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Nonequilibrium Aspects of Quantum Thermodynamics

Mathias Michel

A project supported by the Deutsche Forschungsgemeinschaft

Dissertation Universität Stuttgart
Selbstverlag M. Michel
Stuttgart 2006
Paperback, 239 p., 47 fig.
Also available online.
ISBN-10: 3-00-019902-0
ISBN-13: 978-3-00-019902-8
english blurb
german blurb (Klappentext)

About the book

Questions about the route from a nonequilibrium initial state to the final global equilibrium have played an important role since the early days of phenomenological thermodynamics and statistical mechanics. Nowadays, their implications reach from central technical devices of the contemporary human society, like heat engines, refrigerators and computers to recent physics at almost all length scales, from Bose-Einstein-condensation and superconductors to black holes. This work addresses the foundation of macroscopic laws concerning the decay to equilibrium, e.g. the celebrated Fourier's Law, on microscopic Schröingerian quantum dynamics. Here, a proper treatment requires the usage of modern methods in theoretical physics such as the Theory of Open Quantum Systems, the Kubo Formula in Liouville Space and the novel Hilbert Space Average Method. It turns out that both the relaxation to equilibrium as well as the transport of heat is mainly determined by quantum effects comparable to the role of entanglement in considerations of the global equilibrium within Quantum Thermodynamics. Finally, the foundation of phenomenological thermodynamics on a microscopic theory will hopefully improve our understanding of those most impressive and far-reaching theories and their background and will possibly open the way to overcoming their nanoscopic limits.

Table of Contents

1. Introduction
2. System and Environment
   1. Class of Model Systems
   2. Time-Evolution of Pure and Mixed States
   3. Environment or Heat Bath
3. Open System Approach to Relaxation Processes
   1. Nakajima-Zwanzig Equation
   2. Born Approximation
   3. Redfield, Markov and the Rotating Wave Approximation
   4. The Dissipator
4. Global and Local Properties
   1. Global Stationary Equilibrium State
   2. Approach of Equilibrium
   3. Temperature
   4. Local Energy and Temperature
   5. Open System Equilibrium State
   6. Relaxation Dynamics in QME
5. Quantum Thermodynamic Environment
   1. Finite Environment
   2. Beyond the Born Approximation
6. Hilbert Space Average Method
   1. Hamiltonian Model
   2. Hilbert Space Average
7. General HAM Rate Equation
   1. Short Time Iteration Scheme
   2. Closed Differential Equation
   3. Limits of Applicability
8. Thermalization
   1. Damping Model
   2. The HAM Rate Equation
   3. Comparison to Schrödinger Dynamics
9. Decoherence
   1. Fermi's Golden Rule
   2. Advanced Model System
   3. Several Different Thermalization Times
   4. Decoherence According to Canonical Couplings
   5. Additional Microcanonical Coupling
10. Heat Transport
    1. Fourier's Law
    2. Kinetic Gas Theory for Phonons
    3. Peierls-Boltzmann Equation
    4. Linear Response Theory
    5. Survey of Recent Developments
11. Heat Current
    1. Current Operator
    2. Conserved Quantities
    3. Quasiparticle Interpretation
12. An Open System Approach to Heat Conduction
    1. Heat Transport in Low Dimensional Systems
    2. Fourier's Law for a Heisenberg Chain
    3. Scaling Behavior
13. Perturbation Theory in Liouville Space
    1. Super Operators
    2. Unperturbed System
    3. Local Equilibrium State
    4. Standard Kubo-Formula
    5. Kubo-Formula in Liouville Space
    6. Heat Transport Coefficient
    7. Heat Conductivity of a Model System
14. Reservoir Perturbation Theory
    1. Perturbation Theory
    2. Dependence on the Bath Coupling
    3. Influence of the Bath Coupling
15. Quantum Thermodynamic Approach to Heat Conduction
    1. Mesoscopic Model
    2. Dyson Time-Evolution Expansion
    3. HAM for Transport Scenario
16. Diffusive Behavior from Schrödinger Dynamics
    1. Decay Behavior of a Model System
    2. Fluctuations and Size Effects
    3. Initial States
17. Transport Coefficients
    1. Energy Transport
    2. Heat Transport
    3. Connection between Heat and Energy Transport
18. Conclusion and Outlook
19. Appendices
    1. Hilbert Space Average of Expectation Values
    2. Hyperspheres
    3. Hilbert Space Averages
    4. Hilbert-Space Variance
    5. German Summary -- Deutsche Zusammenfassung
20. List of Symbols
21. Bibliography
22. List of Previous Publications
23. Index

Introduction

Phenomenological thermodynamics and statistical mechanics are part of the most impressive and far-reaching theories of modern physics: Their implications reach from central technical devices of the contemporary human society, like heat engines and refrigerators etc. to recent physics at almost all length scales, from Bose-Einstein-condensates and superconductors to black holes.

After a phenomenological introduction by some early scientists (Celsius, Fahrenheit, etc.) of quantities from our everyday experiences like temperature and pressure, the whole picture dramatically changed with Joule's calculation of the heat equivalent in 1840. This celebrated investigation showed that heat is nothing else but energy. Joule thus opened a connection between early thermodynamics and (by this time already advanced) classical Hamiltonian mechanics.

In 1866 Boltzmann was able to reduce thermodynamics entirely to classical mechanics by identifying the so far phenomenological entropy with the volume of a certain region in phase space (2). Finally, this conjecture led to our modern understanding of thermodynamics. Besides Bolzmann, such famous physicists as Gibbs (12), Ehrenfest (5), Birkhoff (1) and von Neumann (20) tried to prove the celebrated postulate, but did not succeed without using some further assumptions like ergodicity, quasi-ergodicity, molecular chaos etc. which could neither be proven, in general. Nevertheless, thermodynamics has proven to be an extremely effective description of many physical processes.

In those early days most people thought about a thermodynamical system, e.g. a gas, being a classical multi particle system - bouncing balls in a box. With the development of quantum mechanics at the beginning of the 20th century such a classical idea of a gas system became questionable. Of course, that does not categorically mean that everything has to be done quantum mechanically if a simpler effective description is available. Nevertheless, the foundation of thermodynamics on classical Hamiltonian mechanics faces several conceptual problems.

Figure: Foundation of thermodynamics on quantum mechanics (left) or classical mechanics (right), respectively.

To overcome those deficiencies, there have been several recent approaches to thermodynamical behavior from quantum mechanics (24, 29, 15, 14). A very successful approach by Gemmer and Mahler (11, 10, 9, 8) underlines the important role of entanglement between the considered system and the rest of the world for the emergence of thermodynamical behavior. Thus, an extremely non-classical property turns out to be responsible for the everyday experience of enforcement of the equilibrium. Fortunately this foundation comes without any further assumptions and has been called Quantum Thermodynamics (see (10)). Such a quantum mechanical foundation does not only clarify the background of irreversible behavior in terms of a reversible microscopic theory, but also contains the chance to extend some thermodynamical concepts to situations where the preconditions for the theory itself are (in part) violated (cf. Fig. 1.1).

Having established a theory of equilibrium quantum thermodynamics, questions about the stability of the equilibrium arise, whichever microscopic foundation of the theory one prefers. Those questions mainly address the reaction of a system weakly perturbed from the outside, i.e. moved out of its respective equilibrium state. The most important ones refer to the relaxation to equilibrium and the properties of stationary local equilibrium states (18). The latter corresponds to a situation where the system is fixed in a nonequilibrium situation by conflicting constraints from the outside. These constraints may be given by very large, eventually infinite reservoirs of energy, mass, charge etc, featuring different intensive parameters, i.e. temperatures, chemical potentials etc. In contrast, a relaxation to equilibrium emerges from a coupling to such an environment as well, but there are no competiting reservoirs of different intensive parameters present. Thus the system finally reaches the global equilibrium, and will not be forced to stay in a stationary local equilibrium state, featuring energy, temperature or concentration gradients as well as currents of the respective quantities.

While the existence of a final equilibrium is a major topic of equilibrium thermodynamics, the research in nonequilibrium thermodynamics is more interested in the route to this final stationary state as well as the time the system needs for its relaxation process. Furthermore, there could be some differences between the time scale of reaching the thermal equilibrium and the decay of the correlations within the system. Especially for modern quantum information processing (21), decoherence (32, 16, 31, 13) (i.e. decay of correlations) plays a crucial role.

In both situations either the relaxation or the stationary local equilibrium scenario pose central questions about the type of transport of energy, heat, mass, charge etc through the respective system. In our classical world we typically find diffusive (statistical) transport, expressed by such famous results as Fourier's Law (7) of heat conduction and Fick's Law (6) of particle transport. On the other hand some systems feature ballistic transport, i.e. the conductivity diverges (22, 26, 19, 23). Such materials do not show any resistivity for the transport (4), and thus a current of the respective quantity flows in the system without any stimulus, e.g. electrical superconductivity. Interestingly enough, it is the normal transport that appears to be harder to explain, whichever microscopic theory is used. Thus, some researchers claim that a satisfactory derivation of e.g. Fourier's Law from truly fundamental principles was still missing (3).

Besides the growing interest in old concepts like temperature and entropy at the nanoscale, theories of relaxation and transport have recently regained a lot of attention. This renaissance follows not only from the above described fundamental reasons, but also from some practical ones: In a time where electronic circuits and computer chips are getting smaller and smaller, the equilibrium and nonequilibrium thermodynamics (linear irreversible thermodynamics) at small length scales far below the thermodynamic limit (particle number to infinity), gain in importance (28, 25, 30, 17). The rapid miniaturization relies on the controlled theoretical understanding of original macroscopic processes, e.g. transport of energy, heat, charge, mass, magnetization etc. Only from the foundations of a theory its limits of applicability may be inferred. Thus the ongoing technological progress and the lack of a satisfactory microscopic foundation brings one back to rather fundamental questions.

The present study is intended to address some aspects of the far-reaching topics of relaxation and transport. In ``good old tradition'' we will investigate those interesting fields again from first principles - standard quantum mechanics. Besides the use of the theory of open quantum systems, both these topics will be considered by the new and powerful quantum thermodynamical technique the - Hilbert Space Average Method. This background already contains one of the main ideas, namely that the apparently separate fields of relaxation and transport are ruled by the same fundamental principles.

This investigation is basically devided into two central parts: The first part addresses the relaxation processes in small quantum mechanical systems due its coupling to finite as well as infinite reservoirs. The second part deals with transport of heat and energy in relaxation processes as well as stationary nonequilibrium states.

References

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Acknowledgement

It is my pleasure to acknowledge advice and support from several persons who have influenced this work intensively. First of all I would like to thank my supervisor Prof. Dr. G. Mahler for his constant interest in the progress of this work, his constructive criticism, and his support during the last years. It is impossible to honor the impact that Jun.-Prof. Dr. J. Gemmer had on this work. I am indebted to him for suggesting me the very interesting and challenging problem of nonequilibrium quantum thermodynamics, for countless fruitful discussions, for sharing his immense knowledge and expertise with me, for having time to answer my numerous questions, helping me in any complicated situation, and last but not least for his friendship. I thank Prof. Dr. O. Hess (University of Surrey) for writing the second report on my dissertation. Within the Institute of Theoretical Physics I (Universität Stuttgart) I have experienced a stimulating environment for research. I would like to thank my present and former colleagues P. Borowski, Dr. M. Hartmann, M. Henrich, C. Kostoglou, Dr. A. Otte, H. Schmidt, H. Schröder, M. Stollsteimer, J. Teiffel, F. Tonner and P. Vidal for fruitful discussions (not only physical ones). I am grateful for the warm welcome at the Universität Osnabrück, the successful collaboration and many interesting discussions with Apl.-Prof. Dr. J. Schnack, Dr. M. Exler and R. Steinigeweg. Furthermore, I have benefited a lot from discussions with Dr. H.-P. Breuer (Universität Freiburg) and Dr. F. Heidrich-Meisner (University of Tennessee). Financial support by the Deutsche Forschungsgemeinschaft is gratefully acknowledged. I am indebted to Rebekah and Ron Abramski for proofreading parts of the manuscript and thank all my other friends, especially K. Bihlmaier, and my brother for being by my side. Finally, I would like to thank my parents for their extensive support during my studies and especially my father Dr. H. Michel for interesting discussions about physics and my work. Last but not least, I am deeply indebted to Mirjam for her infinite patience, support and love and for proofreading the manuscript.

Stuttgart, September 2006
Mathias Michel