Springer, Berlin, Heidelberg, New York;

Series: Lecture Notes in Physics, Vol. 657

2004, 289 p. Also available online., Hardcover

ISBN: 3-540-22911-6

with a preface by Prof. Dr. Fritz Haake (Essen)

About this Book |

Preface by Prof. Dr. Fritz Haake |

Table of contents |

Introduction |

Further References |

Thanks |

This extended tutorial essay views thermodynamics as an incomplete description of quantum systems with many degrees of freedom. The main goal is to show that the approach to equilibrium - with equilibrium characterized by maximum ignorance about the open system of interest - neither requires that many particles nor is it a precise way of partitioning relevant for the salient features of equilibrium and equilibration. Moreover it is indeed quantum effects that are at work in bringing about universal thermodynamic behaviour of modestly sized open systems. Von Neumann`s concept of entropy thus proves to be much more widely useful than something to be feared, and far beyond truly macroscopic systems in equilibrium.

This monograph views thermodynamics as an *incomplete description*
of many freedom quantum systems.
Left unaccounted for may be an environment with which the system of
interest interacts; closed systems can be described incompletely by
focussing on any subsystem with fewer particles and declearing the
remainder as the environment.
Any interaction with the environment brings the open system to a
mixed quantum state, even if the closed compound state is pure.
Moreover, observables (and sometimes even the density operator) of an
open system may relax to equilibrium values while the closed compound
state keeps evolving unitarily a la Schrödinger forever.

The view thus taken can hardly be controversial for our generation of physicists. And yet, the authors offer surprises. Approach to equlibrium, with equilibrium characterized by maximum ignorance about the open system of interest, does not require excessively many particles: some dozens suffice! Moreover, the precise way of partitioning which might reflect subjective choices is immaterial for the salient features of equilibrium and equilibration. And what is nicest, quantum effects are at work in bringing about universal thermodynamic behavior of modest size open systems. Von Neumann's concept of entropy thus appears as much more widely useful than sometimes feared, way beyond truely macroscopic systems in equilibrium.

The authors have written numerous papers on their quantum view of thermodynamics, and the present monograph is a most welcome coherent review.

Essen, June 2004

Fritz Haake

- Introduction
- Basics of Quantum Mechanics
- Introductory Remarks
- Operator Representations
- Dynamics
- Invariants
- Time-Dependent Perturbation Theory

- Basics of Thermodynamics and Statistics
- Phenomenological Thermodynamics
- Linear Irreversible Thermodynamics
- Statistics

- Brief Review of Pertinent Concepts
- Boltzmann's Equation and H-Theorem
- Ergodicity
- Ensemble Approach
- Macroscopic Cell Approach
- The Problem of Adiabatic State Change
- Shannon Entropy, Jaynes' Principle
- Time-Averaged Density Matrix Approach
- Open System Approach and Master Equation

- The Program for the Foundation of Thermodynamics
- Basic Checklist: Equilibrium Thermodynamics
- Supplementary Checklist

- Outline of the Present Approach
- Compound Systems, Entropy and Entanglement
- Fundamental and Subjective Lack of Knowledge
- The Natural Cell Structure of Hilbert Space

- System and Environment
- Partition of the System and Basic Quantities
- Weak Coupling
- Effective Potential, Example for a Bipartite System

- Structure of Hilbert Space
- Representation of Hilbert Space
- Hilbert Space Average
- Hilbert Space Variance
- Purity and Local Entropy in Product Hilbert Space

- Quantum Thermodynamic Equilibrium
- Microcanonical Conditions
- Energy Exchange Conditions
- Canonical Conditions
- Fluctuations of Occupation Probabilities

- Interim Summary
- Equilibrium Properties
- Local Equilibrium States and Ergodicity

- Typical Spectra of Large Systems
- The Extensitivity of Entropy
- Spectra of Modular Systems
- Entropy of an Ideal Gas
- The Boltzmann Distribution
- Beyond the Boltzmann Distribution?

- Temperature
- Definition of Spectral Temperature
- The Equality of Spectral Temperatures in Equilibrium
- Spectral Temperature as the Derivative of Energy

- Pressure
- On the Concept of Adiabatic Processes
- The Equality of Pressures in Equilibrium

- Quantum Mechanical and Classical State Densities
- Bohr-Sommerfeld Quantization
- Partition Function Approach
- Minimum Uncertainty Wave Package Approach
- Implications of the Method
- Correspondence Principle

- Sufficient Conditions for a Thermodynamic Behavior
- Weak Coupling Limit
- Microcanonical Equilibrium
- Energy Exchange Equilibrium
- Canonical Equilibrium
- Spectral Temperature
- Parametric Pressure
- Extensitivity of Entropy

- Theories of Relaxation Behavior
- Fermi's Golden Rule
- Weisskopf-Wigner Theory
- Open Quantum Systems

- The Route to Equilibrium
- System and a Large Environment
- Time Evolution
- Hilbert Space Average
- Short Time Step Equation
- Derivation of a Rate Equation
- Solution of the Rate Equation
- Hilbert Space Variances
- Numerical Results for the Relaxation Period

- Equilibrium Properties of Model Systems
- Entropy under Microcanonical Conditions
- Occupation Probabilities under Canonical Conditions
- Probability Fluctuations
- Spin Systems
- On the Existence of Local Temperatures
- Quantum Thermometer
- Quantum Manometer
- Adiabatic Following and Adiabatic Process

- Heat Conduction
- Theories of Heat Conduction
- Quantum Heat Conduction

- Quantum Thermodynamic Machines
- Tri-Partite System: Sudden Changes of Embedding
- Work Variables
- Carnot Cycle
- Generalized Control Cycles

- Summary and Conclusion

- Hyperspheres
- Hilbert Space Average Under Microcanonical Conditions
- Hilbert Space Averages and Variances
- Power of a Function
- Local Temperature Conditions for a Spin Chain

References

Index

Originally, thermodynamics has been a purely phenomenological science. Early scientists (Galileo, Santorio, Celsius, Fahrenheit) tried to give definitions 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 fields of physics, like classical mechanics, e.g., was not common in those days. Such a 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.

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. 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 [1], when he connected entropy, a quantity which so far had only been defined phenomenologically, to the volume of a certain region in phase space, an object defined within classical mechanics. This was an enormous step forward, especially from a practical point of view. Taking this connection for granted one could now calculate all sorts of thermodynamic properties of a system from its Hamilton function. This gave rise to modern thermodynamics, a theory the validity of which is beyond any doubt today. Its results and predictions are a basic ingredient for the development of all kinds of technical apparatuses ranging from refrigerators to superconductors.

Boltzmann himself, however, tried to prove the conjectured connection between the phenomenlogical and the theoretical entropy, but did not succeed without invoking other assumptions like the famous ergodicity postulate or the hypothesis of equal "a priory probabilities". Later on, other physicists (Gibbs [2], Birkhoff [3], Ehrenfest [4], von Neumann [5], etc.) tried to prove those assumptions, but none of them seems to have solved the problem satisfactorily. It has been pointed out, though, that there are more properties of the entropy to be explained than its mere equivalence with the region in phase space, before thermodynamics can be reduced to classical mechanics, thus the discussion is still ongoing [6]. The vast majority of the work done in this field is based on classical mechanics.

Meanwhile, quantum theory, also initially triggered by the atomic hypothesis, has made huge progress during the last century and is today believed to be more fundamental than classical mechanics. At the beginning of the 21st century it seems highly unlikely that a box with balls inside could be anything more than a rough caricature of what a gas really is. Furthermore, thermodynamic principles seem to be applicable to systems that cannot even be described in classical phase space. Those developments make it necessary to rethink the work done so far, whether it led to the desired result (e.g., demonstration of ergodicity) or not. The fact that a basically classical approach apparently did so well may even be considered rather surprising.

Of course, there have been suggestions of how to approach the problem on the basis of quantum theory [7-14], but again, none of them seems to have established the emergence of thermodynamics from quantum mechanics as an underlying theory in a conclusive way.

The text at hand can be viewed as a contribution to this ongoing discussion. Thus, on one hand, one might consider this work as a somewhat new explanation for the emergence of thermodynamic behavior. This point of view definitely leaves one question open: Whether or not all macroscopic thermodynamic systems belong to the class of systems that will be examined in the following. The answer to this question is beyond the scope of this text.

Furthermore, this quantum approach to thermodynamics may turn out to be not a one-way road: In fact, this delicate interplay between quantum mechanics and thermodynamics could possibly shed new light on some interpretational problems within quantum mechanics: With the "exorcism" of subjective ignorance as a guiding principle underlying thermodynamic states, the general idea of quantum states representing subjective knowledge might loose much of its credibility.

However, this book might be looked at also from another, less
speculative angle.
Rather than asking how thermodynamic behavior of typical systems
might be explained, one can ask whether the principles of thermodynamics
are a powerful tool for predictions and whether its descriptions might
be applicable to other systems than the pertinent large, many particle
systems.
It is a main conclusion of this work that the answer has to be positive.
For it turns out that a large class of small quantum systems without
any restriction concerning size or particle number show thermodynamic

behavior with respect to an adequately defined set of thermodynamic
variables.
This behavior ("nano-thermodynamics") requires some embedding but is
nevertheless established entirely on the basis of the Scrödinger
gleichung.

So it is left to the reader to decide whether there is room and need for a foundation of thermodynamics on the basis of quantum theory or whether he simply wants to gain insight of how the applicability of thermodynamics can be extended down to the microscopical scale; in both cases we hope the reading will be interesting and clarifying.

This book is not intended to be a review, not even of the most
important contributions, which more or less point into a similar
direction.
Related work includes, in particular, the so called decoherence theory.
We cannot do justice to the numerous investigations; we merely

give a few references [15-18].
It might be worth mentioning here that decoherence has, during the
last years, mainly been discussed as one of the main obstacles
in the implementation of large scale quantum computers
[19], possibly neglecting other aspects of the phenomenon.

Last but not least a short "manual" for the reading of this book shall be given here.

Chapter 2 and 3 are not meant as a full-fledged introduction, more as a reminder of the central topics in quantum mechanics and thermodynamics. They may very well be skipped by a reader familiar with these subjects. Chapter 4 is a collection of historical approaches to thermodynamics with a focus on their insufficiens hereby neglecting their undoubtable brilliance, again this Chapter is not imperative for the understanding of Part II.

Chapter 5 lists the properties of thermodynamic quantities that need to be derived from an underlying theory (quantum mechanics). This derivation is then given in the remainder of Part II. In Chap. 6 the central ideas of this quantum approach to thermodynamics are explained in plain text (no formulas). For a quick survey it might be read without referring to anything else. Starting with Chap. 7 and throughout Part II, these ideas are derived in detail, which will probably only be enlighting if being read from the beginning. Exceptions are Chap. 11 and 14, which are important for the general picture, but have their own "selfcontained" messages.

Chapter 18 mainly consists of numerical illustrations of the

analytically derived principles in Part II.
In order to get an idea of the benefits of the theories derived in
Part II, it might be read as a "stand alone" Chapter.
In Chap. 19 recent results on quantum heat conduction are presented,
it may also be read individually, for it is only loosely connected
to the rest of the book as far as mathematical techniques are concerned.

- [1] Boltzmann, L.
*Lectures on Gas Theory*, Los Angeles, 1964. University of California Press, Los Angeles, 1964. - [2] Gibbs, J. W.
*Elementary Principles in Statistical Mechanics*, New York, 1960. Dover Publications, New York, 1960. - [3] Birkhoff, G. Proof of the Ergodic Theorem.
*Proc. Nat. Acad. Sc. USA*, 17:656-660, 1931. - [4] Ehrenfest, P. and Ehrenfest, T.
*The Conceptual Foundations of the Statistical Approach in Mechanics*, Ithaca, 1959. Cornell University Press, Ithaca, 1959. - [5] Neumann, J. v. Proof of the Quasi-ergodic Hypothesis.
*Proc. Nat. Acad. Sci. USA*, 18:70-82, 1932. - [6] Ben-Menahem, Y. and Pitowsky, I. Introduction.
*Stud. Hist. Phil. Mod. Phys.*, 32:503-510, 2001. - [7] Neumann, J. v. Beweis des Ergodensatzes und des H-Theorems in der neuen Mechanik.
*Z. Phys.*, 57:30-70, 1929. - [8] Pauli, W. and Fierz, Z. Über das H-Theorem in der Quantenmechanik.
*Z. Phys.*, 106:572-587, 1937. - [9] Landau, L. and Lifshitz, E.
*Statistical Physics, Part 1*, volume 5 of*Course of Theretical Physics*, Oxford, 1980. Pergamon Press, Oxford, 3. edition, 1980. - [10] Landau, L. and Lifshitz, I.
*Quantum Mechanics*, volume 3 of*Course of Theoretical Physics*, Oxford, 1977. Pergamon Press, Oxford, 3. edition, 1977. - [11] Schrödinger, E.
*Statistical Thermodynamics*,*a course of seminar lectures, del. in jan. - march 1944*, Cambridge, 1948. Cambridge University Press, Cambridge, 1948. - [12] Lindblad, G.
*Non-equilibrium Entropy and Irreversibility*, volume 5 of*Mathematical physics studies*, Dordrecht, 1983. Reidel, Dordrecht, 1983. - [13] Zurek, W. H. Decoherence and the transition from quantum to classical.
*Physics Today*, 41:Oct 36-44, 1991. - [14] Zurek, W. H. and Paz, J. P. Decoherence, Chaos, and the Second Law.
*Phys. Rev. Lett.*, 72:2508-2511, 1994. - [15] Giulini, D., Joos, E., Kiefer, C., Kupsch, J., Stamatescu, I.-O. and Zeh, H.
*Decoherence and the Appearance of a Classical World in Quantum Theory*, Berlin, Heidelberg, 2003. Springer, Berlin, Heidelberg, 2 edition, 2003. - [16] Braun, D., Haake, F. and Strunz, W. T. Universality of Decoherence.
*Phys. Rev. Lett.*, 86:2913-2917, 2001. - [17] Zurek, W. H. Decoherence, einselection, and the quantum origin of the classical.
*Rev. Mod. Phys.*, 75:715-775, 2003. - [18] Hackermüller, L., Hornberger, K., Brezger, B., Zeilinger, A. and Arndt, M. Decoherence of matter waves by thermal emission of radiation.
*Nature*, 427:711-714, 2004. - [19] Nielsen, M. A. and Chuang, I. L.
*Quantum Computation and Quantum Information*, Cambridge, 2000. Cambridge University Press, Cambridge, 2000.

For a further preview and introduction to the topics in this book see the informative homepage of Jochen Gemmer. Furthermore we would like to inform you about the primary publications of the topics in the present book:

- J. Gemmer, A. Otte, G. Mahler

*Quantum Approach to a Derivation of the Second Law of Thermodynamics*

Phys. Rev. Lett.**86**1927-1930(2001) - J. Gemmer, G. Mahler

*Entanglement and the factorization-approximation*

Euro. Phys. J. D**17**385-393 (2001) - J. Gemmer, G. Mahler

*Limits of effective potential approach: nucleus-electron entanglement in a model of the He+-ion,*

Europhys. Lett.**59**159-165 (2002) - J. Gemmer, G. Mahler

*Distribution of local entropy in the Hilbertspace of bi-partite qauntum systems: Origin of Jaynes' principle*

Europhys. Lett.**62**629-635 (2003) - P. Borowski, J. Gemmer, G. Mahler

*On the concept of pressure in quantum mechanics,*

Europhys. Lett.**62**629-635 (2003) - P. Borowski, J. Gemmer, G. Mahler

*Relaxation to equilibrium under pure Schrödinger dynamic,*

Euro. Phys. J. B**31**249-257 (2003) - M. Michel, M. Hartmann, J. Gemmer, G. Mahler

*Fourier's Law confirmed for a class of small quantum systems,*

Eur. Phys. J. B**34**325-330 (2003)

The authors thank Dipl. Phys. Peter Borowski (MPI Dresden) for the first numerical simulations to test our theoretical considerations and for contributing several figures as well as some text material and Dipl. Phys. Michael Hartmann (DLR Stuttgart) for contributing some sections. Furthermore, we thank cand. phys. Markus Henrich for helping us to design some diagrams and, both M. Henrich and cand. phys. Christos Kostoglou (Institut für theoretische Physik, Universität Stuttgart) for supplying us with numerical data. We have profited a lot from fruitful discussions with Dipl. Phys. Harry Schmidt, Dipl. Phys. Marcus Stollsteimer and Dipl. Phys. Friedemann Tonner (Institut für theoretische Physik, Universität Stuttgart) and Prof. Dr. Klaus Bärwinkel, Prof. Dr. Heinz-Jürgen Schmidt and Prof. Dr. Jürgen Schnack (Fachbereich Physik, Universität Osnabrück). We benefitted much from conversations with Prof. Dr. Wolfram Brenig and Dipl. Phys. Fabian Heidrich-Meisner (Technische Universität Braunschweig) as well as Dr. Alexander Otte and Dr. Heinrich Michel (Stuttgart). It is a pleasure to thank Springer-Verlag, especially Dr. Christian Caron, for continuous encouragement and excellent cooperation. This cooperation has garanteed a rapid and smooth progress of the project. Financial support by the "Deutsche Forschungsgesellschaft" and the "Landesstiftung Baden-Württemberg" is gratefully acknowledged. Last but not least, we would like to thank Björn Butscher, Kirsi Weber and Hendrik Weimer for helping us typesetting the manuscript, proof-reading and preparing some figures.

You are friendly requested to comment on our book and this page under: m.michel@surrey.ac.uk

19th February 2008
© Copyright 2006 by MM