System Approach and Master Equation . . . . . . . . . . . . . .4.8.1Classical Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.8.2Quantum Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .45485052535354Part II Quantum Approach to Thermodynamics5The Program for the Foundation of Thermodynamics. . . . .5.1Basic Checklist: Equilibrium Thermodynamics . . . . . . . . . . . . .5.2Supplementary Checklist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6161646Outline of the Present Approach . . . . . . . . . . . . . . . . . . . . . . . . .6.1Compound Systems, Entropy and Entanglement . . . . . . . . . . . .6.2Fundamental and Subjective Lack of Knowledge . . . . . . . . . . . .6.3The Natural Cell Structure of Hilbert Space . . . . . . . . . . . . . . .656567677System and Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7.1Partition of the System and Basic Quantities. . . . . . . . . . . . . . .7.2Weak Coupling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7.3Effective Potential, Example for a Bipartite System . . . . . . . . .717173748Structure of Hilbert Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8.1Representation of Hilbert Space . . . . . . . . . . . . . . . . . . . . . . . . . .8.2Hilbert Space Average . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8.3Hilbert Space Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8.4Purity and Local Entropy in Product Hilbert Space . . . . . . . . .8.4.1Unitary Invariant Distribution of Pure States . . . . . . . .8.4.2Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .797982858686889Quantum Thermodynamic Equilibrium . . . . . . . . . . . . . . . . . . .9.1Microcanonical Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9.1.1Accessible Region (AR) . . . . . . . . . . . . . . . . . . . . . . . . . . .9.1.2The “Landscape” of Pgin the Accessible Region . . . . .9.1.3The Minimum Purity State . . . . . . . . . . . . . . . . . . . . . . . .9.1.4The Hilbert Space Average of Pg. . . . . . . . . . . . . . . . . . .9.1.5Microcanonical Equilibrium. . . . . . . . . . . . . . . . . . . . . . . .9.2Energy Exchange Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9.2.1The Accessible and the Dominant Regions . . . . . . . . . . .9.2.2Identification of the Dominant Region. . . . . . . . . . . . . . .9.2.3Analysis of the Size of the Dominant Region . . . . . . . . . 1019.2.4The Equilibrium State . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10291919193939597979899 ContentsXI9.39.4Canonical Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103Fluctuations of Occupation Probabilities WA. . . . . . . . . . . . . . . 10410 Interim Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10910.1 Equilibrium Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10910.1.1 Microcanonical Contact . . . . . . . . . . . . . . . . . . . . . . . . . . . 10910.1.2 Energy Exchange Contact, Canonical Contact . . . . . . . . 11010.2 Local Equilibrium States and Ergodicity . . . . . . . . . . . . . . . . . . 11211Typical Spectra of Large Systems . . . . . . . . . . . . . . . . . . . . . . . . . 11311.1 The Extensitivity of Entropy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11311.2 Spectra of Modular Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11411.3 Entropy of an Ideal Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11811.4 The Boltzmann Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11911.5 Beyond the Boltzmann Distribution? . . . . . . . . . . . . . . . . . . . . . . 12112Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12312.1 Definition of Spectral Temperature . . . . . . . . . . . . . . . . . . . . . . . 12412.2 The Equality of Spectral Temperatures in Equilibrium . . . . . . 12512.3 Spectral Temperature as the Derivative of Energy . . . . . . . . . . 12712.3.1 Contact with a Hotter System . . . . . . . . . . . . . . . . . . . . . 12812.3.2 Energy Deposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12913Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13313.1 On the Concept of Adiabatic Processes . . . . . . . . . . . . . . . . . . . . 13313.2 The Equality of Pressures in Equilibrium . . . . . . . . . . . . . . . . . . 13914 Quantum Mechanical and Classical State Densities . . . . . . . 14314.1 Bohr–Sommerfeld Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . 14414.2 Partition Function Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14614.3 Minimum Uncertainty Wave Package Approach . . . . . . . . . . . . 14714.4 Implications of the Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15614.5 Correspondence Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15715Sufficient Conditions for a Thermodynamic Behavior . . . . . 15915.1 Weak Coupling Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15915.2 Microcanonical Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16015.3 Energy Exchange Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16115.4 Canonical Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16115.5 Spectral Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16215.6 Parametric Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16215.7 Extensitivity of Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 XIIContents16Theories of Relaxation Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . 16516.1 Fermi’s Golden Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16516.2 Weisskopf–Wigner Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16616.3 Open Quantum Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16717 The Route to Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16917.1 System and a Large Environment . . . . . . . . . . . . . . . . . . . . . . . . . 16917.2 Time Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17117.3 Hilbert Space Average . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17317.4 Short Time Step Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17417.5 Derivation of a Rate Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 17817.6 Solution of the Rate Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 17917.7 Hilbert Space Variances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18017.8 Numerical Results for the Relaxation Period . . . . . . . . . . . . . . . 181Part III Applications and Models18Equilibrium Properties of Model Systems. . . . . . . . . . . . . . . . . 18518.1 Entropy Under Microcanonical Conditions . . . . . . . . . . . . . . . . . 18518.2 Occupation Probabilities Under Canonical Conditions . . . . . . . 18818.3 Probability Fluctuations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19218.4 Spin Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19318.4.1 Global Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19418.4.2 Local Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19518.4.3 Chain Coupled Locally to a Bath . . . . . . . . . . . . . . . . . . . 19818.5 On the Existence of Local Temperatures. . . . . . . . . . . . . . . . . . . 20018.5.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20218.5.2 Global Thermal State in the Product Basis . . . . . . . . . . 20318.5.3 Conditions for Local Thermal States . . . . . . . . . . . . . . . . 20418.5.4 Spin Chain in a Transverse Field . . . . . . . . . . . . . . . . . . . 20718.6 Quantum Thermometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20918.6.1 The Classical Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20918.6.2 The Szilard Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20918.6.3 Temperature Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . 21018.6.4 Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21018.6.5 Thermodynamic “Uncertainty Relation” . . . . . . . . . . . . . 21218.7 Quantum Manometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21218.7.1 Eigenspectrum of System and Manometer . . . . . . . . . . . 21218.7.2 The Total Scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21418.7.3 Classical Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21518.8 Adiabatic Following and Adiabatic Process . . . . . . . . . . . . . . . . 216 ContentsXIII19Heat Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22119.1 Theories of Heat Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22119.1.1 Linear Response Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . 22219.1.2 Quasi-Particle Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 22319.2 Quantum Heat Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22419.2.1 Model Hamiltonian and Environment . . . . . . . . . . . . . . . 22519.2.2 Current Operator and Fourier’s Law . . . . . . . . . . . . . . . . 22719.2.3 Linear Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 22919.2.4 Heat Conduction in Low Dimensional Systems . . . . . . . 23019.2.5 Fourier’s Law for a Heisenberg Chain . . . . . . . . . . . . . . . 23119.2.6 Implications of These Investigations . . . . . . . . . . . . . . . . 23220 Quantum Thermodynamic Machines . . . . . . . . . . . . . . . . . . . . . . 23520.1 Tri-Partite System: Sudden Changes of Embedding . . . . . . . . . 23520.2 Work Variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23720.3 Carnot Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23720.4 Generalized Control Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23921Summary and Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241Part IV AppendicesAHyperspheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247A.1 Surface of a Hypersphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247A.2 Integration of a Function on a Hypersphere . . . . . . . . . . . . . . . . 248A.3 Sizes of Zones on a Hypersphere . . . . . . . . . . . . . . . . . . . . . . . . . . 250BHilbert Space Average under Microcanonical Conditions . 253CHilbert Space Averages and Variances . . . . . . . . . . . . . . . . . . . . 259C.1 General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259C.2 Special Hilbert Space Averages . . . . . . . . . . . . . . . . . . . . . . . . . . . 262C.3 Special Hilbert Space Variances . . . . . . . . . . . . . . . . . . . . . . . . . . 263DPower of a Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265ELocal Temperature Conditions for a Spin Chain . . . . . . . . . . 267References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 List of Symbolsˆ1ˆ1(µ)∇δijδ(. . . )⊗dAUnit operatorUnit operator in Hilbert space of subsystem µNabla operatorKronecker deltaDirac δ-functionDyadic productInfinitesimal change of an integrable quantity, completedifferentialFinite change of AInfinitesimal change of a non-integrable quantityConvolution of N identical functions fHilbert space average of a quantity fSurface of an n-dimensional hypersphere with radius RCommutator ofˆA withˆBFourier transformation of fMean value of fScalar product in Hilbert spaceSet of all coordinates i = 1, . . . nSet of coordinates in subspace ABSubset of coordinates in subspace JGeneralized spherical coordinatesSpherical coordinates of subspace JSpherical coordinates of subspace ABExpectation value of operatorˆAOperatorOperator in subspace µAdjoint operatorTime-dependent operator in Heisenberg pictureOperator in the interaction pictureBasis state of the gas system gIndex of the energy subspace of the gas system withenergy EgAIndex of the degenerate eigenstates belonging to one en-ergy subspace A of the gas system∆AδACN{f}fO(R, n)ˆA,ˆBF{f}fi|j{ηi, ξi}{ηAB{ηJ{r, φi}{rJ, φJ{rAB, φABˆAˆAˆA(µ)ˆA†ˆAH(t)ˆAI|A, aAab, ξABi, ξJi}ab}i}i}a XVIList of SymbolsA, B/EAijA∗ijAR|B, bBIndex operation under the constraint EgMatrix elements of an operatorˆAComplex conjugate matrix elements of an operatorˆAAccessible regionBasis state of the container system cIndex of the energy subspace of the container systemwith energy EcBIndex of the degenerate eigenstates belonging to one en-ergy subspace B of the container systemLabel for containerDistance measure (Bures metric)Diagonal deviation matrixDimension of the Liouville spaceDimension of special subspace AEnergy of the state |iEnergyZero-point energyTotal energy of container systemEnergy eigenvalues of the container systemTotal Energy of gas system gEnergy eigenvalues of the gas system gOff-diagonal part of a matrix, deviation matrixFidelity between ˆ ρ and ˆ ρFree energyJacobian matrix (functional matrix)Jacobian matrix in subspace JSee FJForceGibbs free energyState density of subsystemsEnergy spectrum of wave package |γState density of the container systemState density of the gas systemState density at energy ELabel for gas subsystemEnthalpyHamilton functionHamiltonianLocal Hamiltonian of subsystem µNext neighbor F¨ orster couplingNext neighbor random couplingNext neighbor non-resonant couplingUnperturbed HamiltonianPlanck’s constantA+ EcB= EbcD2DddAEiEE0EcEcEgEgEFˆ ρˆ ρF(T, V)FFJFABFG(T, p)g(E)g(γ, E)Gc(EcGg(EgG(E)gH(S, p)H(q, p)ˆHˆH(µ)locˆH(µ,µ+1)FˆH(µ,µ+1)RˆH(µ,µ+1)NRˆH0 = h/2πˆ ρˆ ρBAB)A) List of SymbolsXVIIˆHcˆHgHH(µ)iˆIˆIgc|ii||i, j, . . .|i, t|i ⊗ |j = |ijJjjujsˆJˆJ(µ,µ+1)kBˆLHamiltonian of the container system cHamiltonian of the gas system gHilbert spaceHilbert space of subsystem µImaginary unitInteraction operatorInteraction between gas system and container systemBasis stateAdjoint basis stateProduct state of several subsystemsTime-dependent stateProduct state of two subsystemsLabel for a side conditionCurrentEnergy currentEntropy currentCurrent operatorLocal current operator between subsystem µ and µ + 1Boltzmann constantSuper-operator acting on operators ofthe Liouville space(Lindblad operator)Coherent part of Lindblad super-operatorIncoherent part of Lindblad super-operatorTransport coefficient (in general a matrix)Number of micro states accessible for a system; massNumber of levels in lower bandNumber of levels in upper bandTotal number of states in a subspace B, degeneracyNumber of levels under the constraint A, B/ETotal number of states in a subspace A, degeneracyNumber of macro variablesNumber of levels of subsystem µDimension of the total Hilbert spaceNumber of subsystems, particle numberNumber of levels of a subsystemNumber of states in subspace ABSee Ng(EgA)See Nc(EcB)Extensive macro variableVector of all momentum coordinatesMomentum of the µthparticleMomentum operator of the µthparticleProjector projecting out some part ex of the total statespaceˆLcohˆL1,2incLmNc0Nc1Nc(EcN(E)Ng(Egnvarn(µ)ntotNnNAB= NANBNANBXippµˆ pµˆPexB)A) XVIIIList of SymbolspˆP(µ)iiˆPijPQˆ qµqqqµ{qc{qgrRJRABSStotSlins(q, t)s(U, V)sph(n)T(q, t)TT(µ)tTr {. . . }Trµ{. . . }u(q, t)UˆUˆU0(t, t0)UijˆU(t)ˆU(µ)(t)vˆVˆVI(t)VvW(EgW(EcWd(EcWd(EgW(E)W(EgPressureProjector within subspace µTransition operator, projector for i = jPurityHeatPosition operator of the µthparticlePositionVector of position coordinates of N particlesPosition of the µthparticleSet of position coordinates of all container particlesSet of position coordinates of all gas particlesRadial coordinate of a hypersphereRadius of hypersphere in subspace JSee RJEntropyEntropy of system and environmentLinearized von Neumann entropyEntropy densitySpecific entropyn-dimensional hypersphereTemperature fieldTemperatureLocal temperature of subsystem µTimeTrace operationPartial trace over subsystem µEnergy densityInternal energy...

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