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# A quantum approach to thermodynamics [Elektronische Ressource] / vorgelegt von Jochen Gemmer

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A Quantum ApproachtoThermodynamicsVon der Fakultat Mathematik und Physik der Universitat Stuttgart˜ ˜zur Erlangung der Wu˜rde eines Doktors derNaturwissenschaften (Dr. rer. nat.) genehmigte Abhandlungvorgelegt vonJochen Gemmeraus StuttgartHauptberichter: Prof. Dr. G. MahlerMitberichter: Prof. Dr. U. SeifertTag der mundlichen Prufung: 26. Februar 2003˜ ˜Institut fu˜r Theoretische Physik IUniversitat Stuttgart˜2003iiAllnumericalsimulationsinthisthesishavebeencomputedverycarefullybyP.Borowski.Contents1 Introduction 12 Review of the Field 32.1 Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Ergodicity Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 Ensemble Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 Macroscopic Cell Approach . . . . . . . . . . . . . . . . . . . . . . . . . 72.5 The Problem of Entropy Invariance . . . . . . . . . . . . . . . . . . . . . 82.6 Shannon’s Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.7 Time averaged Density Matrix Approach . . . . . . . . . . . . . . . . . . 102.8 Open System Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Structure of a Foundation of Thermodynamics 133.1 Checklist of Properties of thermodynamical Quantities . . . . . . . . . . 133.1.1 Additional necessary Considerations . . . . . . . . . . . . . . . . . 154 Background of the Present Approach 174.

Subjects

##### Astronomie

Informations

A Quantum Approach
to
Thermodynamics
Von der Fakultat Mathematik und Physik der Universitat Stuttgart˜ ˜
zur Erlangung der Wu˜rde eines Doktors der
Naturwissenschaften (Dr. rer. nat.) genehmigte Abhandlung
vorgelegt von
Jochen Gemmer
aus Stuttgart
Hauptberichter: Prof. Dr. G. Mahler
Mitberichter: Prof. Dr. U. Seifert
Tag der mundlichen Prufung: 26. Februar 2003˜ ˜
Institut fu˜r Theoretische Physik I
Universitat Stuttgart˜
2003ii
AllnumericalsimulationsinthisthesishavebeencomputedverycarefullybyP.Borowski.Contents
1 Introduction 1
2 Review of the Field 3
2.1 Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Ergodicity Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.3 Ensemble Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.4 Macroscopic Cell Approach . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.5 The Problem of Entropy Invariance . . . . . . . . . . . . . . . . . . . . . 8
2.6 Shannon’s Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.7 Time averaged Density Matrix Approach . . . . . . . . . . . . . . . . . . 10
2.8 Open System Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 Structure of a Foundation of Thermodynamics 13
3.1 Checklist of Properties of thermodynamical Quantities . . . . . . . . . . 13
3.1.1 Additional necessary Considerations . . . . . . . . . . . . . . . . . 15
4 Background of the Present Approach 17
4.1 Density Operator and Reduced Density Operator . . . . . . . . . . . . . 17
4.2 Compound Systems, Entropy and Entanglement . . . . . . . . . . . . . . 18
4.3 Fundamental and Subjective Lack of Knowledge . . . . . . . . . . . . . . 20
4.4 The natural Cell Structure of Hilbertspace . . . . . . . . . . . . . . . . . 20
5 Analysis of the Cell Structure of Compound Hilbertspaces 23
5.1 Representation of Hilbertspace and Hilbertspace Velocity . . . . . . . . . 23
5.2 Purity and Notation of States . . . . . . . . . . . . . . . . . . . . . . . . 25
5.3 Partition of the Full System into considered System and Surrounding . . 27
5.4 Microcanonical Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5.4.1 Microcanonical Interactions and the corresponding accessible Region 29
g5.4.2 The\Landscape"of P in the accessible Region . . . . . . . . . . 30
g5.4.3 The minimum Purity State and the Hilbertspace Average of P . 30
5.5 Canonical Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5.5.1 The accessible and the dominant Region . . . . . . . . . . . . . . 34
5.5.2 Identiﬂcation of the dominant Region . . . . . . . . . . . . . . . . 34
5.5.3 Analysis of the Size of the dominant Region . . . . . . . . . . . . 35
5.5.4 The canonical equilibrium State . . . . . . . . . . . . . . . . . . . 36
iiiiv Contents
5.6 Single Energy Probabilities and Fluctuations . . . . . . . . . . . . . . . . 38
5.7 Local Equilibrium States and Ergodicity . . . . . . . . . . . . . . . . . . 40
5.8 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.8.1 Numerical Results for microcanonical Conditions . . . . . . . . . 42
5.8.2 Numerical Results for Canonical Conditions . . . . . . . . . . . . 45
5.8.3 Numerical Results for Probability Fluctuations . . . . . . . . . . . 49
6 Typical Spectra of Large Systems 51
6.1 The Extensitivity of Entropy . . . . . . . . . . . . . . . . . . . . . . . . . 51
6.2 Spectra of Modular Systems . . . . . . . . . . . . . . . . . . . . . . . . . 52
6.3 The Boltzmann Distribution . . . . . . . . . . . . . . . . . . . . . . . . . 56
6.3.1 Beyond the Boltzmann Distribution? . . . . . . . . . . . . . . . . 57
7 Temperatures 59
7.1 Deﬂnition of spectral Temperature . . . . . . . . . . . . . . . . . . . . . 60
7.2 The Equality of spectral Temperatures in Equilibrium . . . . . . . . . . . 61
7.3 Spectral Temperature as the Derivative of Energy with Respect to Entropy 63
8 Pressure 67
8.1 On the Concept of adiabatic Processes . . . . . . . . . . . . . . . . . . . 67
8.2 The Equality of parametric Pressures in Equilibrium . . . . . . . . . . . 71
9 Thermodynamical Limit 73
9.1 Weak coupling Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
9.2 Microcanonical Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . 74
9.3 Energy-Exchange Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . 74
9.4 Canonical Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
9.5 Spectral Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
9.6 Parametric Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
9.7 Extensitivity of Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
10 Quantum mechanical and classical State Densities 77
10.1 Similarity of classical and quantum mechanical State Densities . . . . . . 78
10.1.1 Sommerfeld Quantization . . . . . . . . . . . . . . . . . . . . . . 78
10.1.2 Partition Function Approach . . . . . . . . . . . . . . . . . . . . . 78
10.1.3 Minimum Uncertainty Wavepackage Approach . . . . . . . . . . . 79
11 Ways to Equilibrium 85
11.1 Theories of Relaxation Behavior . . . . . . . . . . . . . . . . . . . . . . . 85
11.1.1 Fermi’s golden Rule for external Perturbations . . . . . . . . . . . 85
11.1.2 Fermi’s golden Rule for a coupled Environment . . . . . . . . . . 86
11.1.3 Weisskopf-Wigner Theory . . . . . . . . . . . . . . . . . . . . . . 86
11.1.4 Large Environment Approach . . . . . . . . . . . . . . . . . . . . 86
11.1.5 Numerical Results for the Relaxation Period . . . . . . . . . . . . 93
12 Summary and Conclusion 95Contents v
13 Zusammenfassung 99
13.1 Einleitung und historischer Hintergrund . . . . . . . . . . . . . . . . . . 99
13.2 Axiomatischer Aufbau der Thermodynamik . . . . . . . . . . . . . . . . 101
13.3 Grundprinzipien der Theorie . . . . . . . . . . . . . . . . . . . . . . . . . 102
13.4 Die Zellstruktur des Hilbertraums . . . . . . . . . . . . . . . . . . . . . . 104
13.4.1 Mikrokanonische Bedingungen . . . . . . . . . . . . . . . . . . . . 105
13.4.2 Kanonische Bedingungen . . . . . . . . . . . . . . . . . . . . . . . 106
13.5 Spektren modularer Systeme . . . . . . . . . . . . . . . . . . . . . . . . . 106
13.6 Spektrale Temperatur . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
13.7 Parametrischer Druck . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
13.8 Klassische und quantenmechanische Zustandsdichte . . . . . . . . . . . . 108
13.9 Wege zum Gleichgewicht . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
13.10Schlussbetrachtung und Ausblick . . . . . . . . . . . . . . . . . . . . . . 109
Appendix 113
A Minimization of the eﬁective Interaction, eq.5.24 . . . . . . . . . . . . . . 113
B Conserved Probabilities, eq.5.28 . . . . . . . . . . . . . . . . . . . . . . . 113
C Averages over Hyperspheres, sect. 5.4.3: . . . . . . . . . . . . . . . . . . 114
D From Multiplication to Gaussian, eq. 6.16 . . . . . . . . . . . . . . . . . 115
E Entropy of an ideal Gas, eq. 6.23 . . . . . . . . . . . . . . . . . . . . . . 116
F Stabilized adiabatic Approximation, eq. 8.18 . . . . . . . . . . . . . . . . 117
G Sizes of Hypersphereregions, sect. 5.6 . . . . . . . . . . . . . . . . . . . . 117
H Hilbertspace Averages, sect. 11.1.4 . . . . . . . . . . . . . . . . . . . . . 119
Bibliography 123vi Contents1 Introduction
In the very beginning thermodynamics have been a purely phenomenological science.
Early scientists (Galileo, Santorio, Celsius, Fahrenheit) tried to give deﬂnitions for quan-
tities that where intuitively obvious to the observer, like pressure or temperature and
measured the interconnections between them. The idea that those phenomena could be
directly linked to other ﬂelds of physics, like classical mechanics, etc., was not common
at these days. This 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 of those days.
At the end of the 19’th century, when the atomic theory became popular, people began
to think of a gas as a box with a lot of bouncing balls inside. 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 when he connected entropy, a quantity
which was so far only described phenomenologically, to the volume of a certain region
in phase space, an object deﬂned within classical mechanics [1]. This was an enormous
step, especially from a practical point of view. Taking this connection for granted one
could now calculate all sorts of thermodynamical behavior of a system from it’s Hamilton
function. This gave rise to modern thermodynamics, a theory whose validity is beyond
any doubt today. Its results and predictions are a basic ingredient for the development
of all sorts of technical apparatuses, ranging from refrigerators to superconductors.
Boltzmann himself, however, tried to proof the conjectured connection between the phe-
nomenlogical and the theoretical entropy, but did not succeed without making other
assumptions like the famous ergodicity or the \a priory postulate". Other physicists
(Gibbs, Birkhoﬁ, Ehrenfest, Von Neumann [2{5]) later on then tried to proof those as-
sumptions, but none of this work seems to have solved the situation entirely. It has
even been mentioned that there are more properties of the entropy then its equivalence
with the region in phase space to be explained to reduce thermodynamics to classical
mechanics, thus the discussion is still ongoing [6]. The vast majority of the work done
in this ﬂeld is based on classical mechanics.
Meanwhile quantum theory, also initially triggered of 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 21’th century it seems highly unlikely that a
box with balls inside could be anything more than a rough sketch of what a gas really is.
Furthermore thermodynamical 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
12 1 Introduction
ergodicity) or not.
Of course there have been suggestions how to approach the problem on the basis of quan-
tum theory [7{13], but again, none of them seems to have established the emergence of
thermodynamics from quantum mechanics, as an underlying theory, in a decisive way,
for there is an ongoing discussion about those ideas also.
So this thesis is meant as a contribution to this whole debate about how thermodynamics
can be possibly viewed, as emerging from some underlying theory.
It is organized as follows:
First of all a short and necessarily incomplete overview over the diﬁerent historical at-
tempts to connect thermodynamics to some underlying theory and their problems is
given. Since from this consideration the impression arises that it is even unclear what
needs to be shown and proved to establish such a connection in a satisfying manner, in
the following the program of this task is deﬂned, i.e., one possible list of conjectures,
concerning properties of the entropy and other quantities, is given, that could, proven
item by item, serve as a basis for the techniques of thermodynamics. Those proofs, or
at least approaches to them, are given in the main part of this thesis. In the last part
the connections to other theories are pointed out, e.g., it is explained why the classical
theory works so well from a practical point of view, despite its being unsatisfactory in
the theoretical context.2 Review of the Field
Almost all approaches to a connection between thermodynamics and an underlying the-
ory deal with the irreversibility that seems to be present in thermodynamical phenom-
ena, but is most likely absent, in any underlying theory. So the biggest part of these
approaches are in some sense proofs of the second law of thermodynamics, which states
this irreversibility. They consist of attempts to formulate entropy as a function of quan-
tities, whose dynamics can be calculated in a microscopic picture, in such a way, that the
entropy increases during any evolution until it reaches a maximum that is proportional
to the logarithm of the volume of the accessible phase space (energy shell). This is the
wanted limit because this is the quantity that the entropy has to be identiﬂed with, in
order to get state functions, that are, for the biggest part, in excellent agreement with
experiment. It has not really been appreciated very much that there are other properties
of the entropy that remain to be shown even if the above behavior is established (see
3.1).
One problem of all approaches based on considerations concerning Hamiltonian mechan-
ics is the applicability of classical mechanics itself. To illustrate this, lets consider a
gas consisting of atoms or molecules. In principle such a system should, of course, be
described by quantum mechanics. Nevertheless, for simplicity, one could possibly treat
the system classically, if it started and remained in the Ehrenfest limit, i.e., if the spread
of the wavepackages was small compared to the structure of the potentials which the
particles encounter. Those are given by the particles themselves, which basically repel
¡10each other. If we take the size of those particles to be roughly some 10 m, we have
¡10to demand that the wavepackages should have a width smaller than 10 m in the be-
ginning. Assuming particle masses between some and some hundred protonmasses, and
plugging numbers into the corresponding formulas [14], we ﬂnd that the spread of such
wave packages will be on the order of some meters to 100m after one second, which
means the system leaves the Ehrenfest limit on a timescale much shorter than the one
typical for thermodynamical phenomena. If we demand the packages to be smaller in
the beginning, it gets even worse. Considering this, it is questionable whether any ex-
planation based on considerations of Hamiltonian dynamics in ¡-space (Cartesian space,
spanned by the 6N position and momentum coordinates of a N particle system) or „-
space (Cartesian space, spanned by the 6 position and momentum coordinates of any
particle of the system) can ever be a valid foundation of thermodynamics at all. This
insu–ciency of the classical picture becomes manifest at very low temperatures (freezing
out inner degrees of freedom), and it is entirely unclear why it should become valid at
higher temperatures even if it produces good results.
34 2 Review of the Field
Nevertheless a short, and necessarily incomplete overview, also and mainly including
such ideas, shall be given here.
2.1 Boltzmann Equation
Boltzmann’s work was probably one of the ﬂrst scientiﬂc approaches to irreversibility
(1866) [1]. It was basically meant to explain and quantify the observation that a gas
which is at ﬂrst located in one corner of a volume will always spread over the whole
volume, whereas a gas uniformly distributed over the full volume never suddenly shrinks
to be then concentrated in one corner. This seems to contradict Hamiltonian dynamics
according to which any process that is possible forwards in time, should be also possible
backwards in time.
Boltzmann considered a function f(~q;~v;t) in „-space. „-space is the 6-dimensional space
of the position and the velocity of one particle. f is a density in this space. If f is high
at some point, this means that there are many particles in the corresponding volume-
elementofpositionspace, ?yinginthethesamedirectionwiththesamevelocity. Making
theassumptionof\molecularchaos", i.e., theassumptionthatthevelocitiesandpositions
of particles within cells of „-space that are subject to collisions, are uncorrelated before
the collision, he could come up with an evolution equation (\ Boltzmann equation")
for f, that tends to make f more and more uniform. This is quantiﬂed in the famous
\H-theorem". Boltzmann showed that a function H deﬂned asZ
3 3H(t) = f logf d vd q
could only decrease in time, given the validity of the Boltzmann equation. This way
he somehow established a kind of macroscopic irreversibility but not without additional
assumptions. Furthermore H, though being mathematically reminiscent of later formu-
lations of the entropy, could not really be identiﬂed with the entropy, for it converges
against the logarithm of the accessible volume in „-space, rather than in ¡-space.
2.2 Ergodicity Approach
The basis of this approach, also pursued by Boltzmann, is the assumption that any pos-
sible macroscopic measurement takes a time which is almost inﬂnitely long, compared
to the timescale of molecular motion. Thus, the outcome of such a measurement has to
be seen as the time average over many hypothetical instantaneous measurements. Thus,
if it was true, that a trajectory ventured through all regions of the accessible volume
in ¡-space, no matter where it started, the behavior of a system would be, as if it was
at any point at the same time, regardless of its starting point. This way irreversibility
could be introduced, entropy being somehow connected to the volume, that the trajec-
tory already ventured through, during the observation time.
In order to state this idea in a clearer form, the so called\ergodic hypothesis"has been
formulated: