Quantum thermodynamics: emergence of thermodynamic behavior within composite quantum systems

286 Pages·2004·1.809 MB·English

by  J. Gemmer, M. Michel, G. Mahler (auth.)

Lecture Notes in Physics
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J. Gemmer M. Michel G. Mahler
Quantum Thermodynamics
Emergence of Thermodynamic Behavior
Within Composite Quantum Systems
123
Authors
J.Gemmer M.Michel
Universita¨tOsnabru¨ck G.Mahler
FBPhysik Universita¨tStuttgart
Barbarastr.7 Pfaffenwaldring57
49069Osnabru¨ck,Germany 70550Stuttgart,Germany
J. Gemmer M. Michel G. Mahler , Quantum Thermodynamics, Lect. Notes Phys. 657
(Springer,BerlinHeidelberg2005),DOI10.1007/b98082
LibraryofCongressControlNumber:2004110894
ISSN0075-8450
ISBN3-540-22911-6SpringerBerlinHeidelbergNewYork
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Preface
Thismonographviewsthermodynamicsasanincompletedescriptionofmany
freedomquantumsystems.Leftunaccountedformaybeanenvironmentwith
whichthesystemofinterestinteracts;closedsystemscanbedescribedincom-
pletely by focussing on any subsystem with fewer particles and declaring the
remainder as the environment. Any interaction with the environment brings
theopensystemtoamixedquantumstate,eveniftheclosedcompoundstate
is pure. Moreover, observables (and sometimes even the density operator) of
an open system may relax to equilibrium values, while the closed compound
state keeps evolving unitarily a` la Schro¨dinger forever.
The view thus taken can hardly be controversial for our generation of
physicists. And yet, the authors offer surprises. Approach to equilibrium,
with equilibrium characterized by maximum ignorance about the open sys-
temofinterest,doesnotrequireexcessivelymanyparticles:somedozenssuf-
fice! Moreover, the precise way of partitioning which might reflect subjective
choicesisimmaterialforthesalientfeaturesofequilibriumandequilibration.
And what is nicest, quantum effects are at work in bringing about univer-
sal thermodynamic behavior of modest size open systems. Von Neumann’s
conceptofentropythusappearsasbeingmuchmorewidelyusefulthansome-
times feared, way beyond truely macroscopic systems in equilibrium.
The authors have written numerous papers on their quantum view of
thermodynamics, and the present monograph is a most welcome coherent
review.
Essen, Fritz Haake
June 2004
Acknowledgements
TheauthorsthankDipl.Phys.PeterBorowski(MPIDresden)forthefirstnu-
mericalsimulationstotestourtheoreticalconsiderationsandforcontributing
several figures as well as some text material, and Dipl. Phys. Michael Hart-
mann(DLRStuttgart)forcontributingsomesections.Furthermore,wethank
Cand. Phys. Markus Henrich for helping us to design some diagrams and,
bothM.HenrichandCand.Phys.ChristosKostoglou(Institutfu¨rtheoretis-
che Physik, Universita¨t Stuttgart) for supplying us with numerical data. We
have profited a lot from fruitful discussions with Dipl. Phys. Harry Schmidt,
Dipl. Phys. Marcus Stollsteimer and Dipl. Phys. Friedemann Tonner (In-
stitut fu¨r theoretische Physik, Universita¨t Stuttgart) and Prof. Dr. Klaus
B¨arwinkel, Prof. Dr. Heinz-Ju¨rgen Schmidt and Prof. Dr. Ju¨rgen Schnack
(FachbereichPhysik,Universita¨tOsnabru¨ck).Webenefittedmuchfromcon-
versations with Prof. Dr. Wolfram Brenig and Dipl. Phys. Fabian Heidrich-
Meisner(TechnischeUniversit¨atBraunschweig)aswellasDr.AlexanderOtte
andDr.HeinrichMichel(Stuttgart).ItisapleasuretothankSpringer-Verlag,
especially Dr. Christian Caron, for continuous encouragement and excellent
cooperation. This cooperation has garanteed a rapid and smooth progress of
theproject.Financialsupportbythe“DeutscheForschungsgesellschaft”and
the “Landesstiftung Baden-Wu¨rttemberg” is gratefully acknowledged. Last
but not least, we would like to thank Bjo¨rn Butscher, Kirsi Weber and Hen-
drik Weimer for helping us with typesetting the manuscript, proof-reading
and preparing some of the figures.
Contents
Part I Background
1 Introduction.............................................. 3
2 Basics of Quantum Mechanics ............................ 7
2.1 Introductory Remarks................................... 7
2.2 Operator Representations ............................... 8
2.2.1 Transition Operators.............................. 8
2.2.2 Pauli Operators .................................. 9
2.2.3 State Representation.............................. 9
2.2.4 Purity and von Neumann Entropy.................. 10
2.2.5 Bipartite Systems ................................ 12
2.2.6 Multi-Partite Systems............................. 14
2.3 Dynamics ............................................. 15
2.4 Invariants ............................................. 16
2.5 Time-Dependent Perturbation Theory..................... 18
2.5.1 Interaction Picture ............................... 19
2.5.2 Series Expansion ................................. 20
3 Basics of Thermodynamics and Statistics ................. 21
3.1 Phenomenological Thermodynamics....................... 21
3.1.1 Basic Definitions ................................. 21
3.1.2 Fundamental Laws ............................... 23
3.1.3 Gibbsian Fundamental Form....................... 26
3.1.4 Thermodynamic Potentials ........................ 26
3.2 Linear Irreversible Thermodynamics ...................... 28
3.3 Statistics .............................................. 30
3.3.1 Boltzmann’s Principle, A Priori Postulate ........... 31
3.3.2 Microcanonical Ensemble.......................... 32
3.3.3 Statistical Entropy, Maximum Principle ............. 34
4 Brief Review of Pertinent Concepts....................... 37
4.1 Boltzmann’s Equation and H-Theorem.................... 38
4.2 Ergodicity ............................................. 42
4.3 Ensemble Approach..................................... 43
X Contents
4.4 Macroscopic Cell Approach .............................. 45
4.5 The Problem of Adiabatic State Change................... 48
4.6 Shannon Entropy, Jaynes’ Principle....................... 50
4.7 Time-Averaged Density Matrix Approach ................. 52
4.8 Open System Approach and Master Equation .............. 53
4.8.1 Classical Domain................................. 53
4.8.2 Quantum Domain ................................ 54
Part II Quantum Approach to Thermodynamics
5 The Program for the Foundation of Thermodynamics..... 61
5.1 Basic Checklist: Equilibrium Thermodynamics ............. 61
5.2 Supplementary Checklist ................................ 64
6 Outline of the Present Approach ......................... 65
6.1 Compound Systems, Entropy and Entanglement............ 65
6.2 Fundamental and Subjective Lack of Knowledge............ 67
6.3 The Natural Cell Structure of Hilbert Space ............... 67
7 System and Environment ................................. 71
7.1 Partition of the System and Basic Quantities............... 71
7.2 Weak Coupling......................................... 73
7.3 Effective Potential, Example for a Bipartite System......... 74
8 Structure of Hilbert Space................................ 79
8.1 Representation of Hilbert Space .......................... 79
8.2 Hilbert Space Average .................................. 82
8.3 Hilbert Space Variance.................................. 85
8.4 Purity and Local Entropy in Product Hilbert Space......... 86
8.4.1 Unitary Invariant Distribution of Pure States ........ 86
8.4.2 Application...................................... 88
9 Quantum Thermodynamic Equilibrium ................... 91
9.1 Microcanonical Conditions............................... 91
9.1.1 Accessible Region (AR) ........................... 91
9.1.2 The “Landscape” of Pg in the Accessible Region ..... 93
9.1.3 The Minimum Purity State........................ 93
9.1.4 The Hilbert Space Average of Pg................... 95
9.1.5 Microcanonical Equilibrium........................ 97
9.2 Energy Exchange Conditions............................. 97
9.2.1 The Accessible and the Dominant Regions........... 98
9.2.2 Identification of the Dominant Region............... 99
9.2.3 Analysis of the Size of the Dominant Region ......... 101
9.2.4 The Equilibrium State ............................ 102
Contents XI
9.3 Canonical Conditions ................................... 103
9.4 Fluctuations of Occupation Probabilities W ............... 104
A
10 Interim Summary......................................... 109
10.1 Equilibrium Properties .................................. 109
10.1.1 Microcanonical Contact ........................... 109
10.1.2 Energy Exchange Contact, Canonical Contact........ 110
10.2 Local Equilibrium States and Ergodicity.................. 112
11 Typical Spectra of Large Systems......................... 113
11.1 The Extensitivity of Entropy............................. 113
11.2 Spectra of Modular Systems ............................. 114
11.3 Entropy of an Ideal Gas ................................. 118
11.4 The Boltzmann Distribution ............................. 119
11.5 Beyond the Boltzmann Distribution?...................... 121
12 Temperature ............................................. 123
12.1 Definition of Spectral Temperature ....................... 124
12.2 The Equality of Spectral Temperatures in Equilibrium ...... 125
12.3 Spectral Temperature as the Derivative of Energy .......... 127
12.3.1 Contact with a Hotter System ..................... 128
12.3.2 Energy Deposition................................ 129
13 Pressure.................................................. 133
13.1 On the Concept of Adiabatic Processes.................... 133
13.2 The Equality of Pressures in Equilibrium .................. 139
14 Quantum Mechanical and Classical State Densities ....... 143
14.1 Bohr–Sommerfeld Quantization .......................... 144
14.2 Partition Function Approach............................. 146
14.3 Minimum Uncertainty Wave Package Approach ............ 147
14.4 Implications of the Method .............................. 156
14.5 Correspondence Principle................................ 157
15 Sufficient Conditions for a Thermodynamic Behavior ..... 159
15.1 Weak Coupling Limit ................................... 159
15.2 Microcanonical Equilibrium.............................. 160
15.3 Energy Exchange Equilibrium............................ 161
15.4 Canonical Equilibrium .................................. 161
15.5 Spectral Temperature ................................... 162
15.6 Parametric Pressure .................................... 162
15.7 Extensitivity of Entropy................................. 163