Lecture Notes in Physics

Emergence of Thermodynamic Behavior Within Composite Quantum Systems

By: Jochen Gemmer, M. Michel, GÃ¼nter Mahler

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Hardcover | 21 October 2009 | Edition Number 2

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364 Pages

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24.77 x 16.51 x 1.91

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Over the years enormous effort was invested in proving ergodicity, but for a number of reasons, con?dence in the fruitfulness of this approach has waned. - Y. Ben-Menahem and I. Pitowsky [1] Abstract The basic motivation behind the present text is threefold: To give a new explanation for the emergence of thermodynamics, to investigate the interplay between quantum mechanics and thermodynamics, and to explore possible ext- sions of the common validity range of thermodynamics. Originally, thermodynamics has been a purely phenomenological science. Early s- entists (Galileo, Santorio, Celsius, Fahrenheit) tried to give de?nitions for quantities which were intuitively obvious to the observer, like pressure or temperature, and studied their interconnections. The idea that these phenomena might be linked to other ?elds of physics, like classical mechanics, e.g., was not common in those days. Such a connection was basically introduced when Joule calculated the heat equ- alent in 1840 showing that heat was a form of energy, just like kinetic or potential energy in the theory of mechanics. At the end of the 19th century, when the atomic theory became popular, researchers began to think of a gas as a huge amount of bouncing balls inside a box.

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From the reviews of the second edition: "The aim of this textbook is to show how thermodynamic behavior can emerge for a given finite quantum system. ... The book is written in a conceptually simple language, the focus being rather on the careful interpretation of the concepts at hand and their illustrations on model systems. ... presents some applications to model systems, which allows one to illustrate more precisely previously introduced notions." (Gabriel Stoltz, Mathematical Reviews, Issue 2011 f)

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Non-Fiction Science Physics Quantum Physics & Quantum Mechanics & Quantum Field Theory Thermodynamics & Heat Reference, Information & Interdisciplinary Subjects Research & Information Information theory Cybernetics & Systems Theory

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Published: 14th March 2012

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Background
Introduction	p. 3
References	p. 6
Basics of Quantum Mechanics	p. 7
Introductory Remarks	p. 7
Operator Representations	p. 8
Transition Operators	p. 8
Pauli Operators	p. 9
State Representation	p. 9
Purity and von Neumann Entropy	p. 11
Bipartite Systems	p. 12
Multipartite Systems	p. 14
Dynamics	p. 15
Invariants	p. 17
Time-Dependent Perturbation Theory	p. 19
Interaction Picture	p. 19
Series Expansion	p. 20
References	p. 22
Basics of Thermodynamics and Statistics	p. 23
Phenomenological Thermodynamics	p. 23
Basic Definitions	p. 23
Fundamental Laws	p. 24
Gibbsian Fundamental Form	p. 28
Thermodynamic Potentials	p. 29
Linear Non-equilibrium Thermodynamics	p. 30
Statistics	p. 32
Boltzmann's Principle, A Priori Postulate	p. 33
Microcanonical Ensemble	p. 34
Statistical Entropy, Maximum Principle	p. 36
References	p. 39
Brief Review of Pertinent Concepts	p. 41
Boltzmann's Equation and H-Theorem	p. 42
Ergodicity	p. 46
Ensemble Approach	p. 47
Macroscopic Cell Approach	p. 48
The Problem of Adiabatic State Change	p. 51
Shannon Entropy, Jaynes' Principle	p. 53
Time-Averaged Density Matrix Approach	p. 54
Open System Approach and Master Equation	p. 55
References	p. 61
Equilibrium
The Program for the Foundation of Thermodynamics	p. 65
Basic Checklist: Equilibrium Thermodynamics	p. 66
Supplementary Checklist: Quantum Mechanical Versus Classical Aspects	p. 68
Reference	p. 68
Outline of the Present Approach	p. 69
Typicality and Dynamics	p. 69
On the Analysis of Quantum Typicality	p. 73
Compound Systems, Entropy, and Entanglement	p. 74
Fundamental and Subjective Lack of Knowledge	p. 75
References	p. 76
Dynamics and Averages in Hilbert Space	p. 77
Representation of Hilbert Space	p. 77
Dynamics in Hilbert Space	p. 78
Hilbert Space Average and Variance	p. 81
References	p. 83
Typicality of Observables and States	p. 85
A Class of Accessible Regions	p. 85
Hilbert Space Average of Observables	p. 87
Hilbert Space Variance of Observables	p. 89
Hilbert Space Averages of Distances Between States	p. 91
References	p. 93
System and Environment	p. 95
Partition of the System and Pertinent Notation	p. 95
Comment on Bipartite Systems and Physical Scenarios	p. 98
Weak Coupling and Corresponding Accessible Regions	p. 100
Microcanonical Conditions	p. 102
Energy Exchange Conditions	p. 103
References	p. 105
The Typical Reduced State of the System	p. 107
Reduced State and Typicality	p. 107
Microcanonical Conditions	p. 110
Microcanonical Average State	p. 110
Typicality of the Microcanonical Average State	p. 111
Energy Exchange Conditions	p. 113
Energy Exchange Average State	p. 113
Typicality of the Energy Exchange Average State	p. 114
Canonical Conditions	p. 115
Beyond Weak Coupling	p. 116
References	p. 117
Entanglement, Correlations, and Local Entropy	p. 119
Entanglement, Correlations, and Relaxation	p. 119
Entropy and Purity	p. 123
References	p. 126
Generic Spectra of Large Systems	p. 129
The Extensivity of Entropy	p. 129
Spectra of Modular Systems	p. 131
Entropy of an Ideal Gas	p. 134
Environmental Spectra and Boltzmann Distribution	p. 136
Beyond the Boltzmann Distribution?	p. 137
References	p. 137
Temperature	p. 139
Definition of Spectral Temperature	p. 140
The Equality of Spectral Temperatures in Equilibrium	p. 141
Spectral Temperature as the Derivative of Energy with Respect to Entropy	p. 143
Contact with a Hotter System	p. 144
Energy Deposition	p. 145
References	p. 148
Pressure and Adiabatic Processes	p. 149
On the Concept of Adiabatic Processes	p. 150
References	p. 155
Quantum Mechanical and Classical State Densities	p. 157
Bohr-Sommerfeld Quantization	p. 159
Partition Function Approach	p. 159
Minimum Uncertainty Wave Package Approach	p. 161
Implications of the Minimum Uncertainty Wave Package Approach	p. 170
Correspondence Principle	p. 170
References	p. 171
Equilibration in Model Systems	p. 173
Microcanonical Entropy	p. 174
Canonical Occupation Probabilities and Entropy	p. 176
Probability Fluctuations	p. 180
Spin Systems	p. 182
Global Observables	p. 182
Local Properties	p. 183
Chain Coupled Locally to a Bath	p. 185
References	p. 188
Non-equilibrium
Brief Review of Relaxation and Transport Theories	p. 191
Relaxation	p. 192
Fermi's Golden Rule	p. 192
Weisskopf-Wigner Theory	p. 193
Open Quantum Systems	p. 194
Transport	p. 194
Boltzmann Equation	p. 195
Peierls Approach and Green's Functions	p. 196
Linear Response Theory	p. 197
References	p. 198
Projection Operator Techniques and Hilbert Space Average Method	p. 201
Interaction Picture of the von Neumann Equation	p. 202
Projection on Arbitrary Observables	p. 202
Time-Convolutionless Master Equation	p. 204
Generalization of the Hilbert Space Average	p. 206
Dynamical Hilbert Space Average Method	p. 208
Comparison of TCL and HAM	p. 211
References	p. 212
Finite Systems as Thermostats	p. 215
Finite Quantum Environments	p. 216
Standard Projection Superoperator	p. 217
Correlated Projection Superoperator	p. 220
Accuracy of the Reduced Description	p. 222
References	p. 225
Projective Approach to Dynamical Transport	p. 227
Classification of Transport Behavior	p. 227
Single-Particle Modular Quantum System	p. 228
General TCL Approach	p. 230
Concrete TCL Description and Timescales	p. 231
TCL Approach in Fourier Space	p. 233
Length Scale-Dependent Transport Behavior	p. 234
Energy and Heat Transport Coefficient	p. 238
References	p. 240
Open System Approach to Transport	p. 241
Model System	p. 242
Lindblad Quantum Master Equation	p. 242
Observables and Fourier's Law	p. 244
Monte Carlo Wave Function Simulation	p. 245
Chain of Two-Level Atoms	p. 248
References	p. 253
Applications and Models
Purity and Local Entropy in Product Hilbert Space	p. 257
Unitary Invariant Distribution of Pure States	p. 257
Application	p. 259
References	p. 261
Observability of Intensive Variables	p. 263
On the Existence of Local Temperatures	p. 263
Model	p. 264
Global Thermal State in the Product Basis	p. 265
Conditions for Local Thermal States	p. 267
Spin Chain in a Transverse Field	p. 269
Measurement Schemes	p. 271
Pressure	p. 271
Temperature	p. 274
References	p. 274
Observability of Extensive Variables	p. 275
Work and Heat	p. 275
LEMBAS Scenario	p. 275
Applications	p. 279
Remarks	p. 281
Fluctuations of Work	p. 282
Microcanonical Coupling	p. 284
Canonical Coupling: Constant Interaction Energy	p. 286
Canonical Coupling: High-Temperature Limit	p. 286
Numerical Verification	p. 288
References	p. 289
Quantum Thermodynamic Processes	p. 291
Model of Control	p. 291
Quasi-static Limit	p. 293
Thermodynamic Quantities	p. 293
Otto Cycle	p. 295
Carnot Cycle	p. 297
Continuum Limit of the Particle in a Box	p. 299
Processes Beyond the Quasi-static Limit	p. 300
Solution of the Relaxation Equation	p. 302
Otto Cycle at Finite Velocity	p. 303
Carnot Cycle at Finite Velocity	p. 308
References	p. 313
Hyperspheres	p. 315
Averaging of Functions over Hyperspheres	p. 315
General Hilbert Space Averages and Variances	p. 319
Hilbert Space Averages of Polynomial Terms	p. 319
Hilbert Space Variance of an Observable	p. 321
Special Hilbert Space Averages and Variances	p. 325
General Distances Between States	p. 325
Microcanonical Hilbert Space Averages	p. 326
Microcanonical Hilbert Space Variances	p. 327
Energy Exchange Hilbert Space Averages	p. 329
Energy Exchange Hilbert Space Variance	p. 330
Power of a Function	p. 333
Local Temperature Conditions for a Spin Chain	p. 335
References	p. 338
Index	p. 339
Table of Contents provided by Ingram. All Rights Reserved.

Lecture Notes in Physics

Emergence of Thermodynamic Behavior Within Composite Quantum Systems

At a Glance

Hardcover

Industry Reviews

Shipping

How to return your order

Defective items

You Can Find This Book In

Other Editions and Formats

Paperback

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