NON-EQUILIBRIUM ENTROPY AND IRREVERSIBILITY

Goran Lindblad NON-EQUILIBRIUM ENTROPY AND IRREVERSIBILITY TABLE OF CONTENTS Preface 1. INTRODUCTION AND SUMMARY 2. DYNAMICS AND WORK 3. INFORMATION ENTROPY 3.a Entropy and relative entropy 3.b Gibbs states 3.c Entropy-increasing processes 4. HEAT BATHS 5. REVERSIBLE PROCESSES 6. CLOSED FINITE SYSTEMS 6.a Available work 6.b Recurrences 6.c Entropy functions 7. OPEN SYSTEMS 7.a Markov description 7.b Available work and entropy 7.c Master equation models 8. EXTERNAL PERTURBATIONS 8.a Models of the perturbations 8.b Classical systems 8.c Quantum systems 8.d Effects on the entropy functions 9. THERMODYNAMIC LIMIT 10. THERMODYNAMIC ENTROPY 10.a Thermodynamic processes and entropy 10.b Properties of the entropy functions 10.c Irreversibility and approach to equilibrium 11. MEASUREMENTS, ENTROPY AND WORK 11.a Observations on the system 11.b Information and entropy 11.c Exchange of work and heat 12. OTHER APPROACHES APPENDIX A. QUANTUM MARKOV PROCESSES A.1 Reduced dynamics A.2 Markov processes A.3 Non-passivity of Markov processes A.4 Non-KMS property of Markov processes A.5 Quantum thermal fluctuations APPENDIX B. SENSITIVITY OF HYPERBOLIC MOTION REFERENCES NOTATION INDEX

vii 1 10 20 23 27 32 39 44 47 53 59 62 65 72 74 79 84 89 95 98 105 110 113 118 123 133 136 140 143 148 150 156 163

SUBJECT INDEX

165

SUBJECT INDEX irreversibility 33, 44, 48-9 66, 105, accessibility 15, 96 116 adiabatic limit 43 K-mixing system 76 adiabatic process 96 KMS condition 145 available work 26, 46, 62, 98, 108, 113 KMS state 27, 34 autocorrelation function 143 Liapounov exponents 76 Bloch equations 68, 141 Lie-Trotter formula 69, 139 Carnot cycles 43, 62-3 Loschmidt paradox 2, 47 chaos 75, 80 Markov process 67, 136 complete positivity 133 Markov property 60, 73, 138 correlations 28, 61, 114 master equations 65, 135, 139 cycles 12 Maxwell's demon 18, 113 detailed balance 136 measurements 110-2 dissipation 135 mobility semigroup 14, 97 dissipative part 134 observables 10, 18 distance to P-equilibrium 109 open systems 17, 59, 133 dynamical maps 12, 133 optimal work cycle 46, 98 dynamical semigroup 134 origin of a cycle 13 dual map 12, 135 partial state 21, 60 duration of a cycle 12 passivity 26, 106, 108 entropy, information- 20, 110, 126 preparation of states 32 - , KS- 76, 127 P-entropy 97 - , P- 97 P-equilibrium state 109 entropy, relative 21, 108 quantum regression theorem 67, 137 - , specific 92 quantum stochastic process 67 - , subjective 129 quasi-periodic motion 77 -, thermodynamic 53, 63, 97 recurrences 15, 47, 93 entropy production 65 reduced dynamics 60, 133 equilibrium states 23, 97, 105 relative entropy 21, 108 essentially isolated system 29 reproducible states 38, 96 exponential instability 75, 150 reservoirs 34 external fields 10 reservoir stability 35 fluctuations 58, 91, 111 return to equilibrium 33 generator 134 reversible processes 39, 43 Gibbs states 23 sensitivity to perturbations 31, 78, 150 global mobility 14 specific energy, -entropy 90 Hamiltonian 10 spectral function 145 heat baths 32, 34 spin echo 1, 15 heat transfer 120 stability properties 24, 35 H-theorem 22, 139 stochastic Hamiltonian 28 hyperbolic system 76, 150 stochastic transition 37, 76 I-entropy 20, 110, 126 thermodynamic entropy 53, 63, 97 information 112

- limit 40, 89 - processes 95 - stability 24 time rescaling 34, 37, 135 time's arrow 19 time scales 51 unitary invariance 21 variational principle 24 weak coupling limit (WCL) 34, 135 white noise 29, 74, 148 work 16, 118-22 - cycle 12, 96 - observable 16 Zermelo paradox 47