On reduced dynamics of
quantum-thermodynamical systems
vorgelegt von
Diplom-Physikerin
Akiko Kato
von der Fakult¨at II - Mathematik und Naturwissenschaften
der Technischen Universit¨at Berlin
zur Erlangung des akademischen Grades
Doktor der Naturwissenschaften
Dr. rer. nat.
genehmigte Dissertation
Promotionsausschuß:
Vorsitzender: Prof. Dr. M. Schoen
Gutachter: Prof. Dr. W. Muschik
Gutachter: Prof. Dr. A. Knorr
Tag der wissenschaftlichen Aussprache: 9. Dezember 2004
Berlin 2004
D 83
1
Danksagung
Ich danke herzlich
Herrn Prof. Dr. W. Muschik f¨ur die Betreuung dieser Arbeit und daf¨ur, daß ich
sowohl durch Gespr¨ache mit ihm, als auch in seinen Vorlesungen und Seminaren
viel ¨uber Physik im allgemeinen und ¨uber Thermodynamik im speziellen lernen
konnte.
Herrn Prof. Dr. A. Knorr f¨ur konstruktive Diskussionen und daf¨ur, daß er sich als
zweiter Gutachter zur Verf¨ugung gestellt hat.
Herrn Prof. Dr. M. Schoen als Vorsitzendem des Promotionsausschusses.
den Mitgliedern der Arbeitsgruppe, Dr. Christina Papenfuß, Dr. Sebastian G¨umbel,
Dr. Heiko Herrmann und Dr. Gunnar R¨uckner f¨ur eine sch¨one gemeinsame Zeit am
Institut und f¨ur interessante Einblicke in ihre Arbeitsgebiete.
Herrn Dr. habil. J.-P. Kaufmann f¨ur die fr¨ohlichen Stunden im Musikkreis der
Physik.
Herrn Dipl.-Phys. Andreas Staude als meinem pers¨onlichen Lieferanten von Ferrero-
K¨ußchen.
2
Hiermit versichere ich an Eides statt, daß ich die Dissertation selbst¨andig verfaßt
habe.
Berlin, 3. Oktober 2004
Zusammenfassung
F¨ur die Behandlung irreversibler Prozesse diskreter quantenthermodynamischer Sys-
teme eignet sich die reduzierte Beschreibung. Hierf¨ur betrachtet man bez¨uglich
einer gegebenen Beobachtungsebene verschiedene begleitende Prozesse zur quanten-
mechanischen von Neumann Dynamik. W¨ahrend die kanonische Dynamik auch un-
abh¨angig von einer mikroskopischen Bewegungsgleichung hergeleitet werden kann,
wird die von Neumann Dynamik im Rahmen des Projektionsformalismus in einen
relevanten und einen irrelevanten Anteil bez¨uglich der Beobachtungsebene aufgeteilt.
Als Beispiel f¨ur die Projektionsdynamik geben wir die Robertson Dynamik und die
Fick Sauermann Dynamik an und untersuchen deren Eigenschaften sowie geeignete
Projektionsoperatoren. Hierbei k¨onnen wir den Zugang ¨uber das Mori Skalarpro-
dukt und dem entsprechenden Kawasaki Gunton Operator ¨uber den generalisiert
kanonischen Dichteoperator hinaus auf allgemeine begleitende Operatoren erweit-
ern. Unter Verwendung der kanonischen Dynamik wird die Rate der von Neumann
Entropie bestimmt und analysiert. In der thermodynamischen Anwendung sind
zeitabh¨angige Observablen, die von Arbeitsvariablen abh¨angen, von besonderem In-
teresse. Um den ersten Hauptsatz anwenden zu k¨onnen, m¨ussen die grundlegenden
thermodynamischen G¨oßen, Arbeits- und W¨armeaustausch, bekannt sein. Deswe-
gen fordern wir, daß begleitende Prozesse hinreichend f¨ur die zugeh¨origen Arbeits-
und Flußobservablen sind. Im allgemeinen sind weder die kanonische Dynamik
noch die Projektionsdynamiken hinreichend, so daß wir anschließend einen Ansatz
f¨ur eine hinreichende Dynamik angeben. In Zusammenhang mit der Extended Ther-
modynamics verallgemeinern wir die Forderung nach der hinreichenden Dynamik
auf eine schw¨achere Formulierung. Anhand des Beispiels der kanonischen Dynamik
diskreter Systeme unter thermischem Kontakt analysieren wir die Entropierate und
wenden das Konzept der Kontakttemperatur auf die zugeh¨origen W¨arme¨uberg¨ange
an. In einem anderen Beispiel behandeln wir die Elektron-Phonon-Wechselwirkung
im Festk¨orper. Hier wird die Robertson Projektion durchgef¨uhrt, um den Ein-
teilchenanteil vom Korrelationsanteil f¨ur Mehrteilchen zu trennen. Beide Anteile
sind relevant, um die Elemente der Elektronendichtematrix zu bestimmen. Die Dy-
namik von Teilsystemen sowie deren begleitende Prozesse werden untersucht, auch
im Hinblick auf das Hinreichendsein. Die von Neumann Dynamik eines Teilsystems
wird um einen Term erg¨anzt, der die Entropieproduktion bei innerem W¨armestrom
ber¨ucksichtigen soll.
3
Abstract
Irreversible processes in quantum-thermodynamical discrete systems can be treated
by means of reduced information dynamics. For this, we consider different accom-
panying processes of the quantum mechanical von Neumann dynamics with respect
to a given beobachtungsebene. While canonical dynamics can be derived indepen-
dently from any quantum mechanical dynamics, projection formalism induces the
isolation of the relevant part of the von Neumann dynamics from its irrelevant part
according to the beobachtungsebene. We present the Robertson dynamics and the
Fick Sauermann dynamics as projection dynamics, their properties and appropriate
projectors. We are able to generalise the conventional approach through the Mori
product and the according Kawasaki Gunton operator for the generalised canonical
density operator to any accompanying operator. Starting from canonical dynamics,
it is possible to calculate and analyse the rate of the von Neumann entropy. In ther-
modynamical applications, observables depending on work variables are of special
interest. In order to apply the First Law, the essential thermodynamical quantities
of work and heat exchange have to be known. Thus we demand the sufficiency of
accompanying processes for the according work and flux observables. In general,
neither canonical dynamics nor projected dynamics are sufficient for them, so we
give an ansatz to obtain sufficient dynamics. With regard to the Extended Ther-
modynamics, we generalise the demand of the sufficiency to a weaker formulation.
As an example, we consider the canonical dynamics of discrete systems in thermal
contact, where we analyse the rate of entropy and apply the concept of contact
temperature to the heat exchange. Another example is the electron-phonon inter-
action in a solid. Here, the Robertson projection is used to divide the von Neumann
dynamics into a single particle part and a many particle correlation part, which are
both relevant to determine the according electron density matrix elements. Con-
cerning the dynamics of a subsystem, its accompanying processes are investigated,
also with regard to the suffuciency. The von Neumann dynamics for a subystem
is supplemented with an additional term takig the entropy production by internal
heat fluxes into account.
4
Contents
1 Introduction 8
2 Fundamentals 11
2.1 The Density Operator and its Dynamics . . . . . . . . . . . . . . . . 11
2.2 Beobachtungsebene . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Accompanying Processes of Statistical Operators . . . . . . . . . . . 13
2.4 von Neumann Entropy and Generalised Canonical Operator . . . . . 13
2.5 The Thermodynamical Situation . . . . . . . . . . . . . . . . . . . . 15
3 Dynamics of Statistical Operators 17
3.1 Canonical Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 The Relevant Part of the Density Operator . . . . . . . . . . . . . . 19
3.2.1 Robertson Dynamics . . . . . . . . . . . . . . . . . . . . . . . 20
3.2.2 Fick Sauermann Dynamics . . . . . . . . . . . . . . . . . . . 22
3.2.3 Projector P. . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2.4 Other Mappings P. . . . . . . . . . . . . . . . . . . . . . . . 25
3.2.5 Canonical Dynamics vs. Fick Sauermann Dynamics using
Kawasaki Gunton Operator . . . . . . . . . . . . . . . . . . . 26
3.3 About the Rate of Entropy and Entropy Production . . . . . . . . . 27
3.3.1 Canonical Dynamics . . . . . . . . . . . . . . . . . . . . . . . 27
3.3.2 Fick Sauermann Dynamics . . . . . . . . . . . . . . . . . . . 28
5
CONTENTS 6
4 Sufficiency 29
4.1 Definitions and Illustration . . . . . . . . . . . . . . . . . . . . . . . 29
4.1.1 Sufficiency for Observables . . . . . . . . . . . . . . . . . . . 29
4.1.2 Sufficient Dynamics . . . . . . . . . . . . . . . . . . . . . . . 30
4.2 Investigation of Sufficiency of Dynamics . . . . . . . . . . . . . . . . 32
4.2.1 Canonical Dynamics . . . . . . . . . . . . . . . . . . . . . . . 32
4.2.2 Projected Dynamics of %rel . . . . . . . . . . . . . . . . . . . 34
4.3 A Sufficient Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5 Further Considerations on Sufficiency 39
5.1 Generalised Fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.1.1 Canonical Dynamics . . . . . . . . . . . . . . . . . . . . . . . 39
5.1.2 Robertson Dynamics . . . . . . . . . . . . . . . . . . . . . . . 40
5.1.3 Fick Sauermann Dynamics . . . . . . . . . . . . . . . . . . . 40
5.2 Possible Conditions for an Extended Sufficiency . . . . . . . . . . . . 41
5.3 Weak Sufficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.4 Time Independent Observables . . . . . . . . . . . . . . . . . . . . . 43
6 Examples for Discrete Systems in Thermal Contact 46
6.1 Considering the Partition . . . . . . . . . . . . . . . . . . . . . . . . 46
6.2 Subsystems and their Contact Temperature . . . . . . . . . . . . . . 47
7 An Example: Electron Phonon Interaction 50
7.1 Quantum Well . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
7.2 Some Thermodynamical Quantities . . . . . . . . . . . . . . . . . . . 54
CONTENTS 7
8 The Dissipative Term 57
8.1 Pure Quantum Mechanical Dynamics of a Subsystem . . . . . . . . . 57
8.2 Dynamics of a Subsystem with Reduced Information . . . . . . . . . 59
8.2.1 Some Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 59
8.2.2 Sufficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
8.3 Entropy Production by Heat Conduction . . . . . . . . . . . . . . . . 61
9 Conclusions 65
10 Appendices 67
10.1 Appendix 1: An example . . . . . . . . . . . . . . . . . . . . . . . . 67
10.2 Appendix 2: G¨umbel’s approach . . . . . . . . . . . . . . . . . . . . 69
10.3 Appendix 3: Entropy of Canonical Dynamics . . . . . . . . . . . . . 71
Bibliography 73
Chapter 1
Introduction
In the quantum mechanical approach of thermodynamics, the projection operator
formalism has been successful in many applications. This technique was intro-
duced by Nakajima [Nak58] and Zwanzig [Zwa60], and expanded later by Robert-
son [Rob66] to derive Langevin Mori theory [Fic83], [Bal86] and generalised Fokker
Planck equation [Gra82].
The main argument, why people use reduced information dynamics, is that one
needs evolution equation for only few “relevant” macroscopic variables, neglecting
all “irrelevant” parts regarding the considered experiment [Rau96]. A macroscopic
thermodynamical system has practically an infinite number of microscopic degrees
of freedom. It is not possible to find out its exact quantum mechanical state, not to
mention its density operator %, because we have only restricted information about
the system, represented by a limited number of measuring instruments. It is not
necessary to know the microstate, if we only have measuring devices to obtain a
state which is macroscopically not distinguishable from the exact microstate. There-
fore we do not need the time development of the microstate, but only that of the
macroscopically indistinguishable state which belongs to the microstate. This can
be performed by the reduced description implemented by the projection technique.
Choosing a set, called “beobachtungsebene”, including a limited number of rele-
vant independent observables, we look for a density operator ˆ%different from the
microscopic density operator which generates the correct expectation values with
respect to the chosen beobachtungsebene. Such dynamics are called accompanying
processes according to [Mus94].
This lack of information leads to an other desired characteristic of the projected
description, the irreversibility [Zwa60], [Zub74], [Zub96]. On the one hand, we have
the explicit dependence of the dynamics on memory effects in terms of a time in-
tegral. On the other hand, the method described above do not suffice determining
a unique relevant density matrix. There exist in fact a lot of possible trace class
operators satisfying the same macroscopic conditions. Jayne’s information theory
approach [Jay57a], [Jay57b] tells us, how to choose that density operator Rwhich
maximises the von Neumann entropy, implying minimum and most bias free in-
formation. Consequently, Rhas the well-known form of a generalised canonical
8
CHAPTER 1. INTRODUCTION 9
operator. Assuming the von Neumann dynamics for an isolated quantum mechani-
cal system
˙%=−i
~[H, %],(1.1)
which is a direct conclusion of the Schr¨odinger equation, it is a fact, that the time
derivative of the von Neumann entropy vanishes. The increase of entropy manifested
in the Second Law cannot be explained here, because usual quantum mechanics is
a reversible theory. There are three ways to introduce irreversibility, that means
non vanishing entropy production, into quantum mechanics: First of all the al-
ready discussed reduced description by introducing the projection technique. Here
irreversibility is generated by missing information. In [Schi94], [Kat00a], [Kat99]
projection formalism is used to show the validity of the Second Law under certain
conditions. The second possibility for introducing irreversibility is to change the von
Neumann equation without altering the Schr¨odinger equation. In this case an addi-
tional term appears in the von Neumann equation which is directly connected to the
entropy production [Kau96], [Lin83], chapter 8.3. This kind of entropy production
has nothing to do with missing information as in the case of projector formalism,
but it is generated by special construction of mixed states from pure ones. The third
possibility is to change the Schr¨odinger equation by introducing friction terms, that
means by introducing microscopic irreversibility [Arb96], [Gis81].
Another motivation to use the reduced description concerns the treatment of subsys-
tems of an isolated compound system. Even if we only consider reduced dynamics of
a subsystem, the evolution equation derived from (1.1) still contains the full density
matrix of the compound system, which is unknown in general (chapter 8). This
problem does not arise in projection operator formalism.
When using the projection operator formalism, a problem arises, if the work vari-
ables of the considered system (volume, magnetic field etc.) are time dependent:
Are the expectation values belonging to the time derivatives of the work variables
correctly described by the projections? Or are these time derivatives only repre-
sented approximatively? This question is essential for thermodynamics, because the
power differential appearing in the First Law is described by these time derivatives
of the work variables. This problem is characterised by the concept of “sufficiency”
of the beobachtungsebene, which is treated in chapter 4.
In chapter 2 and 3, we briefly recapitulate the projection operator formalism and the
corresponding dynamics. Next, in chapter 4 and 5, we analyse to what extend we
can describe thermodynamical quantities like work and heat flux in this context. Is
it possible to calculate such quantities from the relevant statistical operator even if
it belongs to a beobachtungsebene that does not include the work or flux operators?
This is the quite fundamental question of sufficiency which must be dealt with, if the
thermodynamical behavior of the quantum system is regarded. In order to examine
work quantities, we will consider operators with explicit time dependence, while
most authors neglect the exchange of work, cf. [Rau96], [Ali88]. In chapter 6 we
show an example of two discrete systems in purely thermal contact, that means no
work nor particle exchange is considered. Chapter 7 includes an example of many
body quantum mechanics, the electron phonon interaction.
At last in chapter 8, we treat dynamics of subsystems derived from (1.1), where
we turn our special attention to the increase of entropy due to internal heat fluxes.
Thermodynamical factors which lead to increasing entropy are internal heat- and
CHAPTER 1. INTRODUCTION 10
material fluxes and other entropy productions. The material flux will be omitted
here. The entropy production we can explain with an additional term in von Neu-
mann dynamics, as shown in [Ber84], [Kau96]. Concerning the heat flux, we will
investigate the connection between the corresponding entropy term and a possible
additional term in (1.1) as a result of time depending probablities of the quantuum
mechanical states.
Chapter 2
Fundamentals
In this chapter we will present basic concepts which are needed for the mesoscopic
description of quantum thermodynamics. Some terms like beobachtungsebene and
accompanying process are introduced, and the thermodynamical situation is ex-
plained that is adequate for the theory of discrete systems.
2.1 The Density Operator and its Dynamics
The vector space of linear operators on Hilbert space is called Liouville space L
[Fic83]. Let us define:
Definition 1
Lob := {X∈ L|X=X+}(2.1)
Lob
tr := {X∈ L|X=X+,|Tr X|<∞} (2.2)
Lob
%:= {X∈ L|X=X+, X ≥0,Tr X= 1}(2.3)
Lob is the set of quantum mechanical observables and Lob
%the set of quantum me-
chanical density operators. Lob and Lob
tr are vector spaces.
The derivative of the density matrix %associated to a quantum mechanical system
yields the von Neumann dynamics [Fic84]
˙%=−iL % +◦
%.(2.4)
Here, Lis the Liouville operator.
Definition 2 Liouville operator belonging to the system whith Hamiltonian H:
L:L → L
L X := 1
~[H, X].(2.5)
11
CHAPTER 2. FUNDAMENTALS 12
◦
%can be justified with time depending probabilities of the eigenstates: the time
derivative of
%=X
m|ϕmi%mhϕm|(2.6)
using Schr¨odinger’s equation yields
˙%=−i
~H% +i
~%H +X
m|ϕmi˙%mhϕm|
|{z }
:=◦
%
.(2.7)
The starting point within the framework of projection dynamics is the classical von
Neumann equation without additional term ◦
%. In chapter 8 we will dwell on ◦
%.
2.2 Beobachtungsebene
The mesoscopic description of a system is based on the choise of a limited set
of observables that are relevant to the considered problem. This set of relevant
observables is called beobachtungsebene [Schw65].
B:= {G1,...,Gn}with n∈N(2.8)
Gi∈ Lob for all i∈ {1,...,n}(2.9)
The observables in the beobachtungsebene do not necessarily commute, but they
must be linearily independent to simplify matters.
Let us introduce the following abbreviation:
G:= (G1...Gn)t.(2.10)
The expectation values of these observables are given by
gi:= Tr (Gi%) =: hGiifor all i∈ {1,...,n}(2.11)
or g:= Tr (G %) =: hGi,(2.12)
if %is the quantum mechanical density operator of the considered system.
In general, the observables Gin the beobachtungsebene depend on some work vari-
ables a1(t), . . . , am(t). This is for instance the case, if we vary the volume of the
considered discrete system with a piston during the experiment. Consequently, the
Hamiltonian becomes a function of the time dependent work variable V(t). The
abbreviation
a:= (a1...am)t(2.13)
will be used henceforth.
CHAPTER 2. FUNDAMENTALS 13
2.3 Accompanying Processes of Statistical Opera-
tors
Considering a macroscopic system it is not possible to determine its exact quantum
mechanical state %, because the only information being available are measured values
of a finite number of observables, carried out by their measurement devices. In
the mesoscopic level of discription we are even not interested in the exact quantum
mechanical state, but in the expectation values g= Tr (G %) of these few observables.
For a chosen set of observables G(a), there exist a lot of quantum mechanical states
that are macroscopically indistinguishable, because their expectation values gare
the same. Therefore, we are free to choose any state ˆ%∈ Lob
%instead of %, if
Tr (G %) = Tr (Gˆ%) (2.14)
is stastisfied for B={G}.
Definition 3 Let be %:R→ Lob
%a quantum mechanical process. A dynamics of a
density operator ˆ%:R→ Lob
%is called accompanying process of %with respect
to B={G}, if
Tr G(t)%(t)= Tr G(t) ˆ%(t)for t∈R.(2.15)
It is implied that each accompanying process is sufficiently often times continuously
differentiable.
2.4 von Neumann Entropy and Generalised Canon-
ical Operator
There exist a method to distinguish one special accompanying process from the
practically unfinite numbers of other accompanying processes. In information the-
ory, we use the Shannon measure to express how uncertain the outcome of an
random experiment is. Suppose there are npossible results F1, . . . , Fnin this ex-
periment, whose probabilities are p1,...,pn, then the Shannon entropy is given by
−kPn
j=1 pjln pj, where kis a constant. The quantum mechanical equivalent can
be defined and be used to determine the ˆ%which maximises this entropy. This ˆ%
represents the greatest uncertainty and is mostly unbiassed.
Definition 4 von Neumann Entropy
S:Lob
%→R+
0
S(X) := −kTr (Xln X) (2.16)
Here, kis the Boltzmann constant.
Definition 5 Let be %the quantum mechanical density operator of a system. Let be
B={G}a beobachtungsebene. That ˆ%∈ Lob
%satisfying (2.14) which maximises the
entropy of the considered system is called generalised canonical operator with
respect to Band will be denoted as RB:
SB:= max
ˆ%S(ˆ%) = −kmin
ˆ%(Tr (ˆ%ln ˆ%)) = −kTr (RBln RB).(2.17)
CHAPTER 2. FUNDAMENTALS 14
If it is clear which beobachtungsebene is meant, we will omit the index B.
Theorem 1 The generalised canonical operator RBwith respect to a beobachtungsebene
B={G}has the following form:
RB=1
Ze−λ·G(2.18)
with the partition function
Z:= Tr e−λ·G.(2.19)
The λare called Lagrangian multipliers.
Proof: See [Gra87] [Gra88]. 2
Consider a thermodynamical process. Let the maximisation take place at every
moment during the time intervall of this process, based on the expectation values
at each time point.
Definition 6 Let be %:R→ Lob
%a quantum mechanical process. That accompany-
ing process of %with respect to B={G}which maximises the entropy for all times
is called canonical.
This process preserves the canonical form of the generalised canonical operator for
all time, so we denote its dynamics by
RB:R→ Lob
%.(2.20)
We will omit the index B, if it is clear which beobachtungsebene to use.
Theorem 2 Let be %the quantum mechanical density operator of a system. Let be
Rthe according generalised canonical operator with respect to a beobachtungsebene
B={G1,...,Gn}={G}. If
s·G−gI6= 0 for all 06=s∈Rn,(2.21)
then the Lagrangian multipliers λare uniquely determined by the n+ 1 constraints
Tr (G R) = Tr (G %) (2.22)
Tr R= 1 .(2.23)
Proof: See [Kat67], [Kau96]. 2
We can rewrite the generalised canonical operator in the following manner:
R=e−λ0e−λ·G.(2.24)
The n+ 1 constraints (2.22) and (2.23) that determine the Lagrangian multipliers
λ0. . . λncan be also be written in terms of the partition function as
g=−∂ln Z
∂λ (2.25)
and λ0= ln Z . (2.26)
CHAPTER 2. FUNDAMENTALS 15
(2.25) is a consequence of Snider’s derivation rule [Sni64]
∂
∂z ef(z)=Z1
0
e(1−µ)f(z)∂f
∂z eµf(z)dµ . (2.27)
This is necessary, if f(z) and ∂f
∂z do not necessarily commute, like here ∂(λ·G)
∂λ =G
and eλ·G[Fic83].
2.5 The Thermodynamical Situation
Thermodynamics of discrete systems deals with the following standard situation:
consider an isolated compound system consisting of two interacting subsystems. Ac-
cording to the First Law of thermodynamics the change of energy in one subsystem
is caused by exchange of work, heat and material with its environment represented
by the second subsystem. Such systems are called Schottky systems according to
[Scho29].
system 1 system 2
-
-
-
˙
W
˙
Q
˙ne
isolated compound system
partition
We will neglect material exchange mostly in this work, except for the example in
chapter 7.
The Hamiltonian Hof the whole system is described by
H=H1+H12 +H2,(2.28)
where H12 is the Hamiltonian describig the interaction between the two subsystems,
denoted in the following as partition. Therefore it is valid
[H1, H2] = 0 ,[H1, H12]6= 0 ,[H2, H12]6= 0 .(2.29)
The compound system is isolated.
˙
H= 0 (2.30)
Because of
Tr (H1%)•= Tr ˙
H1%+ Tr (H1˙%) (2.31)
CHAPTER 2. FUNDAMENTALS 16
we can identify
˙
W1:= Tr ˙
H1%(2.32)
˙
Q1:= Tr (H1˙%) = iTr (% L H1),(2.33)
which are expressions for work and heat exchange between the considered system
no. 1 and its environment consisting of the partition and system no. 2. Thus we
can define:
Definition 7 In the situation described above, we call ˙
H1(a(t)) = ∂H1
∂a ·˙a(t)the
thermodynamical work operator and iLH1the heat flux operator of the first
system.
Using the same definition for all three systems, we have
˙
W:= Tr ˙
H %=˙
W1+˙
W2+˙
W12 (2.34)
˙
Q:= Tr (H˙%) = ˙
Q1+˙
Q2+˙
Q12 .(2.35)
We will use later following two definitions [Mus94]:
Definition 8
The compound system is isolated. :⇔˙
W=˙
Q= 0 (2.36)
Definition 9
The partition is inert. :⇔˙
W12 =˙
Q12 = 0 .(2.37)
Chapter 3
Dynamics of Statistical
Operators
In this chapter we will present a few dynamics with reduced information based on a
chosen beobachtungsebene, among them the canonical dynamics and the projected
dynamics. The dynamics of von Neumann entropy is strongly coupled to the utilised
dynamics. Some entropy rates are discussed as regards their definiteness.
3.1 Canonical Dynamics
The time derivative of the canonical accompanying process of %with respect to
B={G}(2.20) is called canonical dynamics [Mus94]. So the concept of beobach-
tungsebene is essential for this dynamics. In contrast to other dynamics presented
in this work, we do not derive this dynamics from quantum mechanical dynamics
(2.4), but the dynamics is constructed in matching with the measured values of the
relevant observables.
Let us first introduce a term that will allow us to write the canonical dynamics in
an elegant way.
Definition 10 Mori product
(·|·) : Lob ×Lob →R
(A, B)7→ (A|B) := Z1
0
Tr A+e−µλ·GB eµλ·GRdµ (3.1)
This is a generalised version of the Mori product given in [Mor65].
Theorem 3 The generalised Mori product is a real scalar product.
17
CHAPTER 3. DYNAMICS OF STATISTICAL OPERATORS 18
Proof: See [Schi94]. 2
With the abbreviation ∆G:= G−Tr (G %)Iwe get:
Theorem 4 Canonical dynamics is given by
˙
R=RTr R(λ·G)•−RZ1
0
eµλ·G(λ·G)•e−µλ·Gdµ (3.2)
=−RZ1
0
eµλ·G∆ (λ·G)•e−µλ·Gdµ (3.3)
Proof: See [Mus94] or use the derivation rule (2.27). 2
This results in
Theorem 5 Canonical dynamics of the expectation values
˙g=d
dt Tr GR=Tr ˙
G R+Tr G RTr R(λ·G)•
−Tr GRZ1
0
eµλ·G(λ·G)•e−µλ·Gdµ(3.4)
=Tr ˙
G R−G∆ (λ·G)•.(3.5)
From (2.18) and (2.19) we see that the generalised canonical operator depends on
the λand the G(a). Thus we can write the canonical dynamics as follows:
˙
R=∂R
∂λ ·˙
λ+∂R
∂a ·˙a , (3.6)
where the coefficients can be read off in (3.3):
∂R
∂λ =−RZ1
0
eµλ·G(∆G)e−µλ·Gdµ (3.7)
∂R
∂a =−R λ ·Z1
0
eµλ·G∆∂G
∂a e−µλ·Gdµ . (3.8)
(Note that the differentiation with respect to aand λare in fact functional deriva-
tives, to be noted correctly as δ
δa(t)or δ
δλ(t).)
The dynamics of the expectation values is then given by
˙g=∂g
∂λ ·˙
λ+∂g
∂a ·˙a(3.9)
with the coefficients
∂gi
∂λj
=−Gi∆Gj(3.10)
∂gi
∂aj
= I
∂Gi
∂aj!− Gi∆∂G
∂aj!·λ,(3.11)
CHAPTER 3. DYNAMICS OF STATISTICAL OPERATORS 19
as can be seen in (3.5) using the identity
(I|B) = Tr (B R) for B∈ Lob .(3.12)
This is not a dynamics with practical use anyhow, because we need the dynamics
of the Lagrangian multipliers to solve the differential equation (3.5) or (3.6). From
theorem 2 we know that the Lagrangian multipliers are unique, but there is no way
to get out λexplicitly of (2.22) and (2.23). This leads us to the next section where
we discuss another way in deriving a mesoscopic dynamics.
3.2 The Relevant Part of the Density Operator
In this section we will treat dynamics of the relevant part of the density operator.
Density operators of quantum mechanical systems contain much more information
than we need practically for a specific problem. This was why we introduced the con-
cept of reduced information level. The operators in the chosen beobachtungsebene
B={G}do not form a complete base normally, so the density matrix has for this
paticular beobachtungsebene a relevant part, which contributes to the calculation
of expectation values, and an irrelevant part, which does not show any effect on
the trace. Here, specific operators are introduced for the purpose of isolating the
relevant part from the irrelevant part of the density operator taking into account
the chosen beobachtungsebene.
%=%rel +%irrel with (3.13)
%rel ∈ Lob
%,Tr %irrel = 0 (3.14)
Tr G %= Tr G %rel (3.15)
0 = Tr G %irrel (3.16)
The isolation of these two parts is achieved by mappings on the Liouville space.
With this operator, the von Neumann equation (2.4) (whith ◦
%= 0)
˙%(t) = −iL %(t) (3.17)
is transformed into a mesoscopic dynamics of the relevant part of the statistical
operator. The choice of ◦
%= 0 means that we start out here from pure quantum
mechanics without a dissipative term. Increasing entropy in an isolated system is
not achieved by the additional dissipative term in this context. It is rather the
limited available information of the system that produces artificially an increase of
entropy for the observer of the system. If it were possible to determine the exact
quantum mechanical state at one time point, we know its time evolution given by
(3.17), and the entropy should remain constant. In chapter 8 we will consider a few
examples for dynamics with ◦
%6= 0.
In the literature, we can find different sorts of such mappings. Let be %:R→ Lob
%
a quantum dynamical process.
CHAPTER 3. DYNAMICS OF STATISTICAL OPERATORS 20
Nakajima Zwanzig type P:Lob
%→ Lob
%
%7→ P % =%rel and ˙
P= 0 (3.18)
˙%rel =P˙%(3.19)
Robertson type P:Lob
%→ Lob
%
%7→ P % =%rel and ˙
P % = 0 (3.20)
˙%rel =P˙%(3.21)
Fick Sauermann type P:Lob
%→ Lob
%
%7→ P% =%rel (3.22)
˙%rel =P˙%+˙
P % (3.23)
Here, Pis supposed to be linear in all three cases, and idempotent P2=P, because
it is desirable that
P %rel =%rel (3.24)
or P˙%rel = ˙%rel (3.25)
is valid. Since Pis linear and idempotent, it is often called projector. It is necessary
to distinguish between such projectors and orthogonal projectors, which are both
idempotent and selfadjoint with respect to a given scalar product, as we will see
later.
Let us define the operator:
Q:= I−P(3.26)
with the corresponding operator Pin each cases. Qis that mapping which maps %
to %irrel by definition. If Pis idempotent, Qis idempotent, too.
The idea is now to derive dynamics of accompanying processes %rel (definition 3)
using these mappings. In this case the mapping Pcan be also time-dependent.
Linear mappings on the Liouville space Lare called superoperators. An example
of a superoperator is the Liouville operator (definition 2). Here, superoperators
that enable the derivation of the dynamics of the relevant statistical operator are
interesting. Nonlinear mappings delivering the relvant part are treated in [Kat99].
Originally, the projection dynamics was developed by Nakajima and Zwanzig [Nak58],
[Zwa60] with time independent projection operators, to isolate for example the diag-
onal matrix elements of a Hamiltonian from its non diagonal elements. Robertson
[Rob66] introduced time dependent projectors, keeping the dynamics (3.19) un-
chainged, followed by Fick and Sauermann [Fic83], who considered a more general
dynamics (3.23) including the time derivative of the projector.
3.2.1 Robertson Dynamics
The dynamics that is based on equation (3.20) and (3.21) has been treated by
Robertson [Rob66]. Starting out with the von Neumann equation (3.17) and taking
into accout (3.20), it is possible to derive the so called Robertson Dynamics, an
integro-differential equation for %rel .
CHAPTER 3. DYNAMICS OF STATISTICAL OPERATORS 21
Theorem 6 Robertson dynamics
The dynamics of %rel using the linear mapping (3.20), (3.21) is given by
˙%rel (t) = −iP (t)L(t)%rel (t)−Zt
t0
P(t)L(t)T(t, s)Q(s)L(s)%rel (s)ds (3.27)
with ∂
∂s T(t, s) = iT (t, s)Q(s)L(s) (3.28)
T(t, t) = 1 (3.29)
%(t0) = %rel (t0).(3.30)
We will recapitulate the derivation of this dynamics exemplarily, because all the
other following relevant-part-dynamics are derived in the similar way.
Proof: Starting from (3.21) and using the von Neumann equation (3.17) we get
˙%rel (t) = P(t) ˙%(t)
=−iP(t)L(t)%(t)
=−iP(t)L(t)%rel (t)−iP(t)L(t)%(t)−%rel (t).(3.31)
In order to get a differential equation for %rel (t) we must somehow remove %(t) from
this equation. Therefore we calculate the following expression
˙%(t)−˙%rel (t) = −iL(t)%(t) + iP (t)L(t)%(t)
=−iQ(t)L(t)%(t)
=−iQ(t)L(t)%rel (t)−iQ(t)L(t)%(t)−%rel (t).(3.32)
Because of (3.20) we can identify
˙%(t)−˙%rel (t) = d
dt (%(t)−%rel (t)) .(3.33)
Let us define a function T(t, s) with (3.28) and (3.29). We multiply (3.32) by this
function and integrate it from t0to t.
Zt
t0
T(t, s)d
ds %(s)−%rel (s)ds
=−iZt
t0
T(t, s)Q(s)L(s)%rel(s)ds −Zt
t0
∂
∂s T(t, s)%(s)−%rel (s)ds
(3.34)
Bringing the last integral to the left hand side, we can integrate the left term by
using the initial condition (3.30).
%(t)−%rel (t) = −iZt
t0
T(t, s)Q(s)L(s)%rel (s)ds , (3.35)
which we insert in equation (3.31), so we finally obtain (3.27). 2
Robertson derived with this technique the dynamics of the generalised canonical
operator. That is, he assumed in particular that %rel (t) preserves the form of the
generalised canonical operator R(t) for all time:
˙
R(t) = P(t) ˙%(t),(3.36)
CHAPTER 3. DYNAMICS OF STATISTICAL OPERATORS 22
and so do most authors who use this dynamics. Indeed, this is a completely differ-
ent way of deriving the dynamics than in the last section 3.1. The dynamics there
is not based on the quantum mechanical dynamics. In fact, the phenomenological
quantities λ(t) and a(t) are essential for the canonical dynamics, while the dynam-
ics here is derived out of von Neumann dynamics using the projector P, with is
now of fundamental importance. The dynamics (3.27) of the generalised canonical
operator (3.36) is the basis for a series of stochastical dynamics that can be found
in literature: Langevin type equation [Fic83], generalised Fokker Planck equation
[Gra82], perturbation theory approach [Luz00] and so on.
3.2.2 Fick Sauermann Dynamics
The more general case in which a linear Pmaps %(t) to %rel (t) or specially to R(t),
(3.22), has been treated by Fick and Sauermann [Fic83].
Theorem 7 Fick Sauermann dynamics
The dynamics of %rel using the linear mapping (3.22) is given by
˙%rel (t) = −iP(t)L(t) + i˙
P(t)%rel (t)
−Zt
t0P(t)L(t) + i˙
P(t)T(t, s)Q(s)L(s)−i˙
P(s)%rel (s)ds
(3.37)
with ∂
∂s T(t, s) = iT (t, s)Q(s)L(s)−i˙
P(s)(3.38)
T(t, t) = 1 (3.39)
%(t0) = %rel (t0).(3.40)
Proof: Similar to the proof of theorem 6 2
As we have mentioned above, both Fick Sauermann and Robertson dynamics are
based on the dynamics derived by Nakajima [Nak58] and Zwanzig [Zwa60] [Zwa64].
They treated a beobachtungsebene containig only time independent observables
with constant work variables and a time independent operator P(3.18), (3.19).
The Nakajima Zwanzig dynamics is given by
˙%rel (t) = −iP L %rel (t)−Zt
t0
P L e−iQ L·(t−s)Q L %rel (s)ds , (3.41)
as we can conclude immediately from (3.27) or (3.37). This is an integro-differential
equation like all other dynamics in this section. The exponential function appeares
under the integral as a time-evolution operator, because Lis assumed to be ex-
plicitly time independent. There is a formal similarity between this dynamics and
the Robertson dynamics, because in both cases ˙%rel =P˙%is valid. Nakajima and
Zwanzig allowed for time independent projection operator, while Robertson consid-
ered time dependent projectors in principle, only on the condition ˙
P % = 0.
Equation (3.41) is also called “generalised master equation” and was derived in
connection with the perturbation theory. Pis time independent, because %rel is
given there for all times by the diagonal part of %, its eigenvalues, if the unper-
turbed Hamiltonian H0is chosen appropriately. The irrelevant part Q % is then the
nondiagonal part of %.
CHAPTER 3. DYNAMICS OF STATISTICAL OPERATORS 23
3.2.3 Projector P
A prominent example for Pin the Fick Sauermann dynamics is the Kawasaki Gun-
ton operator [Kaw73] which projects the density operator onto its relevant part in
the generalised canonical form with respect to a chosen beobachtungsebene. To do
so, we choose the Mori product introduced in definition 10 and give an orthogonal
projector that projects any observable onto the beobachtungsebene B∪{I}.
Of course it is allowed to use any other inner product in Lthan the generalised Mori
product, as described in [Kau96]. We recommend the following inner product with
is a kind of generalisation of definition 10. This definition includes the definition 10
as one special case.
Definition 11 generalised Mori product
Let be ˆ%∈ Lob
%an accompanying process of the quantum mechanical density operator
%with respect to B={G}.
(·|·) : L×L → R
(X, Y )7→ (X|Y) := Z1
0
Tr ˆ% X+ˆ%µYˆ%−µdµ (3.42)
If we choose ˆ%=R, then we have the “generalised Mori product” in [Kau96]. If
ˆ%=Z−1e−βH , we have the original Mori product [Mor65].
Theorem 8 The generalised Mori product is a scalar product.
Proof:
(i) (F+G|H) = (F|H) + (G|H) for all F, G, H ∈ L
(ii) (λF |G) = λ∗(F|G) for all F, G ∈ L, λ ∈C
(iii)
(F|G)∗=Z1
0
Tr ˆ% F +ˆ%µGˆ%−µ+dµ =Z1
0
Tr ˆ%−µG+ˆ%µFˆ%dµ
=Z1
0
Tr ˆ% G+ˆ%µFˆ%−µdµ = (G|F) for all F, G ∈ L
(iv) Use the spectral representation ˆ%=Pjpj|ψjihψj|. Since ˆ%∈ Lob
%,pj≥0 for
all jis valid.
(F|F) = Z1
0
Tr ˆ% F +ˆ%µFˆ%−µdµ
=Z1
0X
j,k,l,mhψj|ψkipkhψk|F+|ψlipµ
lhψl|F|ψmip−µ
mhψm|ψjidµ
=Z1
0X
j,l
pjhψj|F+|ψlipµ
lhψl|F|ψjip−µ
jdµ
=Z1
0X
j,l
pjpµ
lp−µ
j|hψl|F|ψji|2dµ ≥0
CHAPTER 3. DYNAMICS OF STATISTICAL OPERATORS 24
And
(F|F) = 0 ⇔ hψl|F|ψji= 0 for all l, j ⇔F= 0
2
Definition 12 Mori projector
Let be ˆ%∈ Lob
%an accompanying process of the quantum mechanical density operator
%with respect to B={G}. Let be (·|·)the generalised Mori product with respect
to ˆ%.
PM:Lob →span (B ∪{I})
X7→ PMX:= (X|I)I+ (X|∆G)·(∆G|∆G)−1·∆G(3.43)
where we use the scalar product of definition 11, and
∆G:= G−(G|I)I.(3.44)
Theorem 9 The Mori projector is an orthogonal projector with respect to the gen-
eralised Mori product. It projects observables on B ∪{I}.
Proof:
1. PMPMX=PMXfor all X∈ Lob obviously.
2. PM+=PMbecause of PMX|Y=X|PMYfor all X, Y ∈ Lob
evidently.
3. PMprojects on B∪{I}, see [Kau96] 2
Now we can introduce a uniquely defined mapping PKG acting on Lob
tr which is
connected to the Mori projector in the following way:
Tr PMXY= Tr XPKGY for all X∈ Lob, Y ∈ Lob
tr .(3.45)
Theorem 10 Equation (3.45) is equivalent to
PKG :Lob
tr → Lob
tr
X7→ PKGX
PKGX:= ˆ%TrX+Z1
0
ˆ%µ∆Gˆ%−µˆ% dµ·(∆G|∆G)−1·Tr (∆G X) (3.46)
Proof: Simple calculation 2
Theorem 11 If we choose the accompanying generalised canonical operator Rin
definition 12 and in theorem 10, ˆ%=R, then it is valid
PMX=Tr (R X)I+Tr X∂R
∂g ·G−gIfor all X∈ Lob (3.47)
PKG X=RTr X+∂R
∂g ·Tr (G X)−gTr Xfor all X∈ Lob
tr .(3.48)
The second mapping is called Kawasaki Gunton operator.
CHAPTER 3. DYNAMICS OF STATISTICAL OPERATORS 25
Proof: Use the following equations:
(X|∆G) = −Tr X∂R
∂λ (3.49)
(∆G|∆G) = −∂g
∂λ (3.50)
Z1
0
Rµ∆G R−µˆ% dµ =−∂R
∂λ .(3.51)
2
Theorem 12 The Kawasaki Gunton operator is a projector.
Proof: It can be easily seen that PKG is idempotent. 2
So we can use (3.46) for the Fick Sauermann dynamics (3.37), and especially the
Kawasaki Gunton operator (3.48), if we choose %rel =Rin the dynamics. In this
case, the dynamics is an implicit differential equation:
˙
PKG X= (P X)•−P˙
X
=˙
RTrX+∂R
∂g •
·Tr (GX)−gTrX+∂R
∂g ·Tr ( ˙
G X)−˙gTrX,
(3.52)
hence ˙
Rappears on the righthand side of the equation (3.37), too.
In case that ˙a= 0 is valid, ˙
Gis equal to zero, hence we have
˙
PKG %=˙
R−∂R
∂g ·˙g=˙
R−˙
R= 0 ,(3.53)
so that equation (3.20) is fulfilled, thus we can use this operator for the Robertson
dynamics (3.27), too.
3.2.4 Other Mappings P
Apart from the Kawasaki Gunton operator (3.48), there exist some other linear
mappings which induces the Robertson dynamics (3.27) or the Fick Sauermann
dynamics (3.37) but which are no orthogonal projectors with respect to a scalar
product any more.
Definition 13 generalised Kawasaki Gunton projector
Let be %rel an accompanying process of %with respect to a beobachtungsebene B=
{G}.
PgKG :Lob
tr → Lob
tr
X7→ PgKG X:= %rel Tr X+∂%rel
∂g ·Tr (G X)−gTr X.(3.54)
CHAPTER 3. DYNAMICS OF STATISTICAL OPERATORS 26
This mapping is idempotent as can be easily seen, and satisfies the properties (3.22)
and (3.24), thus it induces the Fick Sauermann dynamics (3.37). The generalised
Kawasaki Gunton operator is also a proper projector for the Robertson dynamics
(3.27), if no time dependent observables are considered. In this case, the equations
(3.20) are valid, as we discussed in the last section.
The Robertson dynamics (3.27) in section 3.2.1 is valid independently from the fact
whether the considered observables are time dependent or not. But it is difficult to
construct an appropriate projector satisfying the nessasary conditions (3.21), (3.20),
and taking account of time dependent observables at the same time. This is why it is
not possible to describe systems with time dependent work variables by Robertson
dynamics so far, although such systems are standard situations in thermodynamics.
Basically, time dependent observables are seldom treated in literature in the context
of reduced-information-dynamics, as can be seen in [Rau96].
3.2.5 Canonical Dynamics vs. Fick Sauermann Dynamics
using Kawasaki Gunton Operator
For both Fick Sauermann dynamics (3.37) using Kawasaki Gunton operator (3.48)
and canonical dynamics (3.2), the maximisation takes place at every moment. In
canonical dynamics one does this by fitting the langrange multipliers to the mea-
sured values. The Fick Sauermann dynamics is derived from a time dependent
projector. So both dynamics allow for same physical issue, namely the dynamics of
generalised canonical operator, but their derivations are completely different.
As mentioned above, canonical dynamics is not based on von Neumann dynamics.
It is a dynamics on its own, derived from the time derivative of the generalised
canonical operator. This leads to the problem, that we must know the dynamics
of Lagrange parameters to obtain the dynamics. Fick Sauermann dynamics is a
projection of the von Neumann dynamics. Such a projected dynamics depends
strongly on the chosen dissipative term ◦
%(which is chosen zero in this chapter).
The projector is known, because it is composed on the relevant observables and
their expectation values, which can be replaced practically by the measured values.
The only difficulty is that the dynamics becomes implicit, because the projector
itself contains the relevant part.
As a result, Rdepends on different variables in the two cases, which makes the
comparison of the two dynamics very difficult [Hag98]. In case of canonical dynam-
ics the generalised canonical operator Ris assumed to be dependent on the work
variables aand the Lagrangian multipliers λ, while Fick Sauermann dynamics is
based upon both, an explicit and a non-explicit time dependence, as Schirrmeister
showed [Schi94].
Ra(t), λ(t)canonical dynamics (3.55)
Rt, a(t)Fick Sauermann dynamics (3.56)
The different dependences can be explained as follows: if we look at the form of
the generalised canonical operator (2.18), it is evident that Rdepends on the λand
on the relevant observables G, which are again depending on the work variables a.
CHAPTER 3. DYNAMICS OF STATISTICAL OPERATORS 27
The macroscopic variables, on which Rin Fick Sauermann dynamics depends, are
aand g.
Ra(t), g(t)(3.57)
The gcan also be calculated with the quantum mechanical density operator %.
Therefore, we have in Fick Sauermann dynamics an implicit dependence on tby
a(t) and an explicit dependence on tby %(t), see (3.56). Though (3.55) and (3.56)
describe the same physical object, they are different functions from the mathe-
matical point of view, to be noted correctly by different symbols e
Ra(t), λ(t)and
b
Rt, a(t).
3.3 About the Rate of Entropy and Entropy Pro-
duction
The time rate of von Neumann entropy (2.16) is strictly connected to the dynamics
of the statistical operator. Since an explicit calculation is only possible, if we use
the generalised canonical form of %rel , we investigate here the rate of entropy and
entropy production for canonical dynamics and for Fick Sauermann dynamics using
Kawasaki Gunton projector.
3.3.1 Canonical Dynamics
The rate of the maximised von Neumann entropy (2.17) belonging to the generalised
canonical operator (2.18) results in [Mus94]:
˙
S=−kTr ( ˙
Rln R) = kTr (λ·G˙
R).(3.58)
Note that if we do not consider work, then the Gbecome time-independent, and
we get the commonly used formulation
˙
S=k λ ·hGi•.(3.59)
Inserting canonical dynamics (3.2) into (3.58), we obtain
˙
S=kTr R(λ·G)•Tr (R(λ·G)) −kTr R(λ·G)•(λ·G).(3.60)
Theorem 13 Let be Rthe generalised canonical operator with respect to a beobach-
tungsebene B. The following conditions are sufficient for the definiteness of the
appropriate rate of entropy, ˙
S≥0.
(i) (λ·G)•=−α λ ·(G−hGiI)with α≥0
(ii) ˙
λ·G=−α λ ·(G−hGiI)with α≥0, if ˙a= 0
(iii) ˙
λ=αTr (G R)2−Tr G2R·λwith α≥0, if ˙a= 0
CHAPTER 3. DYNAMICS OF STATISTICAL OPERATORS 28
Proof: Equation (3.60) can be written as
˙
S=k˙
λ·Tr (G R)2−Tr G2R·λ,(3.61)
if all work variables are constant. From this we get immediately (iii), using the
standard scalar product. Using the generalised Mori scalar product of definition 10,
(3.60) results in ˙
S=−k(λ·G)•∆ (λ·G).(3.62)
From this we get (i) and (ii). 2
The condition of constant work variables ˙a= 0 becomes plausible, when we bear
in mind that the second law deals with isolated systems. The formula (3.61) gives,
on condition that no heat and material transfer with its environment takes place,
the entropy production of the system. A more detailed description is possible, if
we distinguish external work variables from internal ones. In chapter 6 the rate
of entropy is analysed for examples with discrete systems. See also appendix 10.3
containing some remarks on maximised entropy.
3.3.2 Fick Sauermann Dynamics
Since Fick Sauermann dynamics (3.37) using Kawasaki Gunton operator (3.48) is
described by an implicit differential equation, there is no way to calculate the exact
rate of entropy (3.58). Nevertheless, we can insert the righthand side of the dynam-
ics (3.37) into (3.58) and make a few assumption and small time approximation.
Then the expression of ˙
Sdoes not contain ˙
Rany more, and we get the following
theorem [Kat00a], [Kat00b], [Schi94]:
Theorem 14 Let be Tr (R(iLG)) = 0. Using the Fick Sauermann dynamics in
small time approximation we get for the time rate of entropy
˙
S=kiL (λ·G) + e
Qλ·˙
GiL (λ·G) + e
Qλ·˙
G∆t≥0
with e
Q:= I−|∆G)·(∆G|∆G)·(∆G|.
Proof: See [Kat00a]. 2
The condition Tr (R(iLG)) = 0 means that the expectation values of generalised
fluxes are nearly vanishing, if we assume Tr (R(iLG)) ≈Tr (%(iLG)).
Chapter 4
Sufficiency
In thermodynamics observables depend on work variables, normally. To apply the
First Law in the quantum thermodynamical frame of this work, we must ask the
question under what condition the mesoscopic dynamics delivers the correct ex-
pectation values for work observables. We will analyse the dynamics of chapter 3
in this regard, and present a dynamics that describe the generalised works always
correctly.
4.1 Definitions and Illustration
4.1.1 Sufficiency for Observables
In this section it is explained what is meant by so called sufficiency. An accom-
panying process with respect to a beobachtungsebene Bis called sufficient for an
observable not included in B, if its expectation value is correctly given by this
process.
Definition 14 Let Kbe an observable K /∈ B. Let be ˆ%∈ Lob
%an accompanying
process of %with respect to B.ˆ%is called sufficient for K, if
Tr K(t)%(t)= Tr K(t) ˆ%(t)for t∈R.(4.1)
In general, this is not satisfied for an arbitrary observable. However, the accompa-
nying process is apparently sufficient for an observable included in the span of B.
This motivate us to the next definition.
Definition 15 Let be ˆ%∈ Lob
%an accompanying process of %with respect to B. The
set of all observables K∈ Lob for that ˆ%is sufficient will be denoted as B[ˆ%].
K∈B[ˆ%] :⇔Tr K(t)%(t)= Tr K(t) ˆ%(t)for t∈R.(4.2)
29
CHAPTER 4. SUFFICIENCY 30
For example:
Theorem 15 Let be ˆ%∈ Lob
%an accompanying process of %with respect to B. Then
it is valid:
(i) span B ⊆ B[ˆ%]
(ii) ˆ%is sufficient for I.I∈ B[ˆ%]
(iii) let be R∈ Lob
%the generalised canonical operator with respect to B. Then ˆ%is
sufficient for the entropy operator ln R.ln R∈ B[ˆ%]
Proof:
(i) trivial
(ii) trivial
(iii) According to (2.18) the entropy operator is given by
ln R=−λ·G−ln ZI.(4.3)
As a linear combination of Gand I, its expectation value is given correctly by
any accompanying process, and it follows:
Tr (ˆ%ln R) = Tr (%ln R) = Tr (Rln R) = −1
kSB.(4.4)
2
4.1.2 Sufficient Dynamics
In this section we apply the concept of sufficiency in connection with a chosen
dynamics of a statistical operator. This means that the dynamics is sufficient for
the time derivatives of Gfor all time. Its necessity particularly in thermodynamics
becomes clear, when we consider the following example.
Let be the Hamiltonian H(V(t)) a function of the system’s volume which is variable
in time. Let be H(V(t)) included in B. Demanding the sufficiency of the dynamics
of an accompanying process ˆ%means:
Tr ˙
H %!
= Tr ˙
Hˆ%⇔˙
VTr ∂H
∂V %!
=˙
VTr ∂H
∂V ˆ%.(4.5)
Classical thermodynamics tells us:
∂E
∂V S
=−p . (4.6)
CHAPTER 4. SUFFICIENCY 31
So in this situation we demand, that ˆ%describes the system’s pressure and work
correctly.
As we can see, we need such observables which depend on work variables in ther-
modynamics to calculate work. This is an essential quantity according to the First
Law. An accompanying process ˆ%to a chosen G(a) is only useful in thermodynam-
ics, if the expectation values of the time derivatives ˙
G, or of even higher derivatives,
are correctly given by ˆ%. If not, this description of the system does not fit the
real thermodynamical process, because work is not correctly given, and this kind of
description becomes worthless.
As a generalisation of definition 7 we define:
Definition 16 Let be G∈ Lob an observable. ˙
G(a(t)) is called generalised work
operator belonging to G.
Definition 17 Let be ˆ%:R→ Lob
%an accompanying process of %:R→ Lob
%with
respect to B={G}. The dynamics of ˆ%is called sufficient with respect to B, if
˙
G∈ B[ˆ%].
Definition 18 Let be n∈N. Let be ˆ%:R→ Lob
%an accompanying process of
%:R→ Lob
%with respect to B={G}. The dynamics of ˆ%is called sufficient in
n-th order with respect to B, if G(j)∈B[ˆ%]for all j∈ {1,...,n}.
Definition 19 Let be ˆ%:R→ Lob
%an accompanying process of %:R→ Lob
%with
respect to B={G}. The dynamics of ˆ%is called totally sufficient with respect
to B, if G(j)∈B[ˆ%]for all j∈N.
There is an other possibility to handle this problem. Instead of demanding the
sufficiency of Bfor all time derivatives of the G, we can simply include all these
derivatives into the beobachtungsebene. Then an accompanying process would sat-
isfy all constraints per definition. But it is desirable that a beobachtungsebene in-
cludes only observables that are really relevant, that means measurable during the
experiment. Otherwise, we have an infinite number of operators in the beobach-
tungsebene, and it does not make sense to call it “observation level” any more,
because they cannot be all observed or measured.
Theorem 16 Let ˆ%be an accompanying process of %with respect to B. Then these
propositions are equivalent:
˙
G∈ B[ˆ%]
⇔Tr (˙
G %) = Tr (˙
Gˆ%)
⇔Tr (G˙%) = Tr (G˙
ˆ%).(4.7)
Theorem 17 Let ˆ%be an accompanying process of %with respect to B. Let be
n∈N. If n > 1, let be ˆ%sufficient in (n−1)-th order with respect to B. Then these
propositions are equivalent:
G(n)∈ B[ˆ%]
⇔Tr G(n)%=Tr G(n)ˆ%
⇔Tr G(n−1)%(1)=Tr G(n−1) ˆ%(1)
.
.
.
⇔Tr G%(n)=Tr Gˆ%(n).
(4.8)
CHAPTER 4. SUFFICIENCY 32
We have here n+ 1 equivalent equations.
Proof: We can proof this by mathematical induction. If n= 1, we can derive
(2.15)
Tr (G %) = Tr (Gˆ%).
Since ˆ%is sufficient for ˙
G, we immediately get the last equation in (4.7). If (4.8) is
valid for one n∈N, we can derive these n+ 1 equation and get:
Tr G(n+1)%−ˆ%= Tr G(n)ˆ%(1) −%(1)
Tr G(n)%(1) −ˆ%(1)= Tr G(n−1)ˆ%(2) −%(2)
.
.
.
Tr G(1)%(n)−ˆ%(n)= Tr Gˆ%(n+1) −%(n+1).(4.9)
If we set
Tr G(n+1)%= Tr G(n+1) ˆ%,(4.10)
all terms in (4.9) are vanishing and we get (4.8) for n+ 1. 2
To complete our discussion, let us show that the time-derivative of an observable is
also an observable.
Theorem 18
A(t)∈ Lob ⇒d
dtA(t)∈ Lob (4.11)
Proof:
hAφ |ψi=hφ|A+ψi
⇒ h ˙
Aφ |ψi=hAφ |ψi•−hA˙
φ|ψi−hAφ |˙
ψi
=hφ|A+ψi•−h ˙
φ|A+ψi−hφ|A+˙
ψi
=hφ|A+•ψi(4.12)
Thus if Ais an observable, i. e. A=A+, it follows that ˙
A=˙
A+.2
4.2 Investigation of Sufficiency of Dynamics
Here we will test different dynamics of chapter 3 for above defined sufficiency.
4.2.1 Canonical Dynamics
Theorem 19 The accompanying process of maximal entropy Rof %with respect to
Bis not necessarily sufficient.
CHAPTER 4. SUFFICIENCY 33
We will consider a counter-example as a proof, which is at the same time an simple
example for the generalised canonical operator of definition 5.
Proof: Consider a spin 1
2particle in a homogenous magnetic field B. The potential
part of its Hamiltonian in the spin space is
H=−γ~
2
3
X
k=1
σkBk=−γ~
2BzBx−iBy
Bx+iBy−Bz(4.13)
where the σkare Pauli-matrices
σx=0 1
1 0 σy=0−i
i0σz=1 0
0−1,(4.14)
and γis the gyromagnetic ratio [Coh77]. Choosing B={H}as beobachtungsebene,
we get the generalised canonical operator
R=e−µ H
Tr e−µ H
=1
Tr e−µ H exp µγ ~
2BzBx−iBy
Bx+iBy−Bz (4.15)
=1
Z
1
|B|R11 R12
R21 R22 (4.16)
with
Z= 2 coshµγ ~
2|B|
R11 =|B|cosh µγ ~
2|B|+Bzsinh µγ ~
2|B|
R12 = (Bx−iBy) sinh µγ ~
2|B|
R21 = (Bx+iBy) sinh µγ ~
2|B|
R22 =|B|cosh µγ ~
2|B|−Bzsinh µγ ~
2|B|.
To get (4.16) from (4.15) we used the diagonalised Hamiltonian:
BzBx−iBy
Bx+iBy−Bz
=Bz+|B|Bz−|B|
Bx+iByBx+iBy|B|0
0−|B| 1
2|B|
|B|−Bz
2|B|(Bx+iBy)
−1
2|B|
|B|+Bz
2|B|(Bx+iBy)!.
Using (2.25) we calculate the Lagrangian multiplier µ.
∂
∂µ ln Z=γ~
2|B|tanh µγ ~
2|B|=−hHi
⇔µ=2
γ~|B|artanh −2hHi
γ~|B|(4.17)
with h•i := Tr (%•). Inserting this in (4.16) we get:
R=1
2|B| |B|−Bz2hHi
γ~|B|−(Bx−iBy)2hHi
γ~|B|
−(Bx+iBy)2hHi
γ~|B||B|+Bz2hHi
γ~|B|!.(4.18)
CHAPTER 4. SUFFICIENCY 34
Now we are able to calculate some expectation values explicitly, both with the
generalised canonical operator and the exact quantum mechanical density operator,
which is known:
{I, σx, σy, σz}is a basis in the space of 2×2-matrices. For an arbitrary 2×2-matrix
Mit holds [Coh77]:
M=a0I+
3
X
k=1
akσkwith a0=1
2Tr Mand ak=1
2Tr (Mσk).(4.19)
Applying this to the density matrix %=Min our example, we obtain:
%=1
2 I+
3
X
k=1hσkiσk!=1
21 + hσzi hσxi−ihσyi
hσxi+ihσyi1−hσzi.(4.20)
Using (4.13), (4.18) and (4.20), the expectation values result in:
Tr (% H) = −γ~
2hσi·B=hHi(4.21)
Tr (R H) = hHi(4.22)
If the magnetic field changes in time, from (4.13), (4.18) and (4.20) we get further:
Tr %˙
H=−γ~
2hσi· ˙
B=hHihσi· ˙
B
hσi·B=h˙
Hi(4.23)
Tr R˙
H=hHi
|B|2B·˙
B(4.24)
We show now that (4.23) and (4.24) are not identical. Since we need only one
counter-example, let be
˙
Bx=˙
By= 0 ,˙
Bz6= 0 , Bx= 0 , By6= 0 , Bz6= 0 .(4.25)
Then we get
Tr %˙
H= Tr R˙
H⇔By=Bz.(4.26)
This is not necessarily true. 2
In the proof it is shown that even the conventional thermodynamical work (2.32) is
not correctly given with canonical dynamics.
4.2.2 Projected Dynamics of %rel
Here we want to examine the sufficiency of projected dynamics presented in section
3.2. The basic equation of motion is the von Neumann equation with ◦
%= 0. We must
of course consider those which take time dependent work variables into account.
Otherwise, there is no reason of considering sufficient dynamics (see definition 17).
CHAPTER 4. SUFFICIENCY 35
Robertson Dynamics
We noted that there is no known mapping Pfor the Robertson dynamics (theorem
6) taking non-constant work variables in account. Nevertheless we can formulate
an interesting theorem:
Theorem 20 Let be P:Lob
%→ Lob
%linear and idempotent with (3.21) and (3.20),
taking into account non-constant work variables. Let be valid Tr (G P L X) =
Tr (G L X)for all X∈ Lob, then the Robertson dynamics using this mapping is
sufficient.
Proof: With the given conditions it is valid:
Tr (G P L %) = Tr (G L %)
⇔Tr (G P ˙%) = Tr (G˙%)
⇔Tr (G˙%rel ) = Tr (G˙%) (4.27)
2
For example the Kawasaki Gunton projector (3.48) and its generalised version (3.54)
have the property Tr (GP X) = Tr (G X) for all X∈ Lob. But they do not take
time dependent observables into account. This theorem cannot be applied to Fick
Sauermann Dynamics, because the last step of the proof is not possible in that case.
Fick Sauermann Dynamics
Theorem 21 Let %rel be an accompanying process of %with respect to a beobach-
tungsebene B. The Fick Sauermann dynamics of %rel is sufficient, iff
Tr (G(t) ˙%rel (t)) = −iTr (G(t)L(t)%rel (t))
−Zt
t0
Tr G(t)L(t)T(t, s)Q(s)L(s)−i˙
P(s)%rel (s)ds
(4.28)
is satisfied.
This equation can be implicit, if Pcontains %rel .
Proof: Inserting Fick Sauermann dynamics (3.37). into (4.7), we obtain
−iTr(G L %) (t) = −iTr G P L %rel (t) + Tr G˙
P %rel (t)
−Zt
t0
Tr GP L +i˙
P(t)T(t, s)Q L −i˙
P%rel (s)ds .
We eliminate the quantum mechanical density matrix %using
%(t)−%rel (t) = −iZt
t0
T(t, s)Q(s)L(s)−i˙
P(s)%rel (s)ds , (4.29)
CHAPTER 4. SUFFICIENCY 36
which is used in the derivation of the dynamics [Schi94], analogous to (3.35) in the
derivation of the Robertson dynamics.
−iTr (G L %rel ) (t)−Zt
t0
Tr GL(t)T(t, s)Q L −i˙
P%rel (s)ds
=−iTr (G P L %rel ) (t) + Tr G˙
P %rel (t)
−Zt
t0
Tr GP L +i˙
P(t)T(t, s)Q L −i˙
P%rel (s)ds (4.30)
On the other side, we have the dynamics of expectation values using Fick Sauermann
dynamics:
Tr (G˙%rel ) = −iTr (G P L %rel ) (t) + Tr G˙
P %rel (t)
−Zt
t0
Tr GP L +i˙
P(t)T(t, s)Q L −i˙
P%rel (s)ds .
(4.31)
Comparing (4.30) with (4.31), we obtain (4.28). 2
If Pis specially chosen as the generalised Kawasaki Gunton operator (3.54), we get
the following
Theorem 22 Let %rel be an accompanying process of %with respect to a beobach-
tungsebene B. The Fick Sauermann dynamics of %rel generated by the generalised
Kawasaki Gunton operator Pis sufficient, iff
Tr G(t)˙
P(t)T(t, s)Q(s)L(s)−i˙
P(s)%rel (s)= 0 for all s∈[t0, t].
(4.32)
One necessary condition for the sufficiency is
Tr ˙
G Q L %rel (t) = iTr ˙
G˙
P %rel (t),(4.33)
which is equivalent to
Tr h˙
G(−iL%rel −˙%rel )i(t) = Tr ˙
G∂%rel
∂g Tr [G(−iL%rel −˙%rel )] (t).(4.34)
Proof: The generalised Kawasaki Gunton projector has following properties:
Tr (G P X) = Tr (G X) for all X∈ Lob (4.35)
Tr (G˙
P %rel ) = 0 (4.36)
Using these, we obtain from (4.31)
Tr (G˙%rel )
=−iTr (G L %rel ) (t)−Zt
t0
Tr GL(t)T(t, s)Q L −i˙
P%rel (s)ds
−iZt
t0
Tr G˙
P(t)T(t, s)Q L −i˙
P%rel (s)ds (4.37)
We get (4.32) by comparison with (4.28). The time-evolution operator vanishes in
(4.32) for s=t. We can simplify the equation to (4.33) and (4.34) using (3.48) and
(3.52). 2
CHAPTER 4. SUFFICIENCY 37
4.3 A Sufficient Dynamics
In the previous section we have seen that conventional dynamics with reduced infor-
mation are not sufficient in general. Here we will present an ansatz to get sufficient
accompanying dynamics. The idea of [Mus94] is formulated here again more pre-
cisely including a generalisation and a possible interpretation.
Theorem 23 Let ˆ%be an accompanying process of %with respect to B. Let be
(ˆ%−%)•(t) = −˙
h(t) (ˆ%−%)(t) + Y(t) (4.38)
Tr G(t)Y(t)= 0 for t∈R(4.39)
Tr Y(t) = 0 for t∈R(4.40)
with a function of time h(t)and a time dependent operator Y(t). Then the dynamics
of ˆ%is sufficient.
Proof: Using (4.38) we get
Tr G(ˆ%−%)•=−˙
hTr G(ˆ%−%)+ Tr (G Y ) = 0 (4.41)
because of (4.39) and the fact, that ˆ%is an accompanying process. Due to (4.7),
(4.41) is equivalent to
Tr ˙
G(ˆ%−%)= 0 .(4.42)
We need (4.40), because the trace of (4.38) should vanish on each side of the equality.
2
The condition (4.39) may be interpreted like this: since Tr (A B∗) is an inner product
in L,Yis an operator being orthogonal to the chosen beobachtungsebene with
respect to this inner product. One possible operator is Y=%−ˆ%(see the discussion
below).
Theorem 24 Let ˆ%be an accompanying process of %with respect to B. Let be n∈N
and
(ˆ%−%)•(t) = −˙
h(t) (ˆ%−%)(t) + Y(t) (4.43)
Tr G(j)(t)Y(t)= 0 for all j∈ {0,...,n−1}and t∈R(4.44)
Tr Y(t) = 0 for t∈R.(4.45)
Here h(t)is a function of time and Y(t)a time dependent operator. Then the
dynamics of ˆ%is sufficient in n-th order.
Proof: By mathematical induction. The proposition has already been prooved for
n= 1. Let be the proposition valid for one n∈N. Then
Tr G(n)(ˆ%−%)= 0 (4.46)
is valid. The derivation yields
Tr G(n+1) (ˆ%−%)=−Tr G(n)(ˆ%−%)•
=˙
hTr G(n)(ˆ%−%)−Tr (G(n)Y)
= 0 ,(4.47)
CHAPTER 4. SUFFICIENCY 38
if we additionally assume Tr (G(n)Y) = 0. Thus ˆ%(t) is sufficient in (n+1)-th order.
2
The solution of (4.38) and (4.43) is
ˆ%(t)−%(t) = e−h(t)+h(t0)ˆ%(t0)−%(t0) + Zt
t0
eh(s)−h(t0)Y(s)ds(4.48)
=e−h(t)Zt
t0
eh(s)Y(s)ds , (4.49)
if we set
ˆ%(t0) = %(t0).(4.50)
An other possibility to get rid of the first addend in (4.48) is to choose an increasing
positive definite function h.
In the special case of Y= 0 we have ˆ%=%for all time, or at least the difference
between them decreases exponentially in time by choice of a suitable h.
Let us investigate whether this dynamics is given by a projection. In this case
the condition (4.39) suggests itself that Yis given by the irrelevant part of %or
its multiple. But from this, it would follow that we can sum up both terms on
the right hand side of (4.38). Then the irrelevant part %−ˆ%would again decrease
exponentially in time by choice of a suitable h. ˆ%is then nearly equal to %, so it
makes no sence of speaking about projection dynamics.
We can apply the same considerations to the dynamics of (4.43). If Yis the irrele-
vant part of %, the the conditions (4.44) demand that the dynamics is sufficient in
(n−1)-th order:
Tr G(j)Q %= 0 ⇒Tr G(j)P %= Tr G(j)%.(4.51)
So it make no difference if we include the derivatives ˙
G,...G(n−1) into the beobach-
tungsebene, but then we can use again theorem 23.
Chapter 5
Further Considerations on
Sufficiency
5.1 Generalised Fluxes
In chapter 4 we discussed about how work can be calculated in thermodynamics. In
thermodynamical applications we need dynamics, being sufficient for the ˙
G, in order
to get correct expectation values for work. We will see here that a similar problem
already arises even if we consider systems without work exchange, i.e. systems
with time-independent observables. It concerns the generalised flux observables, cf.
definition 7.
Definition 20 Let be G∈ Lob an observable. iLG is called the generalised flux
operator with respect to G.
5.1.1 Canonical Dynamics
Theorem 25 In general, the generalised canonical operator RBdoes not deliver
the correct expectation value for a generalised flux operator iLG, if iLG /∈ B and
G∈ B.
Proof: Let be ˙
G= 0, B={G}. Then
iTr (R L G) = 0 (5.1)
because eλG and Gcommute, and
06= Tr ˙
R G= Tr (R G)•= Tr (% G)•=iTr (% L G) (5.2)
in general. 2
If we choose the Hamiltonian of a subsystem H1for G, this means, that in gen-
eral, neither work (2.32) nor heat flux (2.33) are given correctly by the generalised
canonical operator!
39
CHAPTER 5. FURTHER CONSIDERATIONS ON SUFFICIENCY 40
5.1.2 Robertson Dynamics
Let be H1∈ B. Using the Robertson dynamics (3.27) to obtain the heat flux, we
get:
hH1i•(t) = d
dt Tr (%rel (t)H1)
= Tr (%rel (t)iLH1)−Zt
t0
Tr (H1L T(t, s)Q(s)L %rel(s)) ds . (5.3)
Here, we used the common property (4.35) of the generalised Robertson opera-
tor and of the generalised Kawasaki Gunton operator, which are possible for this
dynamics. In general,
Tr (%(t)iLH1)6= Tr (%rel (t)iLH1) (5.4)
is valid. At this point, we face the same dilemma as in the previous chapter. After
all, heat exchange is a quantity that is as important as work in thermodynamics
according to the First Law. Should we include iLH1into the beobachtungsebene, or
should we demand that the dynamics gives the correct flux? As we can see above,
the second way would make vanish process history in Robertson dynamics.
In general, we have the following situation: B={G}is our beobachtungsebene. The
Robertson dynamics (3.27) with time independent observables delivers the following
dynamics of expectation values:
hGi•(t) = d
dtTr (G%rel (t))
= Tr (%rel (t)iLG)−Zt
t0
Tr (GL T (t, s)Q(s)L %rel(s)) ds . (5.5)
On the other hand,
hGi•(t) = d
dtTr (G %(t)) = Tr (%(t)iLG).(5.6)
Demanding the sufficiency of the dynamics for iLG, or even of all (iL)nG, we would
get a dynamics whose process history is not macroscopically noticeable, which is
consequently von Neumann-like. On the other hand, each Robertson dynamics of
%rel with vanishing process history is sufficient for the generalised fluxes.
5.1.3 Fick Sauermann Dynamics
We observe the same problem in Fick Sauermann dynamics (3.37) with time de-
pendent observables.
hGi•=d
dtTr (G%) = Tr %˙
G+iLG (5.7)
◦
G:= ˙
G+iLG can be interpreted as the observale of work and exchange quantities.
The Fick Sauermann dynamics yields
hGi•(t) = d
dtTr (G%rel ) (t)
CHAPTER 5. FURTHER CONSIDERATIONS ON SUFFICIENCY 41
= Tr %rel ˙
G+iLG(t)
−Zt
t0
Tr GPL +i˙
P(t)T(t, s)QL −i˙
P%rel (s)ds
(5.8)
by using (4.36). Demanding the sufficiency of the dynamics for ˙
G+iLG (or ˙
G, if
we suppress exchange quantities, or iLG, if we suppress work) is equivalent to the
process history becoming macroscopically irrelevant.
5.2 Possible Conditions for an Extended Sufficiency
In this section we investigate the question, under what condition accompanying
processes are sufficient for flux or work observables. For this purpose it becomes
necessary to make assumptions on ◦
%(cf. (2.4)), that has been set zero hitherto
except for the cases of canonical dynamis (3.2) and sufficient dynamics (4.38), (4.43).
Also the quantity ◦
ˆ%:= ˙
ˆ%+iLˆ%of the accompanying process becomes interesting.
Theorem 26 Let be B={G}a beobachtungsebene. Let be ˆ%(t)an accompanying
process of %(t)with respect to B. Let be
◦
ˆ%=◦
%.(5.9)
Then ˆ%is sufficient for ◦
G:= ˙
G+iLG :
Tr %˙
G+iLG=Tr ˆ%˙
G+iLG.(5.10)
Proof:
Tr (G %)•= Tr (Gˆ%)•
⇔Tr ˙
G%+ Tr G˙%−◦
%= Tr ˙
Gˆ%+ Tr G˙
ˆ%−◦
ˆ%
⇔Tr ˙
G%−iTr (GL%) = Tr ˙
Gˆ%−iTr (GLˆ%)
⇔Tr %˙
G+iLG= Tr ˆ%˙
G+iLG (5.11)
2
This proposition gains in importance, if we apply it on subsystems, see section 8.2.2.
Theorem 27 Let be B={G}a beobachtungsebene. Let be ˆ%(t)an accompanying
process of %(t)which is sufficient with respect to B. Let be
◦
ˆ%=◦
%.(5.12)
Then ˆ%is sufficient for iLG :
iTr (% L G) = iTr (ˆ% L G).(5.13)
Proof: Use theorem 26. 2
CHAPTER 5. FURTHER CONSIDERATIONS ON SUFFICIENCY 42
5.3 Weak Sufficiency
If an accompanying process is sufficient for K(definition 14), the expectation value
of an observable K, that is not included in the chosen beobachtungsebene, can
be nonetheless correctly calculated by this process. Here we will present a weaker
formulation of sufficiency.
Let be B={G}a beobachtungsebene, ˆ%(t) an accompanying process.
Tr (G(t)%(t)) = Tr (G(t)ˆ%(t)) ∀t . (5.14)
Since this equation is exact for all time, the time derivatives of both sides are equal.
d
dtTr (Gˆ%) = hGi•=d
dtTr (G%) = Tr ˙
G%−iTr (GL%)
= Tr %˙
G+iLG (5.15)
The expectation values of ˙
G+iLG can be calculated correctly using the dynamics of
the accompanying process, even if they are not included in the beobachtungsebene
theirselves. If the observables are time-independent, we have the expectation value
of iLG . Nonetheless, Tr %˙
G+iLG6= Tr ˆ%˙
G+iLGin general. This is a
weaker form of sufficiency than (4.1).
Definition 21 Let Kbe an observable K /∈ B. Let be ˆ%∈ Lob
%an accompanying
process of %with respect to B.ˆ%is called weakly sufficient for K, if it is possible
to give the dynamics of the expectation value of K, provided that the dynamics of ˆ%
is known.
Definition 22 Let be ˆ%∈ Lob
%an accompanying process of %with respect to B. The
set of all observables K∈ Lob for that ˆ%is weakly sufficient will be denoted as Bw[ˆ%].
K∈ Bw[ˆ%] :⇔ˆ%(t)is weakly sufficient for t∈R.(5.16)
Obviously, it is valid that ˆ%is weakly sufficient for K,Bw[ˆ%]⊆ B[ˆ%] , if ˆ%is sufficient
for Kin terms of definition 14.
Theorem 28 Let be B={G}a beobachtungsebene, ˆ%(t)an accompanying process
with respect to B. Let the observables Gand the system Hamiltonian Hbe time-
independent. Then inLnG∈ Bw[ˆ%]for all n∈N.
Proof:dn
dtnTr (Gˆ%) = hGi(n)=dn
dtnTr (G%) = inTr (%LnG) (5.17)
The last step by complete induction. 2
Theorem 29 Let be B={G}a beobachtungsebene, ˆ%(t)an accompanying pro-
cess with respect to B. The observables Gmay be time-dependent, but let be the
full Hamiltonian of the system time-independent. Then Pn
k=0 n
kikLkG(n−k)∈
Bw[ˆ%]for all n∈N.
CHAPTER 5. FURTHER CONSIDERATIONS ON SUFFICIENCY 43
Proof: By complete induction: o.k. for n= 1 see (5.15). Let be the proposition
correct for one n∈N. Then
d
dtTr %
n
X
k=0 n
kikLkG(n−k)!
=
n
X
k=0 n
kikTr %LkG(n−k+1)+
n
X
k=0 n
kik+1Tr %Lk+1G(n−k)
= Tr %
n+1
X
k=0 n+ 1
kikLkG(n+1−k)!(5.18)
because of
n
X
k=0 n
kikLkG(n−k+1) +
n
X
k=0 n
kik+1Lk+1G(n−k)
=n
0G(n+1) +
n
X
k=1 n
kikLkG(n−k+1)
+
n−1
X
k=0
k+ 1
n−kn
k+ 1ik+1Lk+1G(n−k)+n
nin+1Ln+1G
=n
0G(n+1) +
n
X
k=1 n
kikLkG(n−k+1)
+
n
X
k=1
k
n−k+ 1n
kikLkG(n−k+1) +n
nin+1Ln+1G
=n
0G(n+1) +
n
X
k=1
n+ 1
n−k+ 1n
kikLkG(n−k+1) +n
nin+1Ln+1G
=n+ 1
0G(n+1) +
n
X
k=1 n+ 1
kikLkG(n−k+1) +n+ 1
n+ 1in+1Ln+1G
=
n+1
X
k=0 n+ 1
kikLkG(n+1−k)(5.19)
2
We conclude:
Theorem 30 Let be B={G}a beobachtungsebene, ˆ%(t)an accompanying process
which is sufficient with respect to B. Then ˆ%(t)is weakly sufficient for iLG.
5.4 Time Independent Observables
Let us continue the discussion on the special case of time independent observables.
Including the terms iLG into the beobachtungsebene, we force a vanishing process
history of Robertson dynamics, as discussed in section 5.1. A similar effect is known
in thermodynamics, where there are different types of state spaces depending on
the description of process history [Mus90], [Mus96]. So we name {G, iLG}a large
(or extended) beobachtungsebene and {G}a small beobachtungsebene [Els93].
CHAPTER 5. FURTHER CONSIDERATIONS ON SUFFICIENCY 44
Even though we use the large beobachtungsebene, the process history would not
vanish for the expectation value of iLG, when we use Robertson dynamics:
hiLGi•(t) = −Tr (%rel (t)LLG)−iZt
t0
Tr ((LG)LT (t, s)Q(s)L%rel(s)) ds
=−Tr (%(t)LLG) (5.20)
In general, any dynamics of an accompanying process %rel with respect to a large
beobachtungsebene {G, iLG}has the following properties:
hGi= Tr (%G) = Tr (%rel G) (5.21)
hGi•= Tr (G˙%) = iTr (%LG) = iTr (%rel LG) = hiLGi(5.22)
hGi•• = Tr (G¨%) = −Tr (%LLG)6=−Tr (%rel LLG).(5.23)
In this case, in addition to the expectation values, their first derivatives will be cor-
rectly given by the accompanying process, but not their higher derivatives. To solve
this problem, we can either add LnGinto the beobachtungsebene, or demand the
sufficiency to the dynamics of %rel . In case of the Robertson dynamics, such a totally
sufficient dynamics would indeed make vanish process histories in all derivatives.
Using the Nakajima Zwanzig dynamics (3.41) with the Kawasaki Gunton opera-
tor, we get the same results as above with Robertson dynamics. With the large
beobachtungsebene the dynamics of the expectation values hGidoes not need the
process history, but the second derivative does.
Considering canonical dynamics (3.2), we get the following generalised canonical
operator and its evolution equation for the large beobachtungsebene {G, iLG}:
R=1
Ze−λ·G−µ·iLG (5.24)
˙
R=RTr (RG)−Z1
0
eνλ·GG e−νλ·Gdν·˙
λ
+RTr (RiLG)−Z1
0
eνµ·iLG iLG e−νµ·iLG dν·˙µ(5.25)
Since the operators Gdo not commute in general, we cannot simplify this equa-
tion further. As a simple example let us consider the beobachtungsebene B=
{H12, iLH12}. Here, the Liouville operator is deduced from the full Hamiltonian H
of the system. As explained in section 2.5, H12 is the exchange Hamiltonian, called
Hamiltonian of the partition. The operator iLH12 delivers the value of the parti-
tion’s heat exchange with the two subsystems. Thus iLH12 can be interpreted as the
observable of the heat flux. This is why this large beobachtungsebene is called here
extended beobachtungsebene according to the Extended Thermodynamics [Jou96].
From the generalised canonical operator
R=1
Ze−λ12H12−λQiLH12 (5.26)
follows the canonical dynamics
˙
R=RhH12i−Z1
0
eα(λ12H12+λQiLH12)H12 e−α(λ12 H12 +λQiLH12 )dα˙
λ12
+RhiLH12i−Z1
0
eα(λ12H12 +λQiLH12 )iLH12 e−α(λ12 H12 +λQiLH12)dα˙
λQ
(5.27)
CHAPTER 5. FURTHER CONSIDERATIONS ON SUFFICIENCY 45
and the dynamics of expectation values
hH12i•=hH12i2−Tr Z1
0
eα(λ12H12+λQiLH12)H12
e−α(λ12H12+λQiLH12)dαH12R˙
λ12
+hH12ihiLH12i−Tr Z1
0
eα(λ12H12+λQiLH12)iLH12
e−α(λ12H12+λQiLH12)dαH12R˙
λQ
=−H12 ∆H12˙
λ12 −H12 ∆iLH12˙
λQ
=hiLH12i,(5.28)
hiLH12i•=hiLH12ihH12i−Tr Z1
0
eα(λ12H12 +λQiLH12 )H12
e−α(λ12H12+λQiLH12)dα (iLH12)R˙
λ12
+hiLH12i2−Tr Z1
0
eα(λ12H12+νiLH12 )(iLH12)
e−α(λ12H12+λQiLH12)dα (iLH12)R˙
λQ
=−iLH12 ∆H12˙
λ12 −iLH12 ∆iLH12˙
λQ.(5.29)
If the partition is an inert one, then of course the internal energy hH12iis constant
in time.
Chapter 6
Examples for Discrete
Systems in Thermal Contact
In this chapter we will assume the situation described in section 2.5, that is the
isolated compound system (2.36) of two subsystems, the interaction between them
represented by a partition. Using canonical dynamics of section 3.1, we discuss
some example situations concerning the purely thermal contact without material
and work exchange. Therefore all observables are chosen to be time independent,
and the rate of entropy is given by (3.59).
Another application of canonical dynamics on one dimensional ideal gas and har-
monic lattices is given in [Kat01].
6.1 Considering the Partition
Let us begin with the last example in section 5.4. Let be the beobachtungsebene
given by B={H12, iLH12}. Since there is no time dependence of observables, H12
contains the same information as the sum H1+H2, if the constant total Hamiltonian
is known. The partition is not necessarily assumed to be inert. In this case the
canonical density operator is given by
R(t) = 1
Z(t)e−λ12(t)H12 −λQ(t)iLH12 ,(6.1)
and (3.59) yields the rate of entropy
˙
S(t) = k λ12(t)˙
Q12(t) + λQ(t)d˙
Q12
dt (t)!.(6.2)
Theorem 31 Let be
˙
λ12 =−αλ12 , α > 0 (6.3)
˙
λQ=−βλQ, β > 0 (6.4)
46
CHAPTER 6. EXAMPLES FOR DISCRETE SYSTEMS ... 47
∆H12 6= 0 ,∆iLH12 6= 0 (6.5)
∆iLH12 6k∆H12 (6.6)
the last assumption with respect to the Mori product. Then the rate of entropy (6.2)
is positive definite.
Proof: Inserting the assumptions (6.3) and (6.4) into (5.28) and (5.29), we get
˙
Q12 =αH12 ∆H12λ12 +βH12 ∆iLH12λQ(6.7)
d˙
Q12
dt =αiLH12 ∆H12λ12 +βiLH12 ∆iLH12λQ.(6.8)
The Mori scalar product has the property
A∆B=∆A∆Bfor all A, B ∈ Lob.(6.9)
This, allowing for the conditions in (6.5), (6.3) and (6.4), results in
αH12 ∆H12>0, β iLH12 ∆iLH12>0,(6.10)
and the Schwarz inequality
AB≤ kAk·kBk(6.11)
with the conditions (6.5), (6.6) yields
det αH12 ∆H12βH12 ∆iLH12
αiLH12 ∆H12βiLH12 ∆iLH12>0.(6.12)
From (6.2), (6.7), (6.8) we get
˙
S=kλ12
λQTαH12 ∆H12βH12 ∆iLH12
αiLH12 ∆H12βiLH12 ∆iLH12λ12
λQ.(6.13)
Since the determinants (6.10) and (6.12) are all positive, the quadratic form (6.13)
is positive definite. 2
Actually, it is paradox to make an ansatz for the Lagrange parameters, like the
relaxation ansatz (6.3) and (6.4), because theorem 2 states, that the parameters are
uniquely determined by the n+1 constrains (2.22) and (2.23). However, the theorem
is based upon the uniqueness theorem for solutions of initial value problems, and it
is utterly unknown, how to determine the parameters expicitly. It is not possible
to resolve (2.18) with respect to the parameters. The assumptions (6.3) and (6.4)
are so particular, that unfortunately, it is not possible to show that the derivative
of (6.7) yields (6.8). In [Mus94], the question is investigated how one can derive a
dynamics of the Lagrange parameters out of the canonical dynamics (3.2).
6.2 Subsystems and their Contact Temperature
Let us now consider the beobachtungsebene B={H1, H2}. Then the generalised
canonical operator according to the maximum entropy procedure is given by
R(t) = 1
Z(t)e−λ1(t)H1−λ2(t)H2.(6.14)
CHAPTER 6. EXAMPLES FOR DISCRETE SYSTEMS ... 48
From (3.59) we get the rate of entropy
˙
S(t) = kλ1(t)˙
Q1(t) + λ2(t)˙
Q2(t).(6.15)
If the partition is inert (2.37), we get
˙
S(t) = k(λ1(t)−λ2(t)) ˙
Q1(t).(6.16)
Theorem 32 Let be B={H1, H2}, the compound system isolated, the partition
inert and the system no. 2 in equilibrium. With the identification
λ1(t) + X(t) = 1
kθ1(t), λ2(t) + X(t) = 1
kT2(t)(6.17)
where θ1is the contact temperature of system no. 1,
˙
S≥0 (6.18)
is valid.
Proof: (6.16) results in
˙
S=1
θ1(t)−1
T2(t)˙
Q1(t)≥0.(6.19)
The inequality is induced by the definition of contact temperature [Mus94]. 2
If both systems are not in equilibrium the entropy production caused by heat ex-
change is positive, if the heat exchange ˙
Q1has the form
˙
Q1=k11 (λ1(t)−λ2(t)) , k11 ≥0 (6.20)
as follows from (6.16). For example:
Theorem 33 If H1∆H1=−H1∆H2(6.21)
∆H16= 0 (6.22)
(λ1−λ2)•=−α(λ1−λ2), α > 0 (6.23)
are valid, then the rate of entropy (6.16) for B={H1, H2}with an inert partition
is positive definite.
Proof: According to (3.5), we have
˙
Q1=−H1∆H1˙
λ1−H1∆H2˙
λ2(6.24)
(6.21)
=−H1∆H1(λ1−λ2)•(6.25)
(6.23)
=αH1∆H1(λ1−λ2).(6.26)
Inserting this into (6.16), it is clear that ˙
Sis positive definite because of (6.9). 2
The assumption (6.23) is plausible as a sort of Newton’s law of cooling. (6.21) may
be valid when we bear in mind that the partition is assumed to be inert.
CHAPTER 6. EXAMPLES FOR DISCRETE SYSTEMS ... 49
If the partition is not specified, it is reasonable to extend the beobachtungsebene
for B={H1, H2, H12}with the generalised canonical operator
R(t) = 1
Z(t)e−λ1(t)H1−λ2(t)H2−λ12(t)H12 .(6.27)
Since the compound system is isolated ˙
Q= 0 (2.36), we get from (3.59) the rate of
entropy
˙
S(t) = k(λ1(t)−λ12(t)) ˙
Q1(t) + (λ2(t)−λ12(t)) ˙
Q2(t).(6.28)
Theorem 34 If
H1∆H1+H1∆H2=−H1∆H12(6.29)
H2∆H1+H2∆H2=−H2∆H12(6.30)
∆H16= 0 ,∆H26= 0 ,∆H16k∆H2(6.31)
(λ1−λ12)•=−α(λ1−λ12), α > 0 (6.32)
(λ2−λ12)•=−β(λ2−λ12), β > 0 (6.33)
are valid, then the rate of entropy (6.28) for B={H1, H2, H12}is positive definite.
Proof: Analogically to the proof of theorem 31 and 33, we get
˙
Q1(t)
˙
Q2(t)=αH1∆H1βH1∆H2
αH2∆H1βH2∆H2λ1(t)−λ12(t)
λ2(t)−λ12(t).(6.34)
Since all diagonal elements of the above matrix are positive, as well as its determi-
nant according to (6.11), (6.31), (6.32) and (6.33), we receive the positive definite
quadratic form
˙
S=kλ1(t)−λ12(t)
λ2(t)−λ12(t)TαH1∆H1βH1∆H2
αH2∆H1βH2∆H2λ1(t)−λ12(t)
λ2(t)−λ12(t).
(6.35)
2
Theorem 35 With the identification of the Lagrange parameter in (6.34) by the
contact temperature
λ1(t) = 1
θ1(t), λ2(t) = 1
θ2(t)(6.36)
we obtain
λ12(t) = 1
θ1(t)=1
θ2(t).(6.37)
Proof: From (6.36) we get by definition of contact temperature
˙
Q1=˙
Q2= 0 .(6.38)
Inserting this into (6.34) it follows that (6.37) is valid, because the matrix in (6.34)
is invertible. 2
So the reciprocal temperature λ12 of the partition is identical to its reciprocal con-
tact temperature 1
θ1.
Chapter 7
An Example: Electron
Phonon Interaction
In this chapter, we will consider an example of many body quantum mechanics in
second quantisation. In the first part, we will briefly recapitulate the characteristics
of a semiconductor quantum well, and then analyse the treatment of such a semi-
conductor component by means of projection operator technique [Wal03]. Next,
some thermodynamical quantities in this model are calculated and analysed. All
pictures in this chapter are copied from [Ull04].
7.1 Quantum Well
A single quantum well is composed of three thin semiconductor layers. A narrow
bandgap semiconductor is sandwiched between two layers of large bandgap semi-
conductor. The active layer thickness is there near wavelength of the charge carrier,
thus quantum effects become apparent. An example is the quantum well using GaAs
and AlxGa1−xAs, 0 <x<0.45, as potential barriers, shown in figure 7.1.
Figure 7.1: Scheme of a single quantum well AlGaAs/GaAs/AlGaAs
In the application to quantum laser, one layer is n-type AlGaAs, the other p-
type AlGaAs. The forward biased quantum well emits light in order of eV due to
50
CHAPTER 7. AN EXAMPLE: ELECTRON PHONON INTERACTION 51
interband transitions. In the resonant cavity, this light is reflected back and forth
to enforce induced emission. Besides, there are intersubband and intrasubband
transitions in order of meV. They are caused by interaction of electrons ond phonons,
which is investigated here.
Figure 7.2: Band structure of AlGaAs/GaAs/AlGaAs, E(z)
Figure 7.3: Subband structure of a quantum well, E(z), E(qk)
Figure 7.3 shows the subband structure and the corresponding dispersion relation.
qkdenotes the wave vector which is in-plane for electrons in a quantum well. They
are confined in z direction.
The electron density operator [Nol02]
%ij = Tr a+iaj%(7.1)
can be interpreted as microscopic dipole density describing the electron intersub-
band coherence. The macroscopic polarisation of a single quantum well is then a
CHAPTER 7. AN EXAMPLE: ELECTRON PHONON INTERACTION 52
Figure 7.4: Intersubband and intrasubband transitions, E(kk)
function of these density matrix elements. Waldm¨uller et al. derive differential
equations for %ij to determine them [Wal03]. Here, a+iis the generator of an elec-
tron in subband i,ajdenotes the annihilator of an electron in subband j. Since the
operators are time independent, we calculate the following dynamics
d
dt%ij =d
dtTr a+iaj%=−iTr a+iajL%=iTr %L a+iaj.(7.2)
The last term in (7.2) can be calculated using the Liouville operator Lgiven by the
corresponding Hamiltonian:
H=H0c+H0p+Hcf +Hcc +Hcp (7.3)
=X
i
εia+iai+X
q
~ωqb+
qbq+X
i,j Zdij ·E(z, t)a+iajdz
+X
i,j,m,n
1
2Vijmna+ia+janam
+X
i,j,k,qgij
qa+i
kbqaj
k−qk+g∗ji
qa+i
k−qkb+
qaj
k,(7.4)
εi: energy of an electron in subband i,~ωq: energy of an longitudinal optical phonon
with wave vector q,b+
q/bq: generator/annihilator of an phonon with wave vector q,
dij : intersubband dipole matrix element, E: external electric field, Vijmn: Coulomb
potential, gij
q: Fr¨ohlich coupling matrix. (7.2) can be interpreted as the dynamics
of particle number in subband iif i=j, or else as the dynamics of transition
probability of the electron from subband jto subband i.
In order to make a mean field approximation of this model, one applies the projec-
tion operator technique using Robertson dynamics. As relevant operators, single
electron observables and phonon number operators are chosen, where phonons form
a bath with constant temperature. Therefore the beobachtungsebene is
B=a+iaj, b+
qbq.(7.5)
P% =%MF =1
Z(t)e−Pi,j λij (t)a+iaj−Pqβ~ωqb+
qbq(7.6)
Q% =%Corr (7.7)
CHAPTER 7. AN EXAMPLE: ELECTRON PHONON INTERACTION 53
While the mean field part is supposed to be generalised canonical, no assumption
is made for the correlation part.
The commutation relations of the electron and phonon operators yield:
iL0ca+aab=i
~(εa−εb)a+aab(7.8)
iL0pa+aab= 0 (7.9)
iLcf a+aab=i
~Zdba ·E(z, t)a+bab−a+aaadz (7.10)
iLcca+aab=i
~X
i,j,m,n Vijama+ia+jamab−Vbimna+aa+ianam(7.11)
iLcpa+aab=i
~X
i,j,k,qgia
qa+i
k+qkbqab
k−gbj
qa+a
kbqaj
k−qk
+g∗ai
qa+i
k−qkb+
qab
k−g∗jb
qa+a
kb+
qaj
k+qk(7.12)
The following table contains all contributions to (7.2), that are in the form of
iTr %xLya+aab.
%MF %Corr
L0ci
~(εa−εb)%ab 0
L0p0 0
Lcf i
~Rdba ·E(z, t) (%bb −%aa)dz 0
Lcc i
~Pi,j,m (Vijam (%ib%jm −%im%jb)i
~Pi,j,m VijamTr %Corra+ia+jamab
−Vbimj (%am%ij −%aj %im)) −Vbimj Tr %Corra+aa+iajam
Lcp 0i
~Pi,k,qgia
qTr %Corra+i
k+qkbqab
k
−gbi
qTr %Corra+a
kbqai
k−qk
+g∗ai
qTr %Corra+i
k−qkb+
qab
k
−g∗ib
qTr %Corra+a
kb+
qai
k+qk
For iTr %MF Lcca+aab, the 4-point expectation value has been factorised taking
into account the canonical form of %MF [Fic83], and iTr %MF Lcpa+aabvanishes
assuming the phonon bath.
The mean field part of the wanted differential equations for %ij are now quite clear,
even without using any dynamics. This is a consequence of (7.2), because we can
CHAPTER 7. AN EXAMPLE: ELECTRON PHONON INTERACTION 54
shift the Liouville operator onto the time independent a+iaj. Both the many particle
correlation terms are treated further by means of Robertson dynamics [Wal03].
Interesting fact is, that this approach modifies the actual philosophy of projection
formalism. The quantum mechanical expectation value (7.2) is no longer split into a
relevant and an irrelevant term, but both the projected terms P % and Q% are taken
into account, whereas usually the “irrelevant” part is neglected. Here, the relevant
part represents the mean field part considering only single particle contributions,
the irrelevant part is the many particle correlation part, as can be seen in the table.
Thus, neglecting the irrelevant part means here to abandon the correlations. As a
result, all terms in the above table are exact, approximations are performed only in
the Robertson evaluation of %Corr.
7.2 Some Thermodynamical Quantities
Using the model discussed in 7.1, we are able to calculate some thermodynamical
quantities. We will neglect here the external electric field and confine ourselves to
consider the pure electron phonon interaction. The considered Hamiltonian is now
according to (7.3)
H=H0c+H0p+Hcc +Hcp (7.13)
=X
i
εia+iai+X
q
~ωqb+
qbq+X
i,j,m,n
1
2Vijmna+ia+janam
+X
i,j,k,qgij
qa+i
kbqaj
k−qk+g∗ji
qa+i
k−qkb+
qaj
k.(7.14)
Since the complete system is isolated, we split it into two interacting subsystems of
electrons and phonons according to (2.28):
H1=H0c+Hcc , H2=H0p, H12 =Hcp .(7.15)
Calculating the following commutators
iL0cb+
qbq= 0 (7.16)
iL0pb+
qbq= 0 (7.17)
iLccb+
qbq= 0 (7.18)
iLcpb+
qbq=i
~X
i,j,kgij
qa+i
kbqaj
k−qk−g∗ji
qa+i
k−qkb+
qaj
k(7.19)
iL0cHcc =i
~
1
2X
i,j,m,n
Vijmn (εi+εj−εn−εm)a+ia+janam(7.20)
iL0pHcc = 0 (7.21)
iLccHcc = 0 (7.22)
iLcpHcc =i
~X
a,b,k,qX
i,j,m,n X
l,pgab
qVbjmna+a
kbqa+j
l+p+q−kkan
lam
p
−gab
qVijmaa+j
k−la+i
p+lam
pbqab
k−qk+g∗ba
qVbjmna+a
k−qkb+
qa+j
l+p−kan
lam
p
−g∗ba
qVijmaa+j
k−q−lka+i
p+lam
pb+
qab
k,(7.23)
CHAPTER 7. AN EXAMPLE: ELECTRON PHONON INTERACTION 55
we are able to give expressions for heat exchange between the subsystems:
˙
Q1=˙
Q0c+˙
Qcc =iTr (%LH1)
=i
~X
a,j,k,q
εagja
qTr %a+j
k+qkbqaa
k−gaj
qTr %a+a
kbqaj
k−qk
+g∗aj
qTr %a+j
k−qkb+
qaa
k−g∗ja
qTr %a+a
kb+
qaj
k+qk (7.24)
+i
~X
a,b,k,qX
i,j,m X
l,pgab
qVbjmiTr %a+a
kbqa+j
l+p+q−kkai
lam
p
−gab
qVijmaTr %a+j
k−la+i
p+lam
pbqab
k−qk
+g∗ba
qVbjmiTr %a+a
k−qkb+
qa+j
l+p−kai
lam
p
−g∗ba
qVijmaTr %a+j
k−q−lka+i
p+lam
pb+
qab
k (7.25)
˙
Q2=˙
Q0p=iTr (%LH2)
=iX
i,j,k,q
ωqgij
qTr %a+i
kbqaj
k−qk−g∗ji
qTr %a+i
k−qkb+
qaj
k (7.26)
˙
Q12 =˙
Qcp
∗
=−˙
Q0c−˙
Q0p−˙
Qcc (7.27)
=i
~X
a,j,k,q
εa−gja
qTr %a+j
k+qkbqaa
k+gaj
qTr %a+a
kbqaj
k−qk
−g∗aj
qTr %a+j
k−qkb+
qaa
k+g∗ja
qTr %a+a
kb+
qaj
k+qk
+iX
i,j,k,q
ωq−gij
qTr %a+i
kbqaj
k−qk+g∗ji
qTr %a+i
k−qkb+
qaj
k
−i
~X
a,b,k,qX
i,j,m X
l,pgab
qVbjmiTr %a+a
kbqa+j
l+p+q−kkai
lam
p
−gab
qVijmaTr %a+j
k−la+i
p+lam
pbqab
k−qk
+g∗ba
qVbjmiTr %a+a
k−qkb+
qa+j
l+p−kai
lam
p
−g∗ba
qVijmaTr %a+j
k−q−lka+i
p+lam
pb+
qab
k.(7.28)
On the condition of phonon bath, all mean field parts of the heat exchanges are
vanishing, so we can replace all %by %Corr [Wal03]. Equation ∗in (7.27) is valid,
because the system is isolated, and because we took all Hamiltonian parts into
account (7.13), (7.15):
˙
Q1+˙
Q2=iTr (%L (H1+H2)) = −iTr (%LH12) = −˙
Q12 .(7.29)
It is remarkable, that the partition (section 2.5) is not inert (2.37) in this example,
as can be seen in (7.28). Although Hcc is not included in H12,˙
Q12 contains terms
with Coulomb potential Vijmn. So both electron electron and electron phonon
interactions contribute to the heat exchange ˙
Q12. If there is no electron phonon
interaction (gij =0=g∗mn), all heat exchanges obviously vanish.
The von Neumann entropy (2.16) with respect to the generalised canonical operator
R(t) = 1
Z(t)exp
−X
i,j
λij (t)a+iaj−X
q
β~ωqb+
qbq
(7.30)
CHAPTER 7. AN EXAMPLE: ELECTRON PHONON INTERACTION 56
is given by
˙
S(t) = kX
i,j
λij (t)d
dtTr %a+iaj+kX
q
β~ωq
d
dtTr %b+
qbq
=ki
~X
i,j
λij (t) (εi−εj) Tr %a+iaj
+ki
~X
i,j
λij (t)X
a,b,m,n VabimTr %a+aa+bamaj
−VjamnTr %a+ia+aanam
+ki
~X
i,j X
l,k,q
λij (t)gli
qTr %a+l
k+qkbqaj
k−gjl
qTr a+i
kbqal
k−qk
+g∗il
qTr %a+l
k−qkb+
qaj
k−g∗lj
qTr %a+i
kb+
qal
k+qk
+βk X
qX
i,j,k
iωqgij
qTr %a+i
kbqaj
k−qk−g∗ji
qTr %a+i
k−qkb+
qaj
k,
(7.31)
as can be followed directly from (7.8)–(7.12) and (7.16)-(7.19). The internal energy
of the electrons contribute to the rate of entropy as well as the many particle
potentials Vand g.
Chapter 8
The Dissipative Term
For the mesoscopic dynamics derived from von Neumann dynamics (2.4) we have
set so far the dissipative term equal to zero. On the quantum mechanical level we
wanted no dissipation, and irreversibility arises when we project the dynamics on a
beobachtungsebene, due to the loss of information. In [Kau96], [Ali01], [Lin83] one
can find general treatments of the dissipative term in the von Neumann dynamics.
Here, we will consider some examples of the dissipative term, how they appear in
the quantum mechanical theory, and what consequences they have.
8.1 Pure Quantum Mechanical Dynamics of a Sub-
system
So far in this work, we treated beobachtungsebenen with a finite number of observ-
ables (2.8). Another ansatz is the following. Consider an isolated system consisting
of two (or n, n ∈N) subsystems. One subsystem can be considered as the system
of interest (noted as S), the other as its environment (E), which is probably very
large with bath characteristica. Only the observables of Sare relevant, and we are
interested in eliminating bath quantities. This can be done by setting the Liouville
space L=LS⊗LEas a product of two Liouville spaces. Each observable in Lob
has a part in Lob
Sand in Lob
E. Let us use a more general notation of system no. 1
and system no. 2 in the following:
L=L1⊗L2(8.1)
Theorem 36 With the assumptions given above, the quantum mechanical von Neu-
mann dynamics (3.17) for one subsystem reads as
˙%1=−iL1%1−iTr2(L12 %),(8.2)
and the corresponding Liouville operators are
L1:L1→ L1, L1X:= 1
~[H1, X] (8.3)
L12 :L → L , L12 X:= 1
~[H12, X].(8.4)
57
CHAPTER 8. THE DISSIPATIVE TERM 58
Proof:
˙%1= Tr2˙%=−i
~Tr2[H1⊗I2, %]−i
~Tr2[I1⊗H2, %]−iTr2(L12 %) (8.5)
and
Tr2[I1⊗A2, B] = 0
Tr2((A1⊗I2)B) = A1(Tr2B)
Tr2(B(A1⊗I2)) = (Tr2B)A1for A1∈ L1, A2∈ L2, B ∈ L
yield (8.2). 2
Note that holds:
˙%1(t) = Tr2( ˙%(t)) 6= (Tr2%)•(t).(8.6)
The trace should act on %at every moment. In the last formulation the trace
acts uniquely at one time point so the time evolution of %1no longer contains the
interaction terms with its environment.
A similar dynamics holds for the second subsystem naturally, but we omit it here
to simplify matters. The additional term −iTr2(L12 %) for non isolated systems in
(8.2) corresponds to ◦
%in (2.4).
Theorem 37 Total energy, heat exchange and work exchange of system no. 1 with
its environment is given by
E1=Tr1(%1H1) (8.7)
˙
Q1=Tr1(H1˙%1) = −iTr ((H1⊗I2) (L12 %)) (8.8)
˙
W1=Tr1˙
H1%1(8.9)
Proof:
E1:= Tr1Tr2(%(H1⊗I2)) = Tr1(%1H1) (8.10)
Use the definitions (2.32), (2.33):
˙
Q1=iTr (% L12 (H1⊗I2)) = −iTr1(H1L1%1)−iTr ((H1⊗I2)L12 %) = Tr1(H1˙%1)
(8.11)
˙
W1= Tr1Tr2˙
H1⊗I2%= Tr1˙
H1%1.(8.12)
2
Note that Tr2is not a projector. A projector should map from a vector space
to the same vector space to satisfy P2X=P X. Here we use Tr2:L → L1or
Tr2:L2→C. Strictly speaking %1is not an accompanying process according
to our definition 3, because it does not belong to the same Liouville space like %.
Nevertheless, %1is “sufficient” for all observables G∈ L1:
Tr ((G⊗I2)%) = Tr1(GTr2%) = Tr1(G %1).(8.13)
The dynamics of %1is sufficient for work observables, too:
Tr ((G⊗I2) ˙%) = Tr1(G˙%1).(8.14)
The sufficiency makes sense, because the dynamics of %1is still a quantum mechan-
ical one where no information about the system 1 is yet lost.
CHAPTER 8. THE DISSIPATIVE TERM 59
8.2 Dynamics of a Subsystem with Reduced Infor-
mation
If we only have reduced information about subsystem no. 1 represented by a beobach-
tungsebene B={G}with G1,...,Gn∈ Lob
1, we use mesoscopic dynamics for %1in
place of the full quantum mechanical dynamics (8.2).
8.2.1 Some Remarks
We can adopt the original canonical dynamics (3.2) for subsystems, if we only
change the domain, now Lob
1%instead of Lob
%. This dynamics is so robust, because
it is not based on any quantum mechanical dynamics. However, problems arises,
when we try to derive the dynamics of the generalised canonical operator out of a
projection operator formalism, like in section 3.2. There are two possible ways to
perform this:
(i) One can project either the exact evolution equation (8.2) onto the beobach-
tungsebene as we did it in section 3.2,
˙%tr2
−→ ˙%1
P1
−→ ˙%1rel (8.15)
(ii) or we execute the trace Tr2on the reduced dynamics %rel of the full Liouville
space Lob =Lob
1⊗Lob
2, (3.27) or (3.37).
˙%P
−→ ˙%rel
tr2
−→ ˙%1rel (8.16)
In the following we will explain why it does not make sence to use a projected
dynamics for %1, neither with method (i) nor with (ii).
For (ii), it is clear: if we are only interested in the observables of the first subsystem,
we can choose a beobachtungsebene with relevant observables only of Lob
1, right from
the beginning, and use one of the projected dynamics. More precisely, we choose the
beobachtungsebene B={G1⊗I2,...,Gn⊗I2}and take a projector P:Lob
%→ Lob
%
on the full Liouville space to use (3.27) or (3.37). The trace in (8.16) becomes
unnecessary in this case.
Examining the first possibility (i), we obtain first the following Robertson dynamics
for a subsystem:
The dynamics of %1rel with respect to B1={G}with G1,...,Gn∈ Lob
1using a
linear mapping
P1:Lob
1%→ Lob
1%
˙%17→ P1˙%1= ˙%1rel where (8.17)
%17→ P1%1=%1rel and ˙
P1%1= 0 (8.18)
CHAPTER 8. THE DISSIPATIVE TERM 60
is given by
˙%1rel (t) = −iP1(t)L1(t)%1rel (t)−iP1(t) Tr2(L12(t)%(t))
−Zt
t0
P1(t)L1(t)T1(t, s)Q1(s)L1(s)%1rel (s)ds
−Zt
t0
P1(t)L1(t)T1(t, s)Q1(s) Tr2(L12(s)%(s)) ds , (8.19)
if ∂
∂s T1(t, s) = iT1(t, s)Q1(s)L1(s) (8.20)
T1(t, t) = 1 (8.21)
%1(t0) = %1rel (t0).(8.22)
It can be proved analogously to the proof of theorem 6. However, (8.19) contains
terms with %and H12 that stems from the additional term in (8.2). On that score,
this dynamics is not of practical use, unless the interaction Hamiltonian H12 is
negligible. In this case we obtain the conventional Robertson dynamics. To get rid
of %we must introduce one more projector.
The dynamics of %1rel with respect to B1={G}and B={G⊗I2}with G1, . . . , Gn∈
Lob
1using linear mappings
P1:Lob
1%→ Lob
1%
˙%17→ P1˙%1= ˙%1rel where (8.23)
%17→ P1%1=%1rel and ˙
P1%1= 0 (8.24)
P:Lob
%→ Lob
%
˙%7→ P˙%= ˙%rel where (8.25)
%7→ P % =%rel and ˙
P % = 0 (8.26)
is given by
˙%1rel (t) = −iP1(t)L1(t)%1rel (t)−Zt
t0
P1(t)L1(t)T1(t, s)Q1(s)L1(s)%1rel (s)ds
−Zt
t0
P1(t)L1(t)T1(t, s)Q1(s)Tr2L12(s)%rel (s)
−iZs
t0
L12(s)T(s, u)Q(u)L(u)%rel (u)duds
−iP1(t)Tr2L12(t)%rel (t)−iZt
t0
L12(t)T(t, s)Q(s)L(s)%rel (s)ds
(8.27)
if ∂
∂s T1(t, s) = iT1(t, s)Q1(s)L1(s) (8.28)
T1(t, t) = 1 (8.29)
%1(t0) = %1rel (t0).(8.30)
CHAPTER 8. THE DISSIPATIVE TERM 61
and ∂
∂s T(t, s) = iT (t, s)Q(s)L(s) (8.31)
T(t, t) = 1 (8.32)
%(t0) = %rel (t0).(8.33)
Anyhow, the projector (8.25), (8.26) is the same kind of projector we used for
the conventional Robertson dynamics (3.27). So again, we could have used (3.27)
instead.
Similar results can be obtained regarding the Fick Sauermann dynamics. The con-
clusion is, that we do not need an extra projected dynamics for a subsystem, because
(3.27) and (3.37) already cover this special case.
8.2.2 Sufficiency
Now we will apply theorem 26 to subsystems.
Theorem 38 Let be B={G}a beobachtungsebene with G1,...,Gn∈ Lob
1. Let be
ˆ%1(t)an accompanying process of %1(t)with respect to B. Let be
◦
ˆ%1:= ˙
ˆ%1+iL1ˆ%1= ˙%1+iL1%1:=◦
%1.(8.34)
Then ˆ%1is sufficient for ◦
G:= ˙
G+iL1G. In particular, ˆ%1is sufficient for ˙
H1.
Proof:
Tr (G %1)•= Tr (Gˆ%1)•
⇔Tr ˙
G%1+ Tr (G˙%1) = Tr ˙
Gˆ%1+ Tr (G( ˙%1+iL1%1−iL1ˆ%1))
⇔Tr ˙
G%1−iTr (GL1%1) = Tr ˙
Gˆ%1−iTr (GL1ˆ%1)
⇔Tr %1˙
G+iL1G= Tr ˆ%1˙
G+iL1G (8.35)
And ◦
H1=˙
H1+iL1H1=˙
H1.(8.36)
2
This is of course a stronger proposition than the weak sufficiency in section 5.3.
The sufficiency for the work operator ˙
H1is achieved by the special ansatz that the
quantum mechanical exchange term in the dynamics is equal to the mesoscopic one.
8.3 Entropy Production by Heat Conduction
In section 8.1, we have considered the dissipative term in the dynamics of a subsys-
tem. This term was a direct consequence from the von Neumann dynamics without
CHAPTER 8. THE DISSIPATIVE TERM 62
◦
%. In this section, we will make an ansatz for the dissipative term in the dynamics
of the full system.
Let us assume the von Neumann dynamics (3.17) and the corresponding dynamics
(8.2) for both subsystems. Then the solution of the initial value problem
˙%(t) = −i
~[H(t), %(t)] (8.37)
%(t0) = %1(t0)⊗%2(t0) (8.38)
should be unique. For t=t0, we can rewrite the differential equation (8.37) in two
eqivalent formulations:
(%1⊗%2)•=−i
~[H1⊗I2+I1⊗H2, %1⊗%2]−i
~[H12, %1⊗%2] (8.39)
(%1⊗%2)•=−i
~[H1⊗I2+I1⊗H2, %1⊗%2]−i
~Tr2[H12, %1⊗%2]⊗%2
−i
~%1⊗Tr1[H12, %1⊗%2].(8.40)
Proof: We obtain the first formulation immediately from (3.17) and from the
initial condition (8.38). For the second formulation, we use (8.2) with the same
initial condition:
(%1⊗%2)•= ˙%1⊗%2+%1⊗˙%2
=−i
~[H1, %1]−i
~Tr2[H12, %1⊗%2]⊗%2
+%1⊗−i
~[H2, %2]−i
~Tr1[H12, %1⊗%2]
=−i
~[H1⊗I2+I1⊗H2, %1⊗%2]−i
~Tr2[H12, %1⊗%2]⊗%2
−i
~%1⊗Tr1[H12, %1⊗%2].(8.41)
2
Comparing (8.39) with (8.40), it must hold:
[H12, %1⊗%2] = Tr2[H12, %1⊗%2]⊗%2+%1⊗Tr1[H12, %1⊗%2].(8.42)
In the diagonal representation of %1and %2
%1=X
iϕi
1%i
1ϕi
1(8.43)
%2=X
jϕj
2E%j
2Dϕj
2(8.44)
H12 =X
mnpq |ϕm
1i|ϕn
2ihmnpq
12 hϕp
2|hϕq
1|,(8.45)
(8.42) is equivalent to
X
mnpq |ϕm
1i|ϕn
2ihmnpq
12 (%q
1%p
2−%m
1%n
2)hϕp
2|hϕq
1|
=X
mnpq |ϕm
1i|ϕn
2ihmppq
12 (%q
1%p
2−%m
1%p
2)%n
2hϕn
2|hϕq
1|
+X
mnpq |ϕm
1i|ϕn
2ihqnpq
12 (%q
1%p
2−%q
1%n
2)%m
1hϕp
2|hϕm
1|.(8.46)
CHAPTER 8. THE DISSIPATIVE TERM 63
This cannot be valid in general. Probable causes are the neglect of ◦
%in (8.37), or
an improper initial condition (8.38).
Inserting the von Neumann dynamics (3.17) into the von Neumann entropy (2.16),
it is a fact that the entropy is constant for all time. This conflicts with the following
postulate:
Postulate 1 The isolated compound system cannot be in equilibrium ( ˙
S= 0), if
its subsystems are in nonequilibrium ( ˙
S > 0).
Classical thermodynamics teaches us for example that the entropy of an isolated
system increases, if there are internal material flux or heat flux. To handle this
problem, we can either take into account the ◦
%-term, or use a different entropy
definition, as done for example in [Kam92] or [Gav02].
The heat exchange between the subsystems is in our special interest here. We will
present a suitable term for ◦
%in (2.4) to describe the entropy production caused by
internal heat fluxes, while we omit material and work exchange.
Theorem 39 Let be
˙%=−iL% +iL12X(8.47)
where
[H, H12] = 0.(8.48)
Then it holds:
(i) Tr ˙%= 0
(ii) The compound system is isolated: ˙
Q= 0.
(iii) The partition is inert: ˙
Q12 = 0
(iv)
˙%1=−i
~[H1, %1]−i
~Tr2[H12, % −X] (8.49)
(v)
˙
Q1=−i
~Tr ((H1⊗I2) [H12, % −X]) (8.50)
Proof:
(i) Trivial.
(ii) With (8.48) it is obvious that Tr (H˙%) = 0 .
(iii) With (8.48) it is also obvious that Tr (H12 ˙%) = 0 .
(iv) Use the proof of (8.2) and (8.47).
(v) Use (8.49) to calculate ˙
Q1= Tr1(H1˙%1) .
CHAPTER 8. THE DISSIPATIVE TERM 64
2
◦
%:= iL12Xin (8.47) vanishes, if there is no heat exchange between the subsystems,
that is if H12 = 0. With the dynamics (8.47), the equations (8.39) and (8.40) at
t=t0become:
(%1⊗%2)•=−i
~[H1⊗I2+I1⊗H2, %1⊗%2]−i
~[H12, %1⊗%2−X] (8.51)
(%1⊗%2)•=−i
~[H1⊗I2+I1⊗H2, %1⊗%2]−i
~Tr2[H12, %1⊗%2−X]⊗%2
−i
~%1⊗Tr1[H12, %1⊗%2−X].(8.52)
Both terms are equivalent, if for example
X(t0) = %1(t0)⊗%2(t0) = %(t0) (8.53)
holds. The conjecture is, that
X(t) = %1(t)⊗%2(t) (8.54)
is valid for all time.
Theorem 40 Let be ◦
%=iL12Xdefined by (2.7). From (8.47) follows:
˙
S=−ki
~Tr ([H12, X] ln %).(8.55)
Let be ◦
%1=X
j|ϕj
1i˙%j
1hϕj
1|=−i
~Tr2[H12, % −X] (8.56)
analogous to (2.7). From (8.49) follows for the time rate of the entropy S1:
˙
S1=ki
~Tr ([H12, % −X] (ln %1⊗I2)) .(8.57)
The excess entropy [Mus04] is then given by
˙
S−˙
S1−˙
S2=ki
~Tr (H12 [%−X, ln %−ln (%1⊗%2)]) .(8.58)
Proof: Use (2.7) to show [Mus94]
Tr %(ln %)•= 0 = Tr1%1(ln %1)•.(8.59)
2
Chapter 9
Conclusions
The mesoscopic theory of quantum thermodynamics based on restricted macro-
scopic information is one possible way of describing irreversibility in nonequilibrium
quantum systems. This is performed using a limited set of relevant observables,
called beobachtungsebene. All statistical operators, that yield the same expecta-
tion values of the relevant observables for all time, as the quantum mechanical
density matrix %does, can be used equivalently in the mesoscopic theory. Such
dynamics are called accompanying processes of %.
Canonical dynamics is defined by the time derivative of the generalised canon-
ical operator which maximises the von Neumann entropy with respect to a given
beobachtungsebene B. This dynamics preserves the canonical form of the statistical
operator for all time, and is not derived from any quantum mechanical dynamics.
As long as the time dependence of the Lagrangian multipliers is unknown, this dif-
ferential equation cannot be solved. The other approach makes use of the projection
operator technique. Such a projector isolates the relevant part of the density ma-
trix from its irrelevant part with respect to a given beobachtungsebene. In the ideal
case, this projector is generated by an inner product on the Liouville space, the
generalised Mori product. Assuming different properties of the projector, one can
derive for example the Robertson dynamics or the Fick Sauermann dynamics from
the von Neumann dynamics. Here, the relevant part can be chosen to be generalised
canonical. The rate of von Neumann entropy can be analysed if we presuppose the
canonical form of the statistical operator. One can specify different conditions for
the positivity of this time rate.
In thermodynamical applications, the quantity of work is very essential. Since gen-
eralised work is given as expectation value of the time derivative of a relevant ob-
servable, we demand the sufficiency for accompanying dynamics: This means, that
these work expectation values are correctly given by the considered dynamics, even
though the time derived observables are not themselves included in the beobach-
tungsebene. In general, neither canonical dynamics nor projected Fick Sauermann
dynamics is sufficient. This means, that the irreversible theory using these dynamics
is only an approximative one, because it is not possible to calculate work from other
known quantities. In this case, relevant work observables should be included into
the beobachtungsebene as they are measured. A sufficient accompanying process is
65
CHAPTER 9. CONCLUSIONS 66
obtained using a relaxation ansatz for the difference between %and its accompany-
ing process. Heat exchange is the other quantity, that changes the internal energy
of a system according to the First Law. In general, canonical dynamics is not suf-
ficient for the generalised flux observables. A projected dynamics is only sufficient
for generalised fluxes, if its process history vanishes indentically for all time. We
can extend the concept of sufficiency for ◦
G-observables to that of weak sufficiency.
Applying the canonical dynamics to discrete systems in contact, we are able to use
the contact temperature to interpret Lagrangian multipliers. On some relaxation
conditions, it is possible to show the definiteness of the rate of entropy in different
situations. Another example comes from the field of many particle quantum me-
chanics, and treats the electron-phonon interaction. Here, projection technique can
be used to divide the von Neumann dynamics into two parts with regard to different
physical subject matter: the single particle part and the many particle correlation
part. We can see, that the electron-phonon coupling terms stongly affects the heat
transfer in the system. The rate of entropy contains kinetic parts of Bloch electrons
and potential contributions induced by many body interaction.
The pure quantum mechanical dynamics of a subsystem can be considered as a
reduced dynamics, too. This dynamics is von Neumann like with an additional
dissipative term represented by the interaction Hamiltonian between the subsystem
and its environment. The next step of information reduction is given by the ac-
companying process of the subsystem with respect to a given beobachtungsebene.
In this case, one has to use reduced dynamics based on the full von Neumann dy-
namics, and not on the reduced von Neumann dynamics of the subsystem. We can
conclude the weak sufficiency for subsystems, if the dissipative terms of reduced
von Neumann dynamics and of its accompanying process are equal. In this case,
the accompanying process is sufficient for internal work exchange.
Consider the initial value problem with von Neumann dynamics and two discrete
systems set in contact to each other at the initial time. For an arbitrary interaction
Hamiltonian, this problem leads to some contradictions. So we have to modify either
the initial value by including an interaction term, or we can set an additional term
into the von Neumann dynamics taking the interaction into account. In the special
case of heat exchange between the subsystems, this additional term shall warrant
the entropy production due to internal heat flux. A suggestion for this dissipative
term is made. This certainly is a point which requires further inverstigations. It
is necessary to find an explicit formula for the abstract positive dissipation term,
which is treated so often in literature. With this, the positivity of the entropy
production should be verified.
Chapter 10
Appendices
10.1 Appendix 1: An example
Theorem 41 Consider one particle with spin 1
2in a magnatic field with the beobach-
tungsebene B=x, p, x2, p2, Sz, Hs. Then the generalised canonical operator is
given by
R=e−λ0−λ1x−λ2x2−λ3p−λ4p2−λ5Sz−λ6Hs(10.1)
λ0= ln(ZpZs) = ln ∆x∆p
~+1
2
+∆phxi2
2~∆x+∆xhpi2
2~∆p−1
2ln "1 + ∆x∆p
~−1
2−1#
+ ln 2 cosh ν1
~
2cosh ν2γ~
2|B|−Bz
|B|sinh ν1
~
2sinh ν2γ~
2|B|
λ1=−∆phxi
~∆xln 1 + ∆x∆p
~−1
2−1!
λ2=∆p
2~∆xln 1 + ∆x∆p
~−1
2−1!
λ3=−∆xhpi
~∆pln 1 + ∆x∆p
~−1
2−1!
λ4=∆x
2~∆pln 1 + ∆x∆p
~−1
2−1!
λ5=ν1
λ6=ν2
with ν1, ν2given by
hSzi=−~
2|B|tanh(ν1
~
2)−Bztanh(ν2γ~
2|B|)
|B|−Bztanh(ν2γ~
2|B|) tanh(ν1
~
2)(10.2)
hHsi=−γ~
2|B||B|tanh(ν2γ~
2|B|)−Bztanh(ν1
~
2)
|B|−Bztanh(ν2γ~
2|B|) tanh(ν1~
2).(10.3)
67
CHAPTER 10. APPENDICES 68
Proof: First we introduce linear transformations and consider the following oper-
ator instead of (10.1).
R=e−µ0−µ1(b+(a)−α∗)(b(a)−α)e−ν0−ν1Sz−ν2Hs=RpRs(10.4)
with
b(a) = 1
√2~a(ax +ip), b+(a) = 1
√2~a(ax −ip).(10.5)
Then (b+(a)−α∗),(b(a)−α) have the same properties like the usual creation and
annihilation operators a+, a. Now µ0, µ1, a, α∗, α, ν0, ν1, ν2are to be determined.
Spin part Rs
For the spin part Rswe consider a spin 1
2particle in a homogenous magnetic field
B. The potential part of its Hamiltonian in the spin space is
Hs=−γ~
2
3
X
k=1
σkBk=−γ~
2BzBx−iBy
Bx+iBy−Bz(10.6)
where the σkare Pauli-matrices
σx=0 1
1 0 σy=0−i
i0σz=1 0
0−1.(10.7)
With B1={Sz, Hs},Sz=~
2σzwe get
Rs=e−ν1Sz−ν2Hs
Tr e−ν1Sz−ν2Hs(10.8)
=1
Tr e−ν1Sz−ν2Hs·cosh(ν2γ~
2|B|)
|B|·
e−ν1
~
2|B|+Bztanh(ν2γ~
2|B|)e−ν1
~
2(Bx−iBy) tanh(ν2γ~
2|B|)
eν1
~
2(Bx+iBy) tanh(ν2γ~
2|B|)eν1
~
2|B|−Bztanh(ν2γ~
2|B|)!,
Zs= Tr e−ν1Sz−ν2Hs
= 2 cosh ν1
~
2cosh ν2γ~
2|B|−Bz
|B|sinh ν1
~
2sinh ν2γ~
2|B|.
(10.9)
According to (2.25), we calculate
∂
∂ν1
ln Zs=~
2|B|tanh(ν1
~
2)−Bztanh(ν2γ~
2|B|)
|B|−Bztanh(ν2γ~
2|B|) tanh(ν1~
2)=−hSzi(10.10)
∂
∂ν2
ln Zs=γ~
2|B||B|tanh(ν2γ~
2|B|)−Bztanh(ν1
~
2)
|B|−Bztanh(ν2γ~
2|B|) tanh(ν1
~
2)=−hHsi.(10.11)
Free particle part Rp
For Rp, we use first the condition Tr Rp= 1 with B2={(b+−α∗) (b−α)}. Using
(2.26), we get
µ0= ln Tr e−µ1(b+−α∗)(b−α)= ln 1−e−µ1−1.(10.12)
Rp=e−(µ0+µ1(α∗α−1
2))e−µ1“a
2~x2+1
2~ap2−√2a
~(Re α)x−√2
~a(Im α)p”(10.13)
CHAPTER 10. APPENDICES 69
and B3=x, x2, p, p2result in
ln Zp=µ0+µ1α∗α−1
2=µ0+µ1(Re α)2+ (Im α)2−1
2(10.14)
∂ln Zp
∂Re α= 2µ1Re α=µ1r2a
~hxi ⇒ Re α=ra
2~hxi(10.15)
∂ln Zp
∂Im α= 2µ1Im α=µ1r2a
~hpi ⇒ Im α=r1
2~ahpi(10.16)
Rp=e−(µ0−1
2µ1)e−µ1(a
2~(x−hxi)2+1
2~a(p−hpi)2).(10.17)
With B4=na
2~(x−hxi)2+1
2~a(p−hpi)2owe analyse the following condition us-
ing (10.17) and (10.12):
∂
∂µ1
ln Zp=−1
2−1
eµ1−1=−a
2~∆x2−1
2~a∆p2
⇔µ1= ln "1
2~a∆x2+1
a∆p2+1
21
2~a∆x2+1
a∆p2−1
2−1#.
(10.18)
To determine a, we calculate:
x−hxi=r~
2a(b−α) + (b+−α∗)(10.19)
p−hpi=−ir~a
2(b−α)−(b+−α∗)(10.20)
Dn(x−hxi)2nE= ∆x2
n=~
an+1
2(10.21)
Dn(p−hpi)2nE= ∆p2
n=~an+1
2.(10.22)
It follows
a=∆p
∆x,(10.23)
and inserting this into (10.15), (10.16), (10.18) and (10.12), we get the above theo-
rem.
2
Similar examples with less relevant observables can be found in [Fic83], [Kau96].
10.2 Appendix 2: G¨umbel’s approach
In the outlook of [G¨um04], the following problem is briefly discussed: starting
with an accompanying process, how can one construct another accompanying pro-
cess which is sufficient for some other observables not included in the beobach-
tungsebene?
Tr (% K) = Tr (ˆ% K), K /∈ B
CHAPTER 10. APPENDICES 70
The next theorems show which problems arise if one considers accompanying pro-
cesses interacting with observables outside the beobachtungsebene.
Theorem 42 Let be %1, %2∈ Lob
%accompanying processes of the quantum mechani-
cal density operator %with respect to B={G}. Let be (·|·)1,(·|·)2the generalised
Mori products belonging to %1, %2respectively. Let be PM
1, PM
2the corresponding
Mori projectors. Let be K∈ Lob an arbitrary observable. Then follows
Tr (%1K) = Tr K PKG
1%2=Tr %2PM
1K.(10.24)
Proof:
PM
1K|I2= (K|I)1·(I|I)2+ (K|∆G1)1·(∆G1|∆G1)−1
1·(∆G1|I)2
= (K|I)1+ (K|∆G1)1·(∆G1|∆G1)−1
1·((G|I)2−(G|I)1·(I|I)2)
= (K|I)1
2
From (10.24) follows
Tr (ˆ% K) = Tr K PKG
ˆ%%= Tr % P M
ˆ%K6= Tr (% K) (10.25)
Tr (% K) = Tr K P KG
%ˆ%= Tr ˆ% P M
%K6= Tr (ˆ% K) (10.26)
in general. The last inequality in (10.26) is valid, because we restrict ourselves to a
bounded beobachtungsebene. Normally, the quantum mechanical density matrix %
is related to the “beobachtungsebene” of whole Lob. But the choice of B=Lob would
make any discussion about the “accompanying process” ˆ%nonsensical. Therefore,
the next theorem deals with two beobachtungsebenen B1⊂ B2to get rid of this
problem.
Theorem 43 Let be %1, %2∈ Lob
%accompanying processes of the quantum mechan-
ical density operator %with respect to B1={G},B2={K}respectively. Let be
B1⊂ B2. Let be (·|·)1,(·|·)2the generalised Mori products belonging to %1, %2
respectively. Let be PM
1, PM
2the corresponding Mori projectors. Let be F∈ Lob an
arbitrary observable. Then follows:
Tr (%1F) = Tr F PKG
1%2=Tr %2PM
1F,(10.27)
Tr (%2F)6=Tr F PKG
2%1=Tr %1PM
2F.(10.28)
Proof:
PM
1F|I2= (F|I)1·(I|I)2+ (F|∆G1)1·(∆G1|∆G1)−1
1·(∆G1|I)2
= (F|I)1+ (F|∆G1)1·(∆G1|∆G1)−1
1·((G|I)2−(G|I)1·(I|I)2)
= (F|I)1
PM
2F|I1= (F|I)2·(I|I)1+ (F|∆K2)2·(∆K2|∆K2)−1
2·(∆K2|I)1
= (F|I)2+ (F|∆K2)2·(∆K2|∆K2)−1
2·((K|I)1−(K|I)2·(I|I)1)
6= (F|I)2
CHAPTER 10. APPENDICES 71
2
From (10.27) follows:
Tr (ˆ% F ) = Tr F P KG
ˆ%%= Tr % P M
ˆ%F6= Tr (% F) (10.29)
in general, and from (10.28) we obtain
Tr (F %)6=F P KG
%ˆ%=ˆ% PM
%F.(10.30)
Even if we choose B=Lob, we get the inequality Tr (ˆ% F)6= Tr (% F ). Thus,
G¨umbel’s problem cannot be solved in this manner.
10.3 Appendix 3: Entropy of Canonical Dynamics
We remember: if
h(x, p) = f(x, y)−yp with p=∂f
∂y (10.31)
is valid, then his the Legendre-transform of f, and ∂h
∂p =−yis valid.
According to (2.25) and (4.4), −1
kSBis the Legendre transformed of (−ln Z), if
the work variables are constant [Kat67]
−SB
k(g) = (−ln Z)(λ)−λ·g . (10.32)
We get
∂
∂g −1
kSB=−λ , (10.33)
which is consistent with the entropy formula for constant work variables
˙
S=k λ ·˙g . (10.34)
If we have work variables adepending on time, then the partition function depends
from them, too. This leads to the following theorem.
Theorem 44 If B={G(a)}with explicitly time dependent observables, −1
kSB(a, g)
is not the Legendre transformed of (−ln Z) (a, g).
Proof
Let be the work variables anot constant in time, and let be −1
kSBthe Legendre
transformed of (−ln Z) :
−SB
k(a, g) = (−ln Z)(a, λ)−λ·g . (10.35)
CHAPTER 10. APPENDICES 72
Then
˙
S=∂S
∂a ·˙a+∂S
∂g ·˙g=∂
∂a kln Z+k λ ·g·˙a+k λ ·˙g
=k1
ZTr ∂
∂a e−λ·G·˙a+k∂
∂a Tr λ·G R·˙a+k λ ·˙g
=−k1
ZTr Z1
0
e−(1−µ)λ·Gλ·˙
G e−µλ·Gdµ+kTr λ·˙
G R
+kTr λ·G RTr R λ ·˙
G−kTr λ·G R Z1
0
eµλ·Gλ·˙
G e−µλ·Gdµ
+k λ ·˙g
=kTr λ·G RTr R λ ·˙
G−kTr Rλ·˙
Gλ·G+k λ ·˙g . (10.36)
On the other hand,
˙
S=−kTr ˙
Rln R=kTr λ·G˙
R=k λ ·˙g−kTr λ·˙
G R.(10.37)
Since the expressions are not equal, (10.35) cannot be valid.
2
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Lebenslauf
Name Akiko Kato
Geburtsdatum 25 September 1974
Geburtsort Akashi/Japan
Staatsangeh¨origkeit japanisch
Schulausbildung 1980–1983 Grundschule “Ecole Ile des Sœurs” (Montr´eal/Canada)
1983–1985 Grundschule “Schule Vorhornweg” (Hamburg/Deutschland)
1985–1987 “Gymnasium Othmarschen” (Hamburg)
1987–1989 Gymnasium “Luisenschule” (Essen)
1989–1994 Gymnasium “Goethe-Oberschule” (Berlin)
Juni 1994 Abitur
Studium 1994–1999 Studium der Physik an der TU Berlin
Okt 1996 Vordiplom
Sept 1999 Diplom
Stipendium 1995–1999 Stipendiatin der Studienstiftung des deutschen Volkes
Auszeichnungen Dez 1999 Studienf¨orderpreis der Wilhelm und Else Heraeus-Stiftung
Juli 2000 Erwin-Stephan-Preis der Helene und Erwin Stephan-Stiftung
Lehrerfahrungen 1997–1999 studentische Mitarbeiterin
am Institut f¨ur Mathematik der TU Berlin
1999–2004 wissenschaftliche Mitarbeiterin
an der mathematisch-naturwissenschaftlichen
Fakult¨at der TU Berlin
Forschungsaufenthalt April– sechsw¨ochiger Aufenthalt bei Prof. Dr. David Jou
Mai 2001 Departament de F´ısica, Universitat Aut`onoma de Barcelona
77
