Quantum Thermodynamics: Emergence of Thermodynamic Behavior Within Composite Quantum Systems

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Springer Science & Business Media, Oct 21, 2009 - Science - 346 pages
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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Contents

Introduction
3
References
6
Basics of Quantum Mechanics
7
22 Operator Representations
8
222 Pauli Operators
9
224 Purity and von Neumann Entropy
11
225 Bipartite Systems
12
226 Multipartite Systems
14
References
171
Equilibration in Model Systems
173
161 Microcanonical Entropy
174
162 Canonical Occupation Probabilities and Entropy
176
163 Probability Fluctuations
180
164 Spin Systems
182
1642 Local Properties
183
1643 Chain Coupled Locally to a Bath
185

23 Dynamics
15
24 Invariants
17
25 TimeDependent Perturbation Theory
19
252 Series Expansion
20
References
22
Basics of Thermodynamics and Statistics
23
312 Fundamental Laws
24
313 Gibbsian Fundamental Form
28
314 Thermodynamic Potentials
29
32 Linear Nonequilibrium Thermodynamics
30
33 Statistics
32
331 Boltzmanns Principle A Priori Postulate
33
332 Microcanonical Ensemble
34
333 Statistical Entropy Maximum Principle
36
References
39
Brief Review of Pertinent Concepts
41
41 Boltzmanns Equation and HTheorem
42
42 Ergodicity
46
43 Ensemble Approach
47
44 Macroscopic Cell Approach
48
45 The Problem of Adiabatic State Change
51
46 Shannon Entropy Jaynes Principle
53
47 TimeAveraged Density Matrix Approach
54
48 Open System Approach and Master Equation
55
References
61
Equilibrium
63
The Program for the Foundation of Thermodynamics
64
Equilibrium Thermodynamics
65
Quantum Mechanical Versus Classical Aspects
67
Outline of the Present Approach
68
611 On the Analysis of Quantum Typicality
72
62 Compound Systems Entropy and Entanglement
73
63 Fundamental and Subjective Lack of Knowledge
74
References
75
Dynamics and Averages in Hilbert Space
76
72 Dynamics in Hilbert Space
78
73 Hilbert Space Average and Variance
81
References
83
Typicality of Observables and States
85
82 Hilbert Space Average of Observables
87
83 Hilbert Space Variance of Observables
89
84 Hilbert Space Averages of Distances Between States
91
References
93
System and Environment
94
92 Comment on Bipartite Systems and Physical Scenarios
98
93 Weak Coupling and Corresponding Accessible Regions
100
931 Microcanonical Conditions
102
932 Energy Exchange Conditions
103
References
105
The Typical Reduced State of the System
107
102 Microcanonical Conditions
110
1022 Typicality of the Microcanonical Average State
111
103 Energy Exchange Conditions
113
1032 Typicality of the Energy Exchange Average State
114
104 Canonical Conditions
115
105 Beyond Weak Coupling
116
References
117
Entanglement Correlations and Local Entropy
118
112 Entropy and Purity
122
References
125
Generic Spectra of Large Systems
129
122 Spectra of Modular Systems
131
123 Entropy of an Ideal Gas
134
124 Environmental Spectra and Boltzmann Distribution
136
125 Beyond the Boltzmann Distribution?
137
Temperature
139
131 Definition of Spectral Temperature
140
132 The Equality of Spectral Temperatures in Equilibrium
141
133 Spectral Temperature as the Derivative of Energy with Respect to Entropy
143
1331 Contact with a Hotter System
144
1332 Energy Deposition
145
References
148
Pressure and Adiabatic Processes
149
141 On the Concept of Adiabatic Processes
150
References
155
Quantum Mechanical and Classical State Densities
156
151 BohrSommerfeld Quantization
159
153 Minimum Uncertainty Wave Package Approach
161
154 Implications of the Minimum Uncertainty Wave Package Approach
170
References
188
Nonequilibrium
189
Brief Review of Relaxation and Transport Theories
190
171 Relaxation
191
1712 WeisskopfWigner Theory
192
1713 Open Quantum Systems
193
1721 Boltzmann Equation
194
1722 Peierls Approach and Greens Functions
195
1723 Linear Response Theory
196
References
197
Projection Operator Techniques and Hilbert Space Average Method
201
181 Interaction Picture of the von Neumann Equation
202
183 TimeConvolutionless Master Equation
204
184 Generalization of the Hilbert Space Average
206
185 Dynamical Hilbert Space Average Method
208
186 Comparison of TCL and HAM
211
References
212
Finite Systems as Thermostats
214
191 Finite Quantum Environments
216
192 Standard Projection Superoperator
217
193 Correlated Projection Superoperator
220
194 Accuracy of the Reduced Description
222
References
225
Projective Approach to Dynamical Transport
227
202 SingleParticle Modular Quantum System
228
203 General TCL Approach
230
204 Concrete TCL Description and Timescales
231
205 TCL Approach in Fourier Space
233
206 Length ScaleDependent Transport Behavior
234
207 Energy and Heat Transport Coefficient
238
References
240
Open System Approach to Transport
241
211 Model System
242
213 Observables and Fouriers Law
244
214 Monte Carlo Wave Function Simulation
245
215 Chain of TwoLevel Atoms
248
References
253
Applications and Models
254
Purity and Local Entropy in Product Hilbert Space
257
222 Application
259
References
261
Observability of Intensive Variables
262
2311 Model
264
2312 Global Thermal State in the Product Basis
265
2313 Conditions for Local Thermal States
267
2314 Spin Chain in a Transverse Field
269
232 Measurement Schemes
271
2322 Temperature
274
Observability of Extensive Variables
275
2412 Applications
279
2413 Remarks
281
242 Fluctuations of Work
282
2421 Microcanonical Coupling
284
Constant Interaction Energy
286
2424 Numerical Verification
288
References
289
Quantum Thermodynamic Processes
291
252 Quasistatic Limit
293
2522 Otto Cycle
295
2523 Carnot Cycle
297
2524 Continuum Limit of the Particle in a Box
299
253 Processes Beyond the Quasistatic Limit
300
2531 Solution of the Relaxation Equation
302
2532 Otto Cycle at Finite Velocity
303
2533 Carnot Cycle at Finite Velocity
308
References
313
Hyperspheres
314
General Hilbert Space Averages and Variances
319
B2 Hilbert Space Variance of an Observable
321
Special Hilbert Space Averages and Variances
324
C2 Microcanonical Hilbert Space Averages
326
C3 Microcanonical Hilbert Space Variances
327
C4 Energy Exchange Hilbert Space Averages
329
C5 Energy Exchange Hilbert Space Variance
330
Power of a Function
333
Local Temperature Conditions for a Spin Chain
335
References
338
Index
339
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