A quantum approach to thermodynamics [Elektronische Ressource] / vorgelegt von Jochen Gemmer

Sujets

• Physik
• Astronomie

Informations

• Publié par: universitat_stuttgart
• Publié le: 01 janvier 2003
• Nombre de lectures: 26
• Langue: English

Extrait

A Quantum Approach
to
Thermodynamics
Von der Fakultat Mathematik und Physik der Universitat Stuttgart˜ ˜
zur Erlangung der Wu˜rde eines Doktors der
Naturwissenschaften (Dr. rer. nat.) genehmigte Abhandlung
vorgelegt von
Jochen Gemmer
aus Stuttgart
Hauptberichter: Prof. Dr. G. Mahler
Mitberichter: Prof. Dr. U. Seifert
Tag der mundlichen Prufung: 26. Februar 2003˜ ˜
Institut fu˜r Theoretische Physik I
Universitat Stuttgart˜
2003ii
AllnumericalsimulationsinthisthesishavebeencomputedverycarefullybyP.Borowski.Contents
1 Introduction 1
2 Review of the Field 3
2.1 Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Ergodicity Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.3 Ensemble Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.4 Macroscopic Cell Approach . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.5 The Problem of Entropy Invariance . . . . . . . . . . . . . . . . . . . . . 8
2.6 Shannon’s Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.7 Time averaged Density Matrix Approach . . . . . . . . . . . . . . . . . . 10
2.8 Open System Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 Structure of a Foundation of Thermodynamics 13
3.1 Checklist of Properties of thermodynamical Quantities . . . . . . . . . . 13
3.1.1 Additional necessary Considerations . . . . . . . . . . . . . . . . . 15
4 Background of the Present Approach 17
4.1 Density Operator and Reduced Density Operator . . . . . . . . . . . . . 17
4.2 Compound Systems, Entropy and Entanglement . . . . . . . . . . . . . . 18
4.3 Fundamental and Subjective Lack of Knowledge . . . . . . . . . . . . . . 20
4.4 The natural Cell Structure of Hilbertspace . . . . . . . . . . . . . . . . . 20
5 Analysis of the Cell Structure of Compound Hilbertspaces 23
5.1 Representation of Hilbertspace and Hilbertspace Velocity . . . . . . . . . 23
5.2 Purity and Notation of States . . . . . . . . . . . . . . . . . . . . . . . . 25
5.3 Partition of the Full System into considered System and Surrounding . . 27
5.4 Microcanonical Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5.4.1 Microcanonical Interactions and the corresponding accessible Region 29
g5.4.2 The\Landscape"of P in the accessible Region . . . . . . . . . . 30
g5.4.3 The minimum Purity State and the Hilbertspace Average of P . 30
5.5 Canonical Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5.5.1 The accessible and the dominant Region . . . . . . . . . . . . . . 34
5.5.2 Identiflcation of the dominant Region . . . . . . . . . . . . . . . . 34
5.5.3 Analysis of the Size of the dominant Region . . . . . . . . . . . . 35
5.5.4 The canonical equilibrium State . . . . . . . . . . . . . . . . . . . 36
iiiiv Contents
5.6 Single Energy Probabilities and Fluctuations . . . . . . . . . . . . . . . . 38
5.7 Local Equilibrium States and Ergodicity . . . . . . . . . . . . . . . . . . 40
5.8 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.8.1 Numerical Results for microcanonical Conditions . . . . . . . . . 42
5.8.2 Numerical Results for Canonical Conditions . . . . . . . . . . . . 45
5.8.3 Numerical Results for Probability Fluctuations . . . . . . . . . . . 49
6 Typical Spectra of Large Systems 51
6.1 The Extensitivity of Entropy . . . . . . . . . . . . . . . . . . . . . . . . . 51
6.2 Spectra of Modular Systems . . . . . . . . . . . . . . . . . . . . . . . . . 52
6.3 The Boltzmann Distribution . . . . . . . . . . . . . . . . . . . . . . . . . 56
6.3.1 Beyond the Boltzmann Distribution? . . . . . . . . . . . . . . . . 57
7 Temperatures 59
7.1 Deflnition of spectral Temperature . . . . . . . . . . . . . . . . . . . . . 60
7.2 The Equality of spectral Temperatures in Equilibrium . . . . . . . . . . . 61
7.3 Spectral Temperature as the Derivative of Energy with Respect to Entropy 63
8 Pressure 67
8.1 On the Concept of adiabatic Processes . . . . . . . . . . . . . . . . . . . 67
8.2 The Equality of parametric Pressures in Equilibrium . . . . . . . . . . . 71
9 Thermodynamical Limit 73
9.1 Weak coupling Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
9.2 Microcanonical Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . 74
9.3 Energy-Exchange Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . 74
9.4 Canonical Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
9.5 Spectral Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
9.6 Parametric Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
9.7 Extensitivity of Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
10 Quantum mechanical and classical State Densities 77
10.1 Similarity of classical and quantum mechanical State Densities . . . . . . 78
10.1.1 Sommerfeld Quantization . . . . . . . . . . . . . . . . . . . . . . 78
10.1.2 Partition Function Approach . . . . . . . . . . . . . . . . . . . . . 78
10.1.3 Minimum Uncertainty Wavepackage Approach . . . . . . . . . . . 79
11 Ways to Equilibrium 85
11.1 Theories of Relaxation Behavior . . . . . . . . . . . . . . . . . . . . . . . 85
11.1.1 Fermi’s golden Rule for external Perturbations . . . . . . . . . . . 85
11.1.2 Fermi’s golden Rule for a coupled Environment . . . . . . . . . . 86
11.1.3 Weisskopf-Wigner Theory . . . . . . . . . . . . . . . . . . . . . . 86
11.1.4 Large Environment Approach . . . . . . . . . . . . . . . . . . . . 86
11.1.5 Numerical Results for the Relaxation Period . . . . . . . . . . . . 93
12 Summary and Conclusion 95Contents v
13 Zusammenfassung 99
13.1 Einleitung und historischer Hintergrund . . . . . . . . . . . . . . . . . . 99
13.2 Axiomatischer Aufbau der Thermodynamik . . . . . . . . . . . . . . . . 101
13.3 Grundprinzipien der Theorie . . . . . . . . . . . . . . . . . . . . . . . . . 102
13.4 Die Zellstruktur des Hilbertraums . . . . . . . . . . . . . . . . . . . . . . 104
13.4.1 Mikrokanonische Bedingungen . . . . . . . . . . . . . . . . . . . . 105
13.4.2 Kanonische Bedingungen . . . . . . . . . . . . . . . . . . . . . . . 106
13.5 Spektren modularer Systeme . . . . . . . . . . . . . . . . . . . . . . . . . 106
13.6 Spektrale Temperatur . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
13.7 Parametrischer Druck . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
13.8 Klassische und quantenmechanische Zustandsdichte . . . . . . . . . . . . 108
13.9 Wege zum Gleichgewicht . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
13.10Schlussbetrachtung und Ausblick . . . . . . . . . . . . . . . . . . . . . . 109
Appendix 113
A Minimization of the efiective Interaction, eq.5.24 . . . . . . . . . . . . . . 113
B Conserved Probabilities, eq.5.28 . . . . . . . . . . . . . . . . . . . . . . . 113
C Averages over Hyperspheres, sect. 5.4.3: . . . . . . . . . . . . . . . . . . 114
D From Multiplication to Gaussian, eq. 6.16 . . . . . . . . . . . . . . . . . 115
E Entropy of an ideal Gas, eq. 6.23 . . . . . . . . . . . . . . . . . . . . . . 116
F Stabilized adiabatic Approximation, eq. 8.18 . . . . . . . . . . . . . . . . 117
G Sizes of Hypersphereregions, sect. 5.6 . . . . . . . . . . . . . . . . . . . . 117
H Hilbertspace Averages, sect. 11.1.4 . . . . . . . . . . . . . . . . . . . . . 119
Bibliography 123vi Contents1 Introduction
In the very beginning thermodynamics have been a purely phenomenological science.
Early scientists (Galileo, Santorio, Celsius, Fahrenheit) tried to give deflnitions for quan-
tities that where intuitively obvious to the observer, like pressure or temperature and
measured the interconnections between them. The idea that those phenomena could be
directly linked to other flelds of physics, like classical mechanics, etc., was not common
at these days. This connection was basically introduced when Joule calculated the heat
equivalent in 1840, showing that heat was a form of energy, just like kinetic or potential
energy in the theory of mechanics of those days.
At the end of the 19’th century, when the atomic theory became popular, people began
to think of a gas as a box with a lot of bouncing balls inside. With this picture in mind,
it was tempting to try to reduce thermodynamics entirely to classical mechanics. This
was exactly what Boltzmann tried to do in 1866 when he connected entropy, a quantity
which was so far only described phenomenologically, to the volume of a certain region
in phase space, an object deflned within classical mechanics [1]. This was an enormous
step, especially from a practical point of view. Taking this connection for granted one
could now calculate all sorts of thermodynamical behavior of a system from it’s Hamilton
function. This gave rise to modern thermodynamics, a theory whose validity is beyond
any doubt today. Its results and predictions are a basic ingredient for the development
of all sorts of technical apparatuses, ranging from refrigerators to superconductors.
Boltzmann himself, however, tried to proof the conjectured connection between the phe-
nomenlogical and the theoretical entropy, but did not succeed without making other
assumptions like the famous ergodicity or the \a priory postulate". Other physicists
(Gibbs, Birkhofi, Ehrenfest, Von Neumann [2{5]) later on th