231 11 6MB
English Pages [184] Year 1983
A Supplementary Series te Letters in Mathematical
Physics
Non-Equilibrium Entropy and. Irreversibility By G. Lindblad
Dordrecht / Boston / Lancaster
University of Strathclyde Glasgow
ANDERSONIAN LIBRARY
Adar
ie[i i |Ui|| THCLYDEU
RES
ONES LIBRARY
WITHOAVN of Ee ivi LIRRAR co
ae
Oi ro
a
Non-Equilibrium Entropy and Irreversibility
MATHEMATICAL
PHYSICS
STUDIES
A SUPPLEMENTARY SERIES TO LETTERS IN MATHEMATICAL PHYSICS
Editors: M. FLATO, Université de Dijon, France
M. GUENIN, Jnstitut de Physique Théorique, Geneva, Switzerland R. RAGZKA, Institute of Nuclear Research, Warsaw, Poland J. SIMON, Université de Dijon, France
S. ULAM, University of Colorado, U.S.A. Assistant Editor: J.C. CORTET, Université de Dijon, France Editorial Board:
W. AMREIN, Jnstitut de Physique Théorique, Geneva, Switzerland H. ARAKI, Kyoto University, Japan A. CONNES, 1.A.E.S., France
L. FADDEEV, Steklov Institute of Mathematics, Leningrad, U.S.S.R. J. FROHLICH, LH_E.S., France C. FRONSDAL, UCLA, Los Angeles, U.S.A.
I. M. A.
GELFAND,
Moscow State University, U.S.S.R.
JAFFE, Harvard University, U.S.A.
M. KAC, The Rockefeller University, New York, U.S.A. A. A. KIRILLOV, Moscow State University, U.S.S.R.
A. LICHNEROWICZ, Collége de France, France E. H. LIEB, Princeton University, U.S.A. B. NAGEL, K.T.H., Stockholm, Sweden J. NIEDERLE, Institute of Physics CSAV, Prague, Czechoslovakia A. SALAM, /nternational Center for Theoretical Physics, Trieste, Italy W. SCHMID, Harvard University, U.S.A.
I. E. SEGAL, MLT., U.S.A. D. STERNHEIMER, Collége de France, France I. T. TODOROV, Institute of Nuclear Research, Sofia, Bulgaria
VOLUME
5
Non-Equilibrium Entropy and Irreversibility by
GORAN
LINDBLAD
Department of Theoretical Physics,
Royal Institute of Technology, Stockholm, Sweden
D. Reidel Publishing Company A MEMBER OF THE KLUWER ACADEMIC PUBLISHERS GROUP
Dordrecht / Boston / Lancaster
hn.a hy
eal ~
C|[P
Library of Congress Cataloging in Publication Data Lindblad, Goran, 1940Non-equilibrium entropy and irreversibility. (Mathematical physics studies ; v. 5) Bibliography: p. Includes indexes. 1. Entropy. 2. Irreversible processes. 536'.73 1983 QC318.E57L56 ISBN 90-277-1640-4
I.
Title. II. Series. 83-15953
Published by D. Reidel Publishing Company P.O. Box 17, 3300 AA Dordrecht, Holland Sold and distributed in the U.S.A. and Canada by Kluwer Academic Publishers, 190 Old Derby Street, Hingham, MA 02043, U.S.A. In all other countries, sold and distributed by Kluwer Academic Publishers Group, P.O. Box 322, 3300 AH Dordrecht, Holland
All Rights Reserved © 1983 by D. Reidel Publishing Company, Dordrecht, Holland No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage aud retrieval system, without written permission from the copyright owner Printed in The Netherlands
Dd
Soo Ns LIN
TABLE OF CONTENTS
Preface
_ 1.
vii
INTRODUCTION
2.
DYNAMICS
3.
INFORMATION
AND SUMMARY
AND WORK ENTROPY
3.a
Entropy
and
3.b
Gibbs
3.c
Entropy-increasing
HEAT BATHS
5.
REVERSIBLE
6.
CLOSED
FINITE
6.a
Available
6.b
Recurrences
6.c
Entropy
8.
9.
entropy
states
4.
7.
relative
20 a3
processes
Mi 32
PROCESSES
39
SYSTEMS
functions
44 47 53
description
59
work
OPEN SYSTEMS 7.a
Markov
7.b
Available
7.c
Master
EXTERNAL
work
and entropy
equation
models
62 65
PERTURBATIONS
8.a
Models
8.b
Classical
8.c
Quantum
systems
8.d
Effects
on the entropy
THERMODYNAMIC
of the
perturbations
74
systems
LIMIT
72 79
functions
84 89
10.
143
12s
THERMODYNAMIC
processes
10.a 10.b
Properties
10.c.
Irreversibility
MEASUREMENTS, 11.b
Information
11.c
Exchange
OTHER
APPROACHES
on
MARKOV
PROCESSES
of Markov
property thermal
SENSITIVITY
processes
of Markov
processes
143 148
fluctuations OF HYPERBOLIC
140
MOTION
150 156
REFERENCES
SUBJECT
118
and heat
136
Non-passivity
NOTATION
1t3
processes
A.3
B.
110
the system
133
Markov
APPENDIX
105
dynamics
A.2
Quantum
to equilibrium
123
Reduced
A.5
98
and entropy
A.1
Non-KMS
functions
and approach
of work
QUANTUM
A.4
85
ENTROPY AND WORK
Observations
A.
and entropy
of the entropy
11.a
APPENDIX
vi
ENTROPY
Thermodynamic
INDEX INDEX
163 165
PREFACE The
problem
versible cal
of deriving
microscopic
physics
be digested
dynamics
scientist.
eral
review
work?
As a consequence
which
idea
and work,
which
in such
work.
This
a way
means
be a unique,
in time
in this
entropy allowed
that
is that
done
of energy by a the re-
through
only
in terms
definition
if its value
there
de-
is then
to the available
completely
of this
Hence,
system.
between
the relation
of a state
related
it is simply
entropy
set of problems.
in a set of time-dependent
changes
A consequence
of research.
a bit difficult
is defined
system
it is described
lines
The work
fields.
is no
of dy-
is that
intrinsic
is a family
irreof non-
Instead,
there
one
for each
set of thermodynamic
by the experimenter's
control
of the
there
is constant
formalism.
functions,
of
number
on the concepts
be based
The entropy
fields.
intrinsic
for a closed
versibility
esses
to cyclic
that
quantities.
namical
can
Hamiltonian
external
fined
work
this
to both
common
or classical
macroscopic
trivial
are
of the system
sponse
can
started
kinds).
and the reference
to this
exposure
using rather
a large
used
the arguments
may find
previous
and thermodynamics
dynamics
quantum
some
without
to follow
The
the reader
list
the stan-
from
of various
different
many
from
long
is rather
My ambition
literature,
the existing
can
not to give a gen-
it as far as possible I have
re-
than
too
key aspects
in some
work
of this
contributions
contains
field.
the
of theoreti-
papers
(mainly inequalities
tools from
ideas
more
Why add to this
in this
work
from
on the agenda
is definitely
and to develop
in the course and
goal
differing
an approach
simple mathematical
list
The
of previous
dard treatments,
results
produced
by any
yet another
However,
been
has
and
single
thermodynamics
has
for a century
with
to present
irreversible
system
pro-
through
Vii
the external
fields.
approach
is closely
modelled
the relaxation
properties
are
The present where
ment
were
statistical
mechanics
the given
basic
be a tautology
based
a wide
that
the entropy
It can
of relaxation relevant, formalism. measures sense
will
All
the
The detailed
from
from
second
the
dynamical
sec-
validity, law is
of causality
which
is
description.
With
law it is not’ always
true
to the equilibrium
in a particular
functions
value.
of a finite
model
is beyond
equilibrium
of relaxation
the
rate
is physically present
given
abstract
here
define
and give a well-defined
to equilibrium
for any given
set
processes. working
of conceptual
familiar
the
only the calculation
the entropy
of the deviation
of thermodynamic
in time
is that
a universal
Here
of the second
that
same,
scheme
a form
of the
type of problem
to the notion
number
property
to equilibrium
but this
sense.
namely
increase
be argued
present
if it is to have
interpretation
even
the
property,
as a Markov
such
Vili
behind
in a certain
on a universal
introduced
are
idea
law of thermodynamics,
must
for
set of experiments.
Another ond
in-
of the
relevant
is not
as this
states,
for non-equilibrium
entropy
interpretation
information-theoretic
no a priori
stance
is, for
There
situation.
on this
to improve
tried
I have
work
In this
quantities.
accessible
and the experimentally
concepts
by
the basic
between
relation
tenous
a rather
provide
theory,
ergodic
types
given
that
as for example
treatment,
mathematical
of rigorous
its de-
standard
the
is that
A result
up to the present.
velopment
non-equilibrium
influenced
has
infancy
in its
was
such
that
fact
The
at the time when
possible
not
experiments
tech-
by modern
probed
resolution.
a high time
provide
which
niques
on a type of experi-
the
out of this
and mathematical standard
general
scheme
difficulties,
approaches.
There
ineets most
with
a
of which
is, .for instance,
the recurrence erties not
paradox
of closed
solved
associated
with
quantum
systems.
finite
in the detail
which
matical
rigour
is lacking
tative
arguments
being
aspect
An important
a set which
description
copic
of giving advanced well
must
a more
of large
system.
more
realistic
qualitative
that matheand qualithe
that
quantum
systems.
of the set of experi-
are
intended
to apply,
that
relevant
for a micro-
it likely
I find
foundation
rigorous
by providing
as a better
namics
of the
from
are
of the approximations
of finite
concepts
differ
necessarily
means
I believe
is the specification
the thermodynamic
to which
ments
here
prop-
problems
heuristic
However,
description
in a thermodynamic
involved
This
places,
instead.
periodic
of these
the understanding
improves
approach
present
All
they merit.
in several
used
the almost
of this models
that
field
can
the program only be
of the Hamiltonian
understanding
as
of the experiments
dy-
systems.
Acknowledgements
The
research
Science and
Dr.
and Dr.
behind
Research
this
Council.
work
I wish
0. Berg for comments W.A.
a preliminary
Majewski version.
was
supported to thank
and some
by the Swedish
for the correction
W. Thirring
Professor
references,
Natural
Dr.
of a number
C. Obcemea of errors
in
ee:
ey
be
etn’ 7
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eee
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ae, Serha GES hughteean ie ie n efer anh)
; |
7
«
Yo with
common
, respectively,
origin
H and durations
where t, < So, have a time-
pur We
H,(t) for t-€ [s,.t,1. H
Tore
tec
[ty >So];
H(t)
for pte
[s,,to]-
a work
Conversely,
two
alence,
into
of use
(53st);
A work [s,t]
duration
by
H(t)
=
of given
by I(F),
composition
Bigeltg) hia defined
H(t) equal
cycles
by F is denoted
[s,t]
can
Y, on
[s,u]
and Yo on
cycles
such
[u,t]
up to equiv-
for any choice
that y ~ is:
cycle y defines
in the way
be decomposed,
cycle y on
the time
described
above.
evolution
of S during
Diy)
=
We write
T(y) = T(s,t) the dynamical
Note
that
H(t)
in a finite
maps
are
number of points,
1, 2%. > T(y,) z T(Y9).
indifferent
i.e.
to finite
changes
in
Y>
If the cycles
of composition.
ation
Dy,
=
Yo)
i
Ty,
a
Yo)
SS T(y,)*T(y9)
TUE) eT
D(y,)
concepts
Ys or cours
semighoup
in system
say, lated
a state
p and apply
forces
will
input
restricted
the external
which
the
time
world.
[12]).
Similar
also mobility
to a finite
time
is then
only cycles
y with
throughout
to F.
interval, can
time
ex-
semigroups
a lack
of the
notions
correspond
the experimenter There
of the basic
to the global mob«k-
variables
us to keep track
be assumed
is one
much
it owes
to consider
to the
geneity
restriction
where
cycles
corresponding from
and
work,
be necessary
with
by F. This
(see also Waniewski
theory,
It will sociated
[11]
sane Hees
generated
of the present
ity of Mielnik ist
then
U D(y.)
{TCs ¥ GATES the mobility
origin
a common
Yo have
intervals,
and if D(y,) and D(y) are contiguous
oper-
under a limited
of transformations
The T(y) form a semigroup
[0,T]
keep S isoof time
parameter
D(y) < [t,t]
as-
homot of
to p. This
the following
wherever
it
is applicable.
The range of dynamical the size
of F (and with
experimenter's of motions
control
he can
or the shorter If F is large
ations will responds
t where
the time
idea
appropriate).
S to perform needed
e.g.
during
to perform
then forms.a
of a reversible
increases
The
of S, the
greater larger
a given
a given
if F = Oo » then
be in T(F), which
to the
in T(F) clearly
of the dynamics
force
enough,
maps
all
group.
microscopic
time
with the
is the
set
interval,
transformation.
unitary
transform-
This case system.
cor-
It is more small.
The
interesting
not a group unitary and
difficult
case
and where
this
lack
the size of T(F) when F is
for thermodynamics
it forms
transformations
10 that
to estimate a very
small
of Oo . It will of control
is where
part of the
be argued
of S can
T(F)
be seen
is
set of all
in Chapters
6
as a source
of
irreversibility. Note
that
even
if F is small
effective
reversibility.
evolution
(defined
cays
can
sented
spin-echo
by some y) where
be reversed
by y')
In the
it may be possible
such
by applying that
the
experiment
the sample
a pulsed
initial
to have
the free
magnetization
magnetic
state
[5]
an
field
de-
(repre-
is approximately
re-
stored
Ty’ Ty)"fel = o . In this
case
tically
infinite.
recurrence
F is defined
which
For finite occurs
for any p € E, and A€ odic,
i.e.
by two free
for every
systems
parameters there
0. the sequence
choice
S is prac-
is the phenomenom
of
is almost
o(T(y)"[A])
Then
spectrum.
has a discrete
T(y)
if some
while
peri-
of e« > 0
lo(T(y)"[A]) - p(A)|
0, then
Sr(olu)> e(A)(In a - 1 + a5k)3
If p(A)> au(A) for
Piha SeatAe?
ability,
than yp, and correspond-
if - < b = 0(1), then
it is difficult
p apart
to tell
of low o-prob-
on sets
from u except
too much
not deviate
x € A (with
in favour of the hypothesis
On the other hand,
ingly S1 (plu) is large. o does
of the event
p rather
state
is in the
system
the
the observation
is a strong evidence
p(A))
probability that
, then
from yu and correspond-
ingly S;(p|u) E(BSH), p(H) = E(B,H) We use
= S(B,H) > S;(p).
the notation
Gib.F) = sple.H); Hee} G(F)
(o(BH)s
ul
for every
order
H € F with
for every
is large
enough
tee
is the following.
this
that the range
it is necessary
be possible,
Assumption:
Each
H € F. The
H € F has a discrete
above and has a non-degenerate This upper
hypothesis bound
allowed
temperature
populated
than
the
and
Vim, , , E(B.H) lim, 39
24
states,
the assumption of (g,H)
I
finite
mm
I 8 EXBall) ees
spin
lower
ones.
assumptions made
above
which
to guaran-
is unbounded
EQ (H) > =e systems,
They are
where
B € (0,)}
condition
spectrum
ground level
spectrum.
by less restrictive
Under functions
e.g.
on the energy
ing negative are more
excludes
{E(B,H);
simplest
In
10 shall
6 and
in Chapters
Gibbs
of them.
we expect
the properties to be made
the construction
that
to have
F in order
the set
to restrict
It is necessary states
6 €X0.e)4°H eo Fy.
thus
the higher These
which capable energy
systems
can
have
an
of havlevels be
(see Chapter 6).
Z, E and S are
well-defined
35 E(B4H) = E(B,H) where
the last
2
equality
holds
if and only
if B = ~.
Consequently,
for each H € F and p € E(H), the equation E(p,H)
= E = p(H) > EQ (H)
has a unique
solution
B = B(E.,H) > 0. The
relation
3 ard: 36 S(6,H) = 8 3B E(B,H)
p(8,H)
state
of the equilibrium
with
dimension)
the physical
to obtain
S(B,H)
for identifying
a justification
provides
entropy
the thermodynamic
of the system.
It implies
= lim
As E,(H) has been
to put,
—B >
©
S(p,H).
assumed
dl
sense
In a formal
thermodynamics,
this
while
this
values
be interpreted
the physical
simplicity
be used
of the entropy,
can
by the density
EQ (H) = OS
of states as well
assumption,
able
below.
it is consistent
be Scr
determined
will
to be non-degenerate
as a normalization
9 (4) =i)
that
(3.7)
S(E.H) = S(B(ESH)sH) =.Je0 E(H (yy du @(usH) + Sy(K) Sy (H)
by kp
(multiplied
ere
behaviour
low temperature near
the ground
as the energy
law of
level.
is For
normalization
»
The expression
for the entropy
as the third
(3.7)
then
(given H) for finite
gives
a range
of
temperatures
25
0 < S(B,H)
, although
this
DC)
physical
systems.
(3.1) of Sy and (3.6), a con(2.5) in a cycle y with
= (sth:
W(y,o(s))
=
=p! {S,(0(s)|o(B,H)) - S,(o(t)|o(B,H))} This
bound
for the work
is obtained
formula
Oly). aH,
invariance
the upper
that
above
the assumption
is so for most
Using the unitary venient
0 S(p5H).
Ce
from
not follow
It does jis o
< lim
relation
holds
for all
g.
It immediately
(3.8)
gives
the following
conclusions:
(1) If p(s) = p(B,H) for some B, then W(y)< 0 for all cycles y of origin
H. This
of the second infinite
create work
law,
systems)
(2) If p(s)
which
been
discussed
and Woronowicz
and Wy)
op different
the
Kelvin's
in detail
(mainly
(3) If o(®,H)
initial
= pls).
choice
can
state,
to
an amount
of
on the system.
be reached
of B and y,
the system
expression
identify librium
26
for
In onder
from
then
through
the
initial
the maximal a work
cycle
state
o(s)
work which
can
of origin
H is
= 0 for be extrac-
A(osH) = 8’! S,(olp(B.H)) This
form
[26].
= 0, then.o(t)
from
i.e.
S (ol p(BsH))
has to be performed
ted from
of passivity,
to
IW] =p!
some
has
by Pusz
= o(B3H)
a state
equal
is the property
is sometimes
the RHS with state
(3.9) taken
the available
as a sufficient work
p of S in an environment
reason
in an arbitrary
specified
to non-equi-
by g [27].
In general,
however,
in a trivial,
is accessible
way.
from
p using
This
point will
be clear in Chapter
the discussion
especially
system
the
2, unless
in Chapter
reversible
chapters,
the following
from
state
defined
dynamics
the Hamiltonian behaves
no Gibbs
10.
for
infinite
this
choice
These
relevant
Gibbs as
state
3.c
Under
Entropy-increasing
entropy makes
and the
information
theory
[16],
Penrose
tonic
increase
discarding
[1],
subsystems
They
also
interact
of S is taken
entropy
Penrose
function in some
to
in the
by Wehr] A mono-
[29].
is usually way.
attempts
achieved
elemen-
The most
the correlations
by
between
system.
a system S consisting during
property
staying
reviewed
been
and
is by neglecting
numerous still
while
Goldstein
This
I-
the
of. an.intrinsic.irre-
been
have
information"
of the observed
Consider which
framework.
this
constant.
has
There function
of the new
tary way of doing
the name
use
system
finite
them as measures
entropy
“irrelevant
be men-
subsystems.
for finite
are
I-entropy
to use
see
are
be considered
in fact
of a closed
evolution
of the dynamics.
an alternative
which
will
sometimes
can
de-
and passivity
processes
relative
it impossible
versibility find
time
and Sewell
[17]
some of which we will
by KMS
satisfied
as properties
as well
states
of Gibbs
limits
the unitary
principles
which
states,
KMS
for
thermodynamic
by Thirring
simplicity
4. For
also
reviewed
systems,
infinite
in Chapter
tioned
properties
systems,
for finite
only for
stability
the variational
include
above
scribed
been
have
and which
states [28].
is the various
for
reason
The
[17].
KMS states
choose
we
systems
states
As equilibrium
of a heat bath.
the notion
order to define
in
systems
infinite
also
to consider
be necessary
It will
a finite
time
of two
subsystems
interval.
to be p = Py @ 0». The final
The
state
S=
S, + S5
initial
state
is callea
pp’.
2
used
the notation
With
(3.3)
and
(3.1)
from
it follows
in (3.3),
that
S1 (4) + S15) = Syl) =
(3.10)
= Sr(p') < S(,) + S705) If the correlations
can
be discarded
the
by
in a consistent
of S is redefined So» then
measured
increases
the mechanism
behind
pothesis
in the derivation
of justifying
relations
will
not
commented
upon
in Chapter
as a result the
influence
must
also
be taken
ian.
This
assumptions
[31]
, among
of S, and
interaction.
This
chaos
H-theorem.
by arguing evolution
is
hy-
The dif-
that
the cor-
of S will
be
7. of the
I-entropy
of S can
due to the external
as
a reservoir
the entropy
account.
be modelled
type of model
Kossakowski increase
can
into
entropies
of Boltzmann's
of a system with
sign of the change,
I-entropy
or molecular
the future
of perturbations
nite
perturbations
these
in the value
interaction
if the total
due to the
the Stosszahlansatz
ficulties
A change
and
as the sum of the partial
I-entropy
used
way,
has
However,
others.
in Sy. The dynamical
treated The
maps
term
by Primas
resulting
are
not
happen
In general give a defi-
in the reservoir
in some
by a stochastic been
world.
will
change
also
problems in the [30]
dynamics
the Hamilton-
and causes
an
of the type
ree e F du(x) U(x)pU(x)* where
His
U: X > B(H)
a probability
any map
of this
is a unitary
measure
form
- valued
random
variable
on the space X. Due to (3.1) and
satisfies,
S,(T*Lp]) > Sy(oe)
map
for all
op € E(H)
and
(3.2)
noise"
"white
some
Under
Hamiltonian,
a semigroup
part of the
conditions
on the stochastic
evolution
of the type defined
by (A.4)
and (A.7) is obtained. The generator is of the form (A.7) with
ie
pttved.
2
equality
above
(Sete)
os
(1S.
B = 0
with
S, is a reservoir
where
(3.10)
of
a version
is
This
(3.11).
of (A.7) with the generator
for the solutions
in-
the
that
ror ay
ee
< S,(o(t))
S(e(s))
actually
follows
it then
(3.11)
From
of the perturbation.
the strength
€ measures
where
OV = Ve BUH),
and consequently with AS)= 0 (see (3.13) below). In fact the generator (3.11) is that of an open system in contact with a reser-
lack
is the
system external
world
used
[33]
pointed
nificant
than
in more
detail
the
I-entropy
other
with
systems
where
context
I-entropy
in the
which
[32],
by Tolman
thermodynamic Blatt
to the concept
out that
due
to the
in Chapter which
are
This
8. Here we will
allowed
when
can
sys-
ina
relevant
is in the focus. states
for non-equilibrium perturbations
This
isolated
essentially
exchange
the energy
the
of interaction.
is particularly
in energy.
the change
lack
of an
with
of energy
exchange
an absolute
than
rather
of a closed
of the notion
content
of a detectable
leads
consideration tem
the empirical
cases
In many
AYP).
(see Appendix
temperature
of infinite
voir
be much
the change more
sig-
idea will
be discussed
consider
the changes
exchange
the energy
in
with
is negligible.
Consider
a finite
a large
but finite
system S interacting system
R. The
to be p = Py @ Po. Now
again
taken
state
corresponding
initial
during state
Po is assumed
to the unperturbed
Hamiltonian
a finite of S$ +R
time 1s
to be a Gibbs H, Of Kk:
Po = p(B.H,). 29
know
we
(3.10)
From
that
the
of the partial
I-entropies
of
states
S and R satisfy s AST
where
R AS; dla
as? = S1(o,) - S14).
etc.
But (3.6) says that
S1(p5|P) = - as eo pAE® 0: the
Hence
I-entropy
increase
satisfies
in R always
(343)
aR pace, Provided perform
that work
Hamiltonians
the switching
on/off
on S + R the energy
of the
defined
interaction by the
does
not
unperturbed
is conserved
hee hee and
hence
AS;> - ASS > - pae® = Bae®. Consequently, observe can
if only those
that KES = 0, then
not decrease
processes it can
as a result
are
considered
be concluded
of the
that
interaction
with
where the
we
can
I-entropy
R:
AserbsUes
(3.14)
On the other hand,
if S is in a Gibbs
state
p(BH,),
then as in
(3/13)
AS < BAES. Hence,
in order
(3315) to have AS; > 0 in this
from R to S is necessary. stability of small
30
property external
This
of Gibbs
can
states
perturbations.
case,
be seen under
an energy
transfer
as an expression
the
randomizing
of a
influence
For non-equilibrium fact will
be important
difficult
in this
states in Chapter
case,
of S specified
Of 4151.6.
need
not
8. A rigorous
but a rough
be very large indeed compared states
(3.15)
estimate
to (3.15).
by distributions
hold,
treatment
shows
Consider over
and this seems
that aS; can
for simplicity
the energy
levels
ey
-0f the form
P= dy PPR > where
Pr is the one-dimensional
Let S be perturbed
energy
intervals
by R in such
leading
the number
of states
to a maximal
only one occupied
level
levels
The difference
a way
levels
that
energy
in A. by n(i).
as follows.
one with all
of ey and
A. (where U A. = (0,0))
scrambled,
is calculated
eigenprojection
transfer
Then
The minimal
only
of length
per A, (probability
equally occupied
ey within
[A, | = 6 are
AE = 6. Denote
the maximal
I-entropy
YP, a
value
is obtained
p(i)),
of AS; with
the maximal
(py = oc1)/n(1)
for e, € A.).
is then
AST = }; pli) In n(i). Consequently AS S
I ,max
where
~ In(dAE~)
d is the number
expected leading
the estimate
we obtain
typically to a very
za
(for this class
of states)
,
(3.16)
of states
to increase high upper
per unit
energy
exponentially bound
(3.16)
interval.
with
the
size
for macroscopic
d is of S, systems.
Si
CHAPTER 4.
HEAT BATHS
to achieve
the following
goals:
and
thermodynamics
of classical
isentropic
of reversible
the class
(2) To have in the formalism
in any one
system states.
initial
reproducible
of a set of welldefined,
processes
the observed
of preparing
a method
(1) To have
in order
introduced
are
baths
heat
quasifree
A set of idealized
of tempera-
the notion
ture: to any
Basic
the experimenter most in an
identical
often
possible
formalism
values
a heat
has been
temperature
servations
This
obliterated,
on
it.
i.e.
of carbon
at normal
Clearly
32
one
can
statistical
is a highly
correlations
non-trivial
as-
by the given ex-
of the distant exist
past be-
of future
assumption
of longlived metastable
temperature
prescribed
is then
and the outcome
of the past history
not
state
the memory
of the system
This
such a memory
no
that
prepared
to
it is in
and with
state defined
(Gibbs)
means
can see from the existence retain
F. The
fields
it is
procedures
bath
of given
prepared
In a thermodynamic
of the system while
the past history
tween
states.
of the ageing
sumed to be the equilibrium parameters.
all
systems
the preparation
to restrict
of the external
ternal
in pure
them
to prepare
is
theories
of systems,
microscopic
with
initial
in a welldefined
an ensemble
of
is the ability
system
in statistical
of a state
In dealing
way.
consist
with
contact
the system
it is natural
which
those
to prepare
to represent
taken
often
of a dynamic
the notion
Indeed
state.
analysis
states
(e.g. diamond
as one
which
structure
and pressure).
hope to prove
that
a general
ob-
infinite
system
will
systems
relax
even
to equilibrium
the concept
of relaxation
undefined
in the formalism.
this
is to serve
work
of such relaxation paring some
In fact,
processes.
alone.
For finite
to equilibrium the ultimate
as a conceptual
Thus,
basis
closed
is as yet
objective
of
for the description
the need to have a state pre-
procedure
in order
to construct
the formalism
leads
fundamental
problems
of an almost
philosophical
nature.
The act of preparing process
in an
intuitive
of the distant irreversible ambition
to explain
similar
argument,
on
is wiped
implying
reversible
state
is an
introduction
in a formalism
to constitute
that
the
irreversible
of that word,
The
physics
as the memory of a priori
which
a vicious
introduction
led Krylov
has
some
of reversible circle.
of a priori
to state preparation)
microdynamics,
to
prob-
can not be
[34]
can not be founded
A
to the conclu-
on classical
or
mechanics.
defining
already
noted
introduce
to reduce
as much
procedures
will can
which
a sufficiently
as
of a quantum
in the formalism.
irreversibility
the a priori
be defined
by a class
dynamics
do unavoid-
However,
the state
of highly
it seems
pro-
preparing
idealized
way.
in a microscopic
be specified simple
reason
system
of irreversible
introduction
For this
possible.
processes
2, the measurement
in Chapter
of observables
the algebra
desirable cesses
seem
(which corresponds
As was
ably
out.
procedures
may
sion that statistical quantum
initial
irreversibility on the basis
dynamics
abilities based
past
the
understanding
preparing
microscopic
baths
if left
They also
mathematical
for a rigorous
heat have
treat-
ment.
Obviously to be able
the heat
to absorb
baths
energy
must
be infinite
transferred
from
systems
other
in order
systems
with-
their equilibrium properties. For the same reason they must have a xeturn to equilibrium property. Local disturb-
out changing
33
should
bath.
As the purpose
heat
fermion
fly
degrees makes the notation
B.
or-
higher
for the
expressions
in the
often
used
been
has
and an assumption
systems
for quasifree
functions
der correlation
simple
The
systems.
of open
theory
baths
of heat
model
The quasifree
to equi-
return
temperature
R(gp) for such a heat bath of inverse
in-
We use
choice.
convenient
the most
baths
of the
condensation
of Bose
The occurrence
[35,36].
of freedom
re-
different
between
equipartition
without
off to infinity
the
perturbations
kind as local
is of a rather trivial
librium
consists
Their
systems.
It seems
about
knowledge
our
to a
them
of S.
properties
to do this
or fermion
boson
quasifree
finite
how to couple
is sufficient
properties
laxation
know
where
of systems
the only class
is to force
relaxation
the
the observed
baths
we must
system S and find
small that
of the heat
state,
to a Gibbs
system
(KMS) state of the heat
relax to the equilibrium
ances
of a decay faster than (1 + 1t|)7° for some 6 > 0 for the twotime
show
of the type (A.7) that
decay
functions
is renormalized
the case
in H takes The Gibbs
in the
place
state
interaction eric
34
property,
i.e.
true
time
"most"
the vari-
unless
parameter
by the unperturbed
irreducibility
and does
complex
is more
state
[41]. Hamiltonian
for the WCL
(A.8)).
condition
interactions),
scale
Note that
parameter.
of R is a stationary
for
on a time
place
takes
of S (this is equation an
provided assumptions
certain
(A.10)-(A.12)
rescaled
for S defined
dynamics
satisfies
Hamiltonian
equation
by a master
Hamiltonian)
of R satisfy
system
of an open
dynamics
R is governed
of the form
of S and the temperature
of the reduced
reduced
relaxation
of a time-dependent
Davies
(WCL) are fulfilled.
by the S-R coupling
not give an evolution ation
The
[37-40].
for Davies'
the conditions
(with time-independent
the correlation
on their
which
a reservoir
with
S interacting
the
in the WCL
that
that
limit
on the weak coupling
results could
imply
function
correlation
If the S-R
(which then
is a genevery
initial
state
The
states
of R can
tionary
states
by the reservoir
finite
S (with
KMS
S which under
will
given
remains
state
KMS
state
is thus
able
when
equilibrium
state
reduced
dynamics
that
e,
has
for short)
number
is
in a is availcan
not
of other
staas
interpretation
for their
above
Indeed
0.
considered
perturbation
point
perturbations
under
the
state where
the cases
rest
unique
for the unperthe set of
of the perturbation
precisely
property
due to the The
is
as a function
stability
complex.
state
equilibrium are
stationary
the stationary
small
sufficiently
Models
L is non-degener-
a unique
of having
The exact
property.
same
the
have
has a sta-
[45].
state
KMS
it if the generator
L*[p]=
perturbation
which
evolution
in a time
a bifurcation
is more
(and this
which
and time-independent
clearly
have
The corresponding to a random
dynamics
interaction
a large
essential
The exceptions
states,
equilibrium
satisfy
is a unique
there
dynamics.
parameter
for
of
way.
local
indicated
unique
in general
state
reservoir
device
of the S-R
the unperturbed
satisfying
one
On of them is the dynamical stability,
result
will
(A.7)
i.e.
turbed
[28].
of S will
sense
in the
condition
are
An infinite
the sta-
For every
interactions
preparing
details
which
"near"
tionary
of the type
state
states
will
of the dynamics
is precisely
[43,44].
a small,
that
says
property:
there
of S-R
[42].
among
stabihity
class
KMS
states
state
be characterized
in a reproducible
properties
state,
large
the only
the
stationary
for the WCL of the reduced
of course)
Of course
which
actually
the microscopic
be controlled
unique
Hamiltonian)
a sufficiently
bility
to this
stationary
the Gibbs
ate
relax
at e = Ele
when S + R is subjected of the world
equilibrium
(called
state
X
of S +R
by the temperature of X (if defined), a process which corresponds to the "heat death" of our part of the uni-
would be determined verse. we must
This
is actually
consider
not
the
the situation
interesting where
aspect
the coupling
here.
Instead
to X is very
35
weak
one.
bath,
heat
given
so small
It may of this
can
tem relaxes
to equilibrium
ture
is defined
rate
(defined
doping time
needed
lattice
with
for the
fields.
At the
during
the much
preparation
even same
paramagnetic
for very
time
shorter
low temperatures system
needed
The
field.
In this state
sys-
tempera-
relaxation
is obtained
of the ‘initial
the spin time
spin
the
and a suitable
impurities.
the
in solids
when
magnetic
coupling)
spin-lattice
by the
to a few minutes
36
in a strong
by the crystal
the crystal
is obtained
of the sample
polarization
S - R
of the
the strength
where
In NMR experiments
be controlled.
realization
an experimental
to recall
procedure
preparing
is
perturbation
relevant.
is still
be of interest
state
interaction initial
the WCL
that
of the
strength
this
that
It is assumed
strength.
on their
bound
is an upper
there
that
from X provided
perturbations
the uncontrolled
it to dominate
use
he can
then
Ap to a
the coupling
control
can
if the experimenter
Thus,
unperturbed
the
be near
will
state
stationary
the new
e, and
small
by X, then so has Lyle) for sufficiently
unperturbed
evolution
for the
state
stationary
if Lp has a unique
case,
In the generic
fer hp > 0 in the WCL.
and
¢ 0 and H corresponding
for all
at
of the
in spite
of T(F),
maps
unitary
the
using
time
state
to this
to return
be made
if it can
p, and
rium
at t = 0 in an equilib-
is prepared
If the system
irreversibility.
of
problem in the definition
conceptual
poses a severe
by the
induced
or
spontaneous
of recurrences,
The existence
fields,
systems.
macroscopic
for some
non-trivial
finite,
regular
or disturbances
qualitatively
new
sys-
due to the
features.
This
approach
seems
in small
systems
of how this
necessary e.g.
may
for the type
if we want
internal
pe done
will
of experiments
of the description
(but not
recurrences
to take
now
in molecules.
be given.
considered
unique) into
the time
from S in a given
relaxation
relaxation
The basic
here
time
initial
let F, H be fixed
ment.
Define,
A sketch is that
homogeneity
must
for them
to tell
a given
amount
always
be
us which
account.
needed
t and
scale
There
processes
idea
the time
is only an approximation.
a specified
Consider
to treat
to extract
state. and
Let
left
p be the
state
of work
of S at time
out of the notation
for the mo-
for Q(p)< E < p(h)
mek oH Intostd (ose) and
S,(T(y)"Lel) < S(osy) principle
due to the variational
reads
(3.6) and (2.6), which
p(T(y)[H]) = e(H - W(y)). (6.12) and (6.14)
When using mal
work
property
When
cycles.
this argument
p = o(B,H),
is restricted
H € F, then,
by the
to opti-
passivity
(6.6),
QU o>F .H)
=
E(p3H),
and hence
S(o3F) < S(p3FsH) = S(BsH) = S,(o). (2) This follows from (6.3), (6.14) and the following which follows immediately from (3.7)
(6.15)
S(p',H) < S(B,H) # E(B',H) < E(B5H). (3) The first
statement that
If F is so large that
p(p,H)
€ 2(p,F),
follows for every
from
(6.2%
p € Eo there
relation
(6. 14) and’ (615).
1F
is (8,H € F) such
then
S(e3F)+< S(@st) follows
from
(2).
Furthermore
S1() = S{@.H) by the unitary
invariance
of Sy. From
(1) follows
that
55
S1(o) = S(psF). (4) Put 6 = Yk AP»
0e= Qo, 3H)» and leave out F in the notation.
(6.5) says that
Ya ARQ,£ 9 = QpsH), and (6.10) a.
that for each k there
E(B) 5H). From
is a unique B, = B(Q) sH) with
(3.7) and (6.14) follows
that
S(o,) < S(B, sH). In the same way there E(B,H) From
= Q,
is a unique B = B(Q,H) S(ésH)
(3.6) and the previous
0 < dy
This holds
= Stpsh): statements
follows
that
ApSy(e(B, 5H) |o(BsH))=
as es
follows
Q)
that
YK Apsley.) Q0aHEF
Q
of gp. As all
values
for finite
(without phase transitions! )
system
For a finite
constant
ed
we can
finite
the 8, By are
a
find
that
C such
_ 98 >>
0
u € [min(Q, .Q); max(Q, 5Q)]. and consequently
forall
Yk A
(Q a
It follows
Q,)
é
ad 2e/Gy
that
S(o,) 0 we obtain
Oo
< S(p). But then bk dS (oy) = S(p)
S(pPk )
)-for.ali_k. implies that S(o,) = S(pee
processes
The set of thermodynamic this
are
chapter
optimal
cycles
work
that
of different
the reversible
in Chapter far one
can
system
only.
are
not equivalent,
and
isentropic
processes.
The reason
is
processes
of Chapter
5 are
not
necessarily
in-
in the dynamics
cluded
origin
the
For example,
respects.
define
in general
they do not
in some
imperfect
of
functions
and entropy
used
here.
These
will
defects
be repaired
10, while the goal of this chapter was to show just how proceed
In view of the of irreversibility it is natural
using
the Hamiltonian
influence
of the recurrences
and on the definition
to ask what
dynamics
effect
of a finite
on the concept
of the entropy
fluctuations
functions,
in the system may
57
have
in this
averages,
respect.
it is evident
to produce
work.
observations tem,
As the work
Indeed,
capable
a case
that
which
not
the fluctuations
are
relevant
of resolving
it is only the recurrences which
which
are
entropy
be exploited
defined
11.
above
from
work
only when
on the
sys-
the
given
initial
out of the system
to underline are
directly
In the formalism
in defining
useful
be used
performed
predictable
to obtain
of difficulty
It is perhaps
functions
them are
in Chapter
state
versibility.
of ensemble
can
is treated
a source
in terms
fluctuations
above
can
is defined
and
the notion
of irre-
again
the
deterministic,
that
non-fluctuating
quantities. We note bility
response
form
ate
kernel
is simple
no microscopic namics
being
cation
of the
58
[50].
which
performed
The
relation
as the temperature
dynamics reversible response
in this or
irreversible
kernel.
of linlinear
by a phenomthe pass-
response
to be a
can
be defined
then
on the system between
is fixed.
case,
It was
the
to satisfy
this
Irreversibility
state.
models
is given
is designed
of the work
new.
in the context
In such
from
of irreversi-
is far from
forces
is defined
in the forces.
the non-equilibrium
entropy
work
to external
The work
as non-recoverability
the definition
and collaborators
thermodynamics
integral
property.
of basing
of recoverable
of the system
enological
quadratic
idea
by J. Meixner
irreversible
ivity
the
on the notion
introduced ear
that
There
energy
and
is of course
the phenomenological depending
to cre-
on the
dyspecifi-
CHAPTER The
7.
OPEN SYSTEMS
that
description
that
It turns
out
those
given
namely tion
runs
7.a
Markov
into
analogous
functions
of entropy
In this chapter the subject
unexpected
quantum
of open
models
in section systems,
of such
a descrip-
such
equations,
master
in
systems
defined
to those
for the simplest
by Markovian
among
[52]
& Kossakowski
[1], § 3.6).
be a thermodynamic
terms
Ingarden
and
(see also Penrose
others
6.c.
[51]
& Sudarshan
Mehra
[33],
by Blatt
taken
of view was
difficulties.
description
heat
to a single
weakly
again
a finite
system S coupled
bath R = R(®).
Chapters
4 and 5 dealt with the case where the
Consider
induced
to the relaxation
ters
allowed 2 and 6.
correlations
only
dynamics
namics
by the S - R interaction.
be shown
how the thermodynamic
state
of S can
be defined
be neglected.
can
idealized
of trying
of the
(as before),
are
while
made.
provided
condition
in chap-
entropy that
of a
the S - R
is actually
the difficulties
satis-
of defining
system S + R and deriving
of S (which is not feasible
ing assumptions
Now the fields
situations.
to solve
infinite
This
compared
way described
It will
in very
Instead
slowly
or changes
to act on S in the arbitrary
non-equilibrium
fied
in time
of S$ is constant
Hamiltonian
F are
for
models
better
provide
systems
point
This
processes.
irreversible
will
(non-Hamiltonian)
open
to the conclu-
authors
led many
to define
in trying
met with
were
systems
for closed
irreversibility sion
which
and paradoxes
problems
except
The set F denotes
the dynamics
the
the reduced
dy-
in the WCL) the followthe Hamiltonians
of S
of R and the S - R interaction
59
are the same for all values an evolution namics
for S + R and
the expression
well-defined.
Chapters
for the work
the dynamics
places.
notation
in this
There
cycles
chapter
is no need
6.b).
The
in the dynamics
The objective
circum-
by S + R is to
property mentioned
in
and thermodynamics
system
(2.7) (and hence the derived
(2.7) depends
dynamics
in (2.7))
specific,
instant
states
necessary
sense
that the work functions
[t,)
determine
relative’to
of macroscopic
size,
Pos However,
the simplest,
indeed
This
t. Note
the work
even
that
if
that p(t) de(see Appen-
assumption
is non-redundant
is
in the
is true if F is large enough
say of atomic
then
are
(2.7) for all y € T(F) with D(y) ¢
1f °F 3 Oos which
system,
the work
satisfied
The Markov
description
p(t) uniquely.
S ,ceig.
if S is a small
definition).
reduced
and the
{og(t)s t € D(y)},
is clearly
of S has the Markov property
if the
descrip-
on po(t) only for ally €
This hypothesis
complete
the dy-
entropy)
pg (u) uniquely for all u > t and all y € r(F)
dix A.2 for a more
to define
reduced
thermodynamic
of pg(t) only at each
that it depends
T(F) with D(y) < [t,0). termines
To be more
on the whole set of partial
while we must demand the reduced
only the
5
above.
is to treat
p, of S (defined
functions
of Chapter
made
approach
of S + R using is fixed).
(t = © in the
processes
by the assumptions
state
on the duration
to be infinite
B of R (which
to be functions
out of the notation
for a limitation
parameter
60
performed
left
reversible
of the open
tion given by the partial
also
these
dy-
of S + R is assumed
and
as S + R is taken
of section
included
assumed
Under
(return to equilibrium)
F be fixed
of the work
namics
A.1.
reduced
4 and 5.
Let in most
(2.7)
Every y € I'(F) defines
a non-Hamiltonian
in Appendix
Furthermore,
have the relaxation
are
hence
for S as described
stances
of the fields.
in this
the only
may
be a natural
dimensions.
functions case
practical
can
not
the Markov way
assumption
If S is a system be expected assumption
to ensure
that
the
is
work
it will
Consequently
property.
the demanded
have
functions
places
in most of this chapter and in several
be taken for granted further on.
description
sequential
for the work
there ii
functions,
for the states
equivalence
observational
is the following Ks
(7.1)
Pog = Pg ® Pp(B) where
Pg is defined
In the equilibrium glected
state
Pp(B) is the Gibbs
and
in (2.7)
can
of S + R the correlations
states
The assumptions ing way:
It is possible
the S - R correlations
tion
leads
to trouble
The assumption the prediction I-entropy.
Let
be neglected.
can
the two
of the evolution the time
be formulated
in the follow-
the S - R boundary
be seen
will
that
may
to choose
that
be ne-
(7.2)
above
introduced
of R.
weak:
is sufficiently
if the S - R interaction
Po, p(BsH) = Po(B5H) @ p(B)
the
for
that
means
This
of S and the work
state
of the future
the prediction
the equilib-
leave
must
R unchanged.
system
infinite
of the
state
transfer
energy
and the consequent
action
are incon-
the S - R inter-
Furthermore,
functions.
in
that
implies
dynamics
of S + R the S - R correlations
a complete
rium
of the reduced
property
The Markov
That
in such
this
a way
idealiza-
below.
sides
of (7.1)
of S gives
evolution
are
a basic
for
equivalent inequality
for
give
T*(t = s)Log(s) © pp(B)] = Pgyglt) ~ Pglt) & Palé) in the
The
increase
due
to the neglect
the I-entropy
I-entropy
of S + R given
of the S - R correlations,
by (3.10),
which
and the change
of R given by (5.1) give, for alls
is in
~% vandal
y € T(F) with D(y) = [s,t], O(y) =H, 61
AS a
: aS* PS
age S,(e(t)) - Sy(e(s)) -
- B{W(y,p(s)) + (p(t) - p(s))[H]}> 0 where 7.b
p stands Available
The energy nite.
for
Pg from
work
(7.3)
now on.
and entropy
Q defined
in Chapter
It is convenient
to use
6 now
refers
instead
to S + R and
the finite
is infi-
quantity
(6.7)
A(oo,p3H) = sup{W(yseg.p)3 y € T(F), Oly) = H}. Note
that
this work
0 < BAP.
where
the
is extracted
23H)
RHS can
from S + R. From
(3.8)
follows
£ Sy (Po pg]Popp (BoH))
be written,
due
to (7.2),
St (Poi plPoyp(BsH)) = S:(plo(BsH)) + S;(op|op(B)). The
part belonging
assumptions nificantly
above. from
to R can
not be used
In fact,
as R is assumed
equilibrium,
(5.2)
to produce
work,
by the
not to deviate
sig-
gives
S1 (Pp | p(B) ) =a)
in the thermodynamic
limit.
Hence,
expressed
in the
partial
state
of S
0 < BACpsH)< S;(p|o(B5H)). Now let p be one relaxing provide
o=
properties, a Carnot
the reversible
cycle as an optimal
p(BsH,), H, € F, then equality BA( (BH, ) 5H)
62
of the Gibbs
a
(7.4) states
of S.
processes work
holds
S1(o(B5H,)|e(B5H))
Due to the assumed
of Chapter
cycle.
In fact,
5 will if
in (7.4): L735)
due to the existence
is the "limit" vy:
as t +=
H(t)
Tt
of the optimal
work
cycle
of origin
H which
of the set ty},
= 4 for t = 0,t
t H it AH - Hy) for t € (0,1), and which
gives
and
in the
hence
equality
in (7.3)
second
inequality
We can now define
(7.6)
by the argument
of Chapter
5
of (7.4).
the thermodynamic
entropy of the open sy-
stem S in the following way (F and B are fixed) Definition:
(7.7)
S(o) = S(B,H) + BIp(H) - E(B5H).- A(o3H)] Proposition:
S(p) has the following
properties.
(1) S(p) does not depend on the origin H of the cycles. (2) Sp) < S(p) for all
p. Equality
holds when
po € G(8,F).
(3) For any y € r(F,H), D(y) = [s;t], S(p(s)) < S(p(t)) + as R
(7.8)
as® = - p{W(y,o(s)) + (o(t) = o(s))(HI}. The
isentropic
(7.8),
processes,
correspond
(4) For all sets
which
to optimal
are
those
where
equality
holds
in
work cycles.
{p, E Eg, hy 2 0; Ye , = Ls
La S(O.) £ SOK AQP) Proof:(1)
By the assumed
relaxation
property
there
are
optimal
work cycles y € I'(F,H), 1, € I(F SH)» such that
T(y)*Co] = o(B5H)
63
T(y,)"Le] = e(B5H,)(The dynamical
maps
are
now
y' = Y; * Yo where Y => Carnot cycle of the form W(y',p)
=
Wy) 50)
non-unitary!)
(7.6).
i
Define
a work
cycle
O(y;) = H and y, is a reversible
Wy
From
(7.5) follows
that
2e(B5H,))
= Wy, 50) + (p - e(B,H,))CH -aHbe
+ 6°'S,(o(B,H,)|0(B5H)) » and hence
that
BA(o3H)> S(BsH) - S(B3H,) + + B{A(p3H,)1 + e(H - Hy) + E(B,H,) - E(B,H)} Exchanging
H and H, gives
the opposite
inequality,
hence
equality
holds.
(2) Using the relaxing
lim t-
property as in (1), we can put
o(t) = (63H)
ao
and p(s) = p in (7.3) to obtain
for all y € [(F,H) then
there
and hence the inequality.
is, due to the
passivity
of o, the trivial
cycle H(t) = H for all t, hence equality (3) (6.3)
is equivalent
If p = 0(f,H), optimal
HE
work
holds.
to
A(o(t)3H) + W(y,0(s)) < A(p(s) 3H) for s < t, which, with (7.7), gives (7.8) directly. (7.8) corresponds
64
to equality
in (6.3),
i.e.
Equality in
to optimal
F,
cycles.
(4) This follows Note
ing way,
that
directly
from
the definition
introducing
(6.5), (7.7)
a relative
(6.7) and (7.7).
can
be written
in the follow-
entropy for S (compare
(3.9) and
ro) S(o|o(B,H)) The inequality the less
reduced
(7.8)
the deviation
near
of relaxation.
reversibility
entropy
a positive
Alicki
heat conduction
We now
turn
not make
sense
un-
to the
problem
assumption.
equilibrium
this
applied
when
the fields
It is then
using
This was done by Spohn
who defined erties.
rate
from
for this
it does
through
models
S stays
to the
equation.
of S is Markovian.
equation
By assumption,
law for S + R expressed
of S. Obviously
a justification
Master
compared
is the second
description
the dynamics
of finding 7.c
= BA(p3H).
possible
the WCL’ Markovian
vary
to find master
[53] and Spohn & Lebowitz
production
concept
to models
of heat
[24],
its
and discussed
slowly |
prop-
engines
and
[54,55].
es For any y = {H(t)} € T(F,H) with D(y) = [0,1], which defin the evolution
o(t) of the state, we write t
AS(y,p(0)) = i dt o(t)
(7.9)
H(t))] o(t) = Ss, (o(t))] + $2(t)[1n0(B A simple
calculation
gives
AS = - BW(y,p(0)) + S,(0(0) |o(B5H)) - S,(e(t)|o(BH)). In the WCL
the state
ing t large
enough
we
of S will
relax
to equilibrium,i.e.
by choos-
have
o(t) ~ o(B,H) 65
AS = - BW(y,0(0)) + S,(p(0)|e(B5H)). When
(0)
that AS is the total
I-entropy
(7.3) and (7.5)
from
= p(B5H,), Hy € F, then it follows increase
aS = p[A(o(0)sH) - W(y,0(0)] = aS7** > 0 then
varies
slowly
the WCL
gives
a master
instantaneous
in (A.7) and has the t
for all
i.e.
L(t) Cote, Hit) yi, =0, Then
the expression
(7.10)
(7.9)
can
be written
ees aaadsl Ral tip Arcrecl )Fo(esH(t))3)| From
the H-theorem
(3.5)
and
(A.13)
it follows
that
g t= 0 for all
sit) > 0 o is the (7.11)
that
irreversibility
ter equation When
isfy scaled
the
I-entropy reduced
the fields
(7.11). time
defined
in the
(7.10)
are
Provided
that
of the WCL
(6) in [41])
does not satisfy
Equation
measure
of S given
to vary then
at a rate
o can
the variation
Now the form
in general.
is not
not be expected
takes
of Davies
which
place
and Spohn
[41]
of the generator
Consequently in terms
to sat-
on the
is more complex than in the previous (7.10)
of the
by the WCL mas-
holds.
(A.13) can no longer be interpreted
66
[53].
is a consistent
the results
evolution.
by Spohn
dynamics
allowed
to the relaxation,
give a semigroup
tion
production
as long as
slow compared
t
(7.11)
I-entropy
means
to
corresponding
state
of the Hamiltonian,
value
For each
of t [54].
a function the Gibbs
of annihilating
(A.8)
property
the dissipative
where
(A.12),
is of the form given
fixed t the generator
of relaxation,
to the rate
equation
is now
part lq of the generator
the
compared
If H(t)
restill
(equa-
case and it
the H-theorem
of the I-entropy
production
o.
For arbitrary as the ment
rescaling
of this
group
variation
of the time
case.
i.e.
satisfying
the Markov
property
Instead dynamics Markov
there
(A.11).
will
of section
The question
thermodynamics. tionary fields
have
A minimal
filled
and
state,
then
if there
open
system.
dent
that
an
This
For the models stochastic
comes
of work
from
correlations
sponse
[56]
Markov
for repeated
property
is equivalent
in the state
H-theorem
(A.13)
The evolution
to the stationary
and
from
dynamics
of the system which
case.
called
is given
as well
quantum
in terms
the higher
as the implies
(see (A.11)). loss
the
it is evi-
in this
property
to a monotonic
rethat
The
of information
is expressed
in the
is given
by a master
[25]. of such
Markovian
(A.12) where the dissipative
now £4xed, independent of H(t). given constant
is not ful-
This describes
This
with
t > o and the
property
property
measurements
fields.
and tract-
of the sta-
when
bath,
quantum
the
is consistent
be defined
(A.9).
with
be extracted
maps always form a semigroup
contained
equation
can
reduced
simplicity
rate
heat
can
the Markov
to the time-dependent
the dynamical
the
entropy
of the quantum regression theorem order
can not have
Markovian
limits
relaxation
of irreversible
processes
of dynamical
dynamics
If the passivity
amount
no thermodynamic
as
by a semi-
A.
if the model
obtained
is a finite
work
model
a treat-
be given
that the dynamics
to find
results
permit
conditionis the passivity
values.
infinite
not
an approximate
arises
states
constant
not
only for mathematical
then
"equilibrium"
does
7.a and Appendix
for S, we may attempt
ability.
no rigorous
not be a family
to derive
chosen
are
of S will
This means
of trying
property,
there
in the WCL
The evolution
in general,
maps
rates
systems
part Ly of the generator
Then the stationary
H is not the Gibbs state p(B,H)
is
state for a
in general
(except
67
with the for B = 0). This in itself is no disaster, but together that evolution (A.12) for a time-dependent Hamiltonian, it implies of the Bloch
example
B > 0. This
is easy
equations
in NMR when
the magnetic
the rate
of magnitude
as
in conflict
with
rium
form of the dissipative
if the WCL
holds,
(7.10)
fying similar
of (7.11)
iS
2:
then
as every This adaption
entropy
the thermodynamic
S(p.F)
that
the equilib-
with
in Appendix
thermodynamics
is obtained satis-
It is quite in [59].
discussed
model
the model
A.
This type
H(t).
[58].
of NMR
If F =
is passive. l-entropy
for S is the
ae S,(o)>
state
is a Gibbs
model,
state.
however,
has another
disease.
of the dissipative
fast
in the S - R correlations.
relaxation itself
of S , making
an
in a highly
the entropy
The
part of the generator
involves
68
Markovian
These
described
are
the Hamiltonian
expresses
non-
this
then
considered,
for arbitrary
probabilistic implies
a particular
from
effect.
cycles.
work
part of the generator
equations
Bloch
to the classical
The validity 0
by definition,
the modified
includes
with
of the
the derivation
inconsistent
and these
is consistent
which
A model
order
order
of the heat
for the relevant
Ho) are
second
of the reservoir,
state
that
same
is clearly
of the state
deviations
properties
other
have
also
models
result
is not valid
be a small
will
passivity
This
by a fixed
(defined
state
Gibbs
of dissipation. property
example
particular is of the
if only small
that
It is evident
are
work
due to the fact
(A.12)
equation
master
of the calculations
perform
can
the passivity
It is of course
bath.
can
in this
the system
at which
the rate
field
that
It is found
A.3.
in Appendix
given
The details
strong.
be chosen arbitrarily
in the
to see
when
is not passive
the system
implicit
singular
production
assumption
of an
This
dependence infinite
instantaneous to the value
infinitely
assumed
behaviour
of o on the
for most
of
initial
state
states
[53].
Furthermore,
not possible interact
in general
with
apparatus.
other
the dynamics to specify
systems,
One may express
consistency, Hy € F and
which
can
semigroups
ively,
through
how S will
any measuring
as a lack
in the following
way
it is
of dynamical
way.
Choose
His
i = 1,2,
of dynamical
the WCL
form
maps
defined
satisfying
(A.8).
by H, and Hos respect-
L = L(H) for each value Then,
by the Lie
- Trotter
[60]
epebt,
is the semigroup
tien) hp tt/2n))"
generated
by
(L(H, ) + L(H,)).
— Po|
On the other ciently
property
(A.5) where the generator
Lech =arim
L =
this
with
that
let
be the
formula
in a consistent
in particular
be seen
T.{t) = exp[tL(H.)],
of H has
is so singular
hand,
regular
T(t) represents
the Lie
- Trotter
Hamiltonian
formula
evolution
the evolution
applied
to a suffi-
of § + R would
generated
give that
by
L(H) # 5(L(H,) + L(Hp)) in general, of
there
S + R must
means
that
of S which there
be very
the higher are
singular order
defined
is no proper The
is a contradiction
in
inconsistencies
in this
can
not
process
shows
model.
correlations
[56]
stochastic
which
that
This
for the
of the two models
singularity
reduced
be defined. associated
the dynamics
dynamics
Consequently with
descibed
this model.
above
pose
no
69
taken
which
S =k
the correlations absent.
It seems
feature
interaction
that
even
the
stationarity
the problems
the
systems
fluctuations.
above
couples
response
the two aspects [50,61],
it imposes describe
do not occur. of the
and
postulated
stochastic
processes
see
small
deviations
as a
this
is no
There
system
system
theory
stochastic
in terms
to external
forces
in
thermal
the fluctuation-dissipation
property.
a case which did not lead to problems
process
the spontaneous
in a way which
from
processes
A stochastic
it is not difficult
the passivity
are
subsystems.
but it can describe
In linear
thermodynamics
70
except when
property
of a quantum
and commutative
response
case,
the commutative
state
of interacting
described
not define
always
of the
with
the dissipation
One can
not be fulfilled.
can
states
For classical
ticular
quantum
non-Markovian
property and
the
description
of the partial
theorem
Markov
no strict
hence
in [56]
The
implies
condition
of the holistic property of quantum dynamics:
consistent
does
This
and
general
for the more introduced
have
can
the dynamics
the
with
field has to be consistent state.
equilibrium
of this
the passivity
in an equi-
Hamiltonian.
interaction
by the
of S in a time-dependent
evolution that
defined
correlations
of
"memory"
complete
in the past,
moment
at a specified
state
librium
that S + R was
information
the
involves
the dynamics
to be to accept
system S. The
for the open
evolution
a non-Markovian
seems
problems
these
The only way of avoiding
states.
of the equilibrium
the passivity
with
is consistent
of open quantum systems
to equilibrium
tion of the relaxation
descrip-
to be no Markovian
seems
there
however,
case,
the general
In
or B = 0.
from R is insignificant
transfer
long as the energy
as
thermodynamics
with
consistent
is also
model
equation
master
The Markovian
of dissipation.
of the rate
to be on the scale
be
must
of H(t)
of variation
the rate
where
in the WCL,
problems
is consistent to see
Of course,
a given
that
with in par-
such models
equilibrium
state,
in the quantum case either.
are
thermal
(2) The small defined
by a Markovian
for quantum and
of simple
there
cations
The modelling for
systems
Markov
property
correspond specify
a few degrees
For systems
constant.
the microscopic
However,
size
appli-
observable,
as the
of the system
of view.
points
is relevant
process
the physical
of Boltzmann's
the assumption
stochastic
state
in physical
value
general
set
the
For them
of freedom.
and thermody-
it restricts
of the two
by a stochastic
of macroscopic
to something
as
due to the small
or a more
as
e.g.
process,
randomness
exclusiveness
of the dynamics
is insignificant
entropy
between
models.
selfconsistent
with
stochastic
is inconvenient
systems
is a natural
here.
equation.
contradiction
The apparent namics
master
but
description,
in an open system, which can
fluctuations
by a stationary
be described
often
property
stationarity
is no obvious
there
a thermodynamic
This case demands
system.
(sub-)
but finite
in a large
equilibrium
far from
of a state
preparation
by the
conditioned
to equilibrium
relaxation
(1) The transient
of processes:
two types
the following
between
tinguish
to dis-
necessary
it seems
dynamics,
on the microscopic
based
which
ones
the present
especially
models,
general
For more
model
does
information is much
too
of a not
really
needed
to
large.
71
CHAPTER 8.
EXTERNAL
noted
that when
the actual
value
of the Hamiltonian,
under can
random
small
provided
by the energy
have
this
stability
that
even
very
of the states. I-entropy, tion
property
This
group
of the present
property
However,
of problems
is lacking
Hamiltonian
systems,
8.a The
Models
F.
include
If F is treated
be, at least,
different
zero
unperturbed
work
cycles
will
be an
in I(F)
on the evolution increase
in the
The object for these
treatment
of this
set
for classical
and even about
some
other
systems.
fluctuations correctly
point
energy shall
interaction
for the energy
72
dynamical
intrin-
2 and 6, as a destruc-
is known
a bit more
to believe
by the
justification
systems
do not
of the perturbations
perturbations
the set will
while
reason
semigroup.
mathematical
for quantum
production
states
as an
of Chapters
of the mobility
a rigorous
of classical
be seen
to
stability
I-entropy
influence
is to give some
chapter
statements.
classes
will
and, in the formalism
of any
This
may be amplified
a sizeable
influence
the
is every
there
and
perturbations
small
on
control.
is stable
the state
Non-equilibrium
transfer.
of S and have
sic dynamics
(3.15)
by the bound
be expressed
then
of the dynamics.
perturbations
world
corresponding
state
S is in a Gibbs
We already
it will
baths,
no effective
has
the experimenter
which
X) over
heat
part of the external
that
from
to perturbations
be subject
the
from
the system S is separated
Even when
(called
PERTURBATIONS
levels.
as a quantum
fluctuations.
levels
be able
between
in the fields
S and
A similar
F which
conclusion
field,
If F serves
of S , which
to perform
F which
work
there
to couple
it must do if the on S, then
provides holds
then
define
there
a line width
if F represents
in Chapter
studied
is so weak
tion
as
the experimenter
the
exchange
the energy
is practically
ture
world
ternal
of S stays
not
of S.
In NMR experiments
fluctuations out
evolutions. Appendix
not
in
[63]
be taken
(multiple
Consequently,
this
a Markov
7.a)
the attendant for granted.
able to take a Markovian
effect
pulse)
In spite
evolution
not too fast
e.g.
[62].
It was
for non-Markovian of
(in the sense of the
caveat,
as a model
do
will
perturbed
of the S - X correlations, of this
a
by suitable
techniques
is typical
description
not mean
of information.
lack
field
of the evolution
neglect
form
Sometimes
be removed
of this
in the applied magnetic that
state
Consequently
does
This
completely.
of X can
in spite
initial
be unknown.
of perturbations,
types
A and section
tem S, with
influence
for the
model
of the
reproducible.
be neglected
averaging
for certain
that
not
are
of the fields,
manipulations
pointed
energy
of X and the precise
in general
will
can
the correlations
part of the perturbing
local
by the ex-
that S + X will
We assume
state
the macroscopic
the S - X correlations
just
tempera-
the average
while
the preparation
during
equilibrium
Furthermore,
perturbations.
of the S - X interaction
that
that
a mathematical
is to find
problem
of the external
reach
to assume
constant.
The first action
manner
in an uncontrolled
with
be driven
S could
Otherwise
zero.
an energy
a definite
part of X with
each
with
be neglected
interactions
we have
In addition
and the fields.
baths
heat
reproducible
for" by the
in S unaccounted
change
the
during
if he found
the data
reject
would
interac-
the
can
of energy
exchange
the total
In fact,
experiment.
baths
heat
the
equilibrium
reach
not
that S + X will
in-
is that
here
case
relevant
7, the
F a minimal
with
interaction
to the
In contradistinction
that
S.
perturbs
which
world
the external
with
teraction
We conclude
S and
defining
setup
is in any experimental
there
constituents.
up of atomic
for S made
a container
it seems
sysmust
reason-
of the "irreducible
73
not
to semigroup
lead
which
the generator
has
is non-decreasing singular tum white A). This time
that
is insignificant. there
that
with
if a finite
which
to be a good
seems
the perturbations.
reason
to make
added
S and X
between
(with 6 = 0!)
of Markov
into
this
7. This
is true
account,
a fact
choice
to the deterministic
(for
models
of
for a model model
the corresponding
systems
to
of the equilibrium
in Chapter
is taken
For classical
term
is a diffusion
and Appendix
is restricted
exchange
incompatibility
described
of
is no quan-
For this type of model
exchange
energy
[64-66]
a model
the perturbation
the
I-entropy
the action
as there
(see
of such
the energy
can then be neglected.
8 > 0) and thermodynamics even
temperatures
By (3.15)
is no problem
They represent
temperature,
the validity
so short
intervals
states
of infinite
part of
the
models
and
in [30,31]
the dissipative
For these
(3.12)).
for finite
noise means
(3.11).
(equation
reservoirs
where
evolutions
the form
treated
perturbations
noise
white
with
Hamiltonians
the
have mentioned
we
models
Markovian
of such
As an example
can
part which
fields.
of the external
by manipulations
be removed
that
i.e.
of the perturbations,
of the effect
part"
in phase
evolution
Space. 8.b
Classical
Provided
with
systems a stochastic
may try to find consequences
an answer
of these
model
to the following
perturbations
in S1) for the irreversibility The conclusion nal
world
which
are
Poincaré that
of the external
recurrence
is essential
for
essentially
closed
theorem
led some
the
increase
investigators
behaviour
(in the sense
are
one
of S?
perturbations
irreversible
back at least as far as E. Borel
What
(and the consequent
‘energy exchange with the environment).
74
problem:
of the dynamics
the uncontrollable
perturbations
to the
due to the extereven
of having
for systems
a negligible
This line of ideas goes
[67,68].
He calculated
the effect
of small
fluctuations
on distant sphere lead
stars)
on the trajectories
gas model.
He found
exponential
instability
of the work
of Krylov
Sinai
and the hard
on billiards
been
It has
order
to have
a stochastic
[71].
Instead
an
on
dependence
motion
unpredictable
scribe
motion
mean
that
system,
tity of new
a classical
The
system
dynamical phase
measure
on
measure
contains
ify one
point
mation.
The
space an
infinite
rate
In ergodic
creation
an
though
continuous
amount space
on the
an
or "observable"
fact.
is given
which
For by a
to spec-
i.e. amount
amplify
level
ad
to Lebesgue
relative
infinite
can
quan-
of information
of uncertainty,
takes
to
be taken
can
mathematical
state
initial
of the dynamics
instability
tainty to a "macroscopic" ly constant
resulting
absolutely
in the phase
to de-
give a well-defined
will
due to the following
is possible
infinitum
used
observations
of repeated
sequence
information.
the
to make
properties
of the motion
observation
additional
each
to a sen-
rise
give
is often
chaos
systems
dynamical
suffices
which
in
not necessary
can
(stochastic)
unpredictability
infinite
of
deterministic.
are
of motion
in an
The term
unpredictable
with
long-term
The
[71,72].
feature
the description
with
in classical
conditions
initial
This
for the work
are
of the motion
instability
sitive
the equations
motion
[69]).
[49,70].
perturbations
external
that
of turbulence,
gas
pre-
important
an
in connection
e.g.
claimed,
sphere
been
have
by Berry
it is the basis
and
[34]
as
unpredictable
arguments was
trajectories
close
essentially
of the dynamics
in the hard
of the molecules
of initially
the review
(see e.g.
times
numerous
(due to events
of molecules
Similar
of the perturbations.
a result
field
the collisions
become
they rapidly
that
and argued
that
divergence
exponential
to an
sented
in the gravitational
this
of inforuncer-
at an asymptotical-
[72].
theory
the asymptotic
information
gain
is given
75
the
expo-
the Liapounov
to Pesin
due
formula
by a simple
given
KS entropy
and the
nents
between
is a relation
there
measures
variant
in-
systems with smooth
dynamical
For smooth
hyperbolic.
called
often
are
instability
exponential
this
with
Systems
[71,74].
above
mentioned
of the trajectories
divergence
tic exponential
the asympto-
give
which
exponents
Liapounov
of the
in terms
lated
be formu-
alternatively
can
of the dynamics
instability
The
[73].
of the dynamics
determinism
the formal
of
in spite
quality
unpredictable
a genuinely
and consequently
KS-entropy
for the
value
a positive
have
property
K-mixing
Systems with
[49,73].
invariant
entropy
- Sinai
by the Kolmogorov
E7 Wer Ss It can chaotic tems
not
over
the whole
indicates
where
be expected
that
phase
in the energy
where
cal
approximation,
first
drete
spectra
stability
K-mixing quantum
can
properties
for such KS entropy
neither
the Liapounov
of randomness
system thus
76
a state
it can
not
given
provide
on
simple
into take
sys-
subregions on differ-
to exist
a thresh-
by numeri-
is often
are
called
exponents.
the
several
for
that has
infinite
with
characterization
described
above.
dis-
of in-
No non-trivial
of KS entropy
suggestions
infinite
not defined
energy
systems
of the concepts
made
is by noting
of finite
uniformly
as calculated
Hamiltonian
although
been
Trajectories
lack
quantum
systems,
[56,75-77]. are
seems
threshold
be no asymptotic
been
have
exponents)
values,
on the lines
have
work
be divided
there
This
are
[46].
of finite
there
generalizations
will
systems
appear.
the stochastic transition In the case
space
non-zero
models
Numerical
(and the Liapounov
For Hamiltonian
old
realistic
space.
the phase
the KS entropy
ent values.
that
or open
for quantum
or
for a systems
systems
and
Another
way of seeing
for any
realistic
quantum
a finite
I-entropy
and
uncertainty
needed
for
the
best (i.e.
almost
periodic
with
number
also
of frequencies,
is,
there
systems
For these
of
predictability
a good long-term
sense,
in a formal
at least
a finite
periodic)[69,73].
conditionally
called
is quasi-periodic
the motion
system where
Hamiltonian
classical
integrable
completely
for a finite
true
is also
This
ergodic.
and at
periodic
is almost
the motion
that
noted
we already
fact,
In
in t € (0,0).
process
stochastic
non-deterministic
a genuinely
the motion.
When searching
the
sensitive
their
case,
can
one
divergence
entropy
which
intrinsic
to the system,
the noise
term.
only
hold
a stationary
proaches
with
tionary
states
perturbations
will
distribution.
"equilibrium"
have
strong
[78,79].
of the dynamics
of the Gibbs
reminiscent
of that
The example
in Appendix
ap-
For dynamical the sta-
small
property
under
This
stability
states
of
obviously
can
property
hyperbolic
a stability
strenth
the system
until
is
which
of the
increase
linear
interval,
time
a sufficiently
systems
ways
a finite
the I-entropy
a coefficient
it is independent
i.e.
approximatively
This
during
in time with
term
diffusion
B). In some approximation
increase
show a linear
I-
in the
of a small
the addition
of expo-
rate
the
simple
From
increase
and the
(in
perturbations
between
relation
classical
to the quantum
candidate.
to be a-likely
from
(Appendix
of the chaotic
to generalize
on external
of trajectories
results
to the dynamics will
dependence
a direct
see
nential
be possible
seems
of Borel)
spirit
models
it might
which
systems
for some property
random
is in some
described
in Chapter
a.
will
depend
parameter can
on the
satisfies
e of the
t in a characteristic
be considered
entropy
strength
B indicates
as closed,
to a given accuracy, a bound
way.
in the
that the value of S(o(t))
perturbation This sense
implies
and the time that
of having
during an interval
the
system
a constant
(0,t)
I-
if the noise
(valid for all t) of the form
77
(8.1)
0 and k(p,t) is a non-decreasing of t which has a finite limit as t >= (this is (B.7)).
where
pertur-
external
from
instability
an exponential
ing a system with
of protect-
difficulties
the practical
underlines
clearly
of (8.1)
The form
bations.
from
ever,
spectively.
to distinguish
errors,
in practice
from a regular
Hamiltonian
systems
re-
difficult
be very
may
trajectory
a stochastic
In fact,
the dis-
of view
points
and truncation
due to noise
blurred
becomes
tinction
and computational
the physical
How-
behaviour.
asymptotic
due to the different
and absolute
fined
is well-de-
systems
and quasi-periodic
chaotic
between
difference
the
that
noted
We already
complex.
is more
of perturbations
effect
long
of an extremely
one
the
motion
quasi-periodic
with
systems
dynamical
For classical
period:
Classical exponential will
may
nents
be averaged
lead out
[81,82].
This
a time
interval
of length
understood which
may
at all.
Some
as a guide
serve
state
far from
ternal
noise,
mately
linearly
From
78
equilibrium. the
I-entropy
with
time
the exponential
the example
When can
have
imagination a local
be expected
divergence B it can
divergence over
place
been
performed
[83]. be in a
is perturbed to increase
interval
to be
seem
not
instability
the dynamics
in a certain
in Appendix
to take does
process
experiments
system with
expo-
local
to the recurrence
comparable
for the
instabilty
(non-stochastic)
of regular
be expected
of this
computer
Let a quasi-periodic
for which
at least
the details
However,
periods.
can
averaging
motion
The linear
to give the asymptotic
is characteristic
which
of trajectories
local
This
instability.
to a global
approximately
a local,
[80].
of trajectories
divergence
not necessarily
have
may
approxi-
of length
of trajectories be conjectured
be ex-
t,
say,
is relevant. that
the
I-entropy
production
lar to Pesin's system
form
the noise ponents rence
(8.1).
are
[74].
will
a time
span
the
still
of the
can not be seen
unless
time,
the regin-
diver-
linear
far
is still
the system
in the quasiperiodic
regime
The bound on e
B (see (B.9)).
in Appendix
recur-
periods,
the
from
that
provided
long
ex-
A logarithmic
apparent.
expected
is then
that the local
recurrence
become
simi-
indeed.
is the behaviour
by the model
indicated
than
again
This
equilibrium.
is then
shorter
low bound
system will
of trajectories,
gence
I-entropy
in a much
longer
I-entropy
of the
crease
out
exponents
for having a closed
(8.1) with t = t. Provided give a very
of the
nature
of the local
The condition
constant
not averaged
periods
average
The quasiperiodicity
satisfies
Over
from
formula
of approximately
general
ular
is some
will then be of the form (B.10) (8.2) Cy is independent
where
the local
when
ly small
periods
recurrence
and the
exponents
to be exponential-
be expected
of t. Cy can
are
large. 8.c
Quantum
systems
is known
of relevance small
random
Thus
must
unavoidably
types
be even
can
conjectural
more
under
of behaviour
which
the picture
little
be sketched than
that
above.
There systems In both
seems
with
integrable
with
the possible
avout
Here
systems.
quantum
of finite
perturbations.
case
for this given
to the case
return
We now
discrete
classical
quantum
very
long
reason
to be no a priori spectra systems
and classical recurrence
to believe
differ
in principle
in their
sensitivity
cases
periods
almost
from
which
quantum
completely
to perturbations.
periodic
may exist,
that
evolutions do not differ
19
from
work
recent
tensive chaos
(or quantum
stochasticity)
label
for a large
number
quite
small
time
scale
The periods
of the system
present
problem
case
one
of the time
the WCL
deals
periods.
This
with
that,
Note
treatment scale
a time
is not
the
re-
the size this
Consequently
systems.
parameter,
of spontaneous
fast with
extremely
for large
nature
by its very
to the recurrence
compared
the periods
the
H, is that
Hamiltonian
9 for an estimate).
(see Chapter
to the rescaling
than
increase
can
is a very weak
condition
a given
is shorter
of a quantum
for the observation
condition
obvious
in a system with
instability, currences.
in
phenomena
chaotic
molecules. [84].
The most
relevant
impelled by
been
largely
of seemingly
observation
the experimental
has
work
The theoretical
instability.
exponential
of the classical
analogue
a quantum
to find
attempts
compatible
and not obviously
of different
is a
concept
This
[80,84,85].
of quantum
concept
defined
vaguely
the very
on
in the ex-
discussed
been
has
dynamics
quantum
such
characterize
which could
The properties
time scale.
on a realistic
behaviour
observable
in their
above
described
dynamics
the unstable
due of the long
interesting
here.
A considerably
stronger
that
the correlation of a quantum
property
the K-mixing system
implies
of a classical
functions
the spectrum
for quantum
chaos
is obtain-
(for which no rigour is claimed).
argument
ed through the following We know
condition
a Lebesgue system
[73].
of the correlation
spectrum
for
In the case
function
r(t) = p(T(t)[A]” A) (see (A.16))
is given by the resonance
fo, k] = ACE Hence,
80
a quantum
frequencies
eae E,)3 EL € Sp. hid system
can
only be expected
to mimic
the classical
unstable
motion
of this
spectrum.
the
time
during
scale
ing energy
a time
There
t and
is then
short
to reveal
an uncertainty
the maximal
separation
the discreteness
relation
5E between
between neighbour-
levels
TOOE
KR.
(8.3)
If an average
or minimal
expect
a combination
to see
On the other
demands
that
of the energy
in order
behaviour.
time
the cer-
evolution
is several
times
the
by the width
given
function
in S at all, satisfy
must
state
of a chaotic
the observation
for SE we may
any evolution
in the given
The observation
distribution,
is chosen
and chaotic
to see
of the correlation
time"
"relaxation
separation
of regular
uncertainty
inequality.
opposite
level
hand,
energy
the total tainly
too
i.e.
eho lbtii= pCH))53)* Boulos to show a stochastic
In order
scale,
on the energy
when
crossings this
may
systems
highly
to recurrences
conserved
or nearly
that
analytic
constants
with regularity
levels
can
to regular
There
are
motion)
of
classical
then
some
of (8.3) and or almost of
to the existence
besides
the existence
(complete
(quasi-periodic
degeneracy
be coupled
observables
systems
of motion
This
periods.
with
and avoided
which can lead to violations
conserved
for classical
spectra.
nearest
As an antithesis
[84,85].
corresponding
degenerate
of short
of the energy
degeneracy
Recall
quantum
separations
large level
varied
are
have
levels
to non-degenerate
the fields
behaviour,
systems
even
leading
repulsion"
"level
distribution
to be associated
is believed
behaviour
an even
time of (oc-
number
spacing between
with a fairly uniform
Stochastic
neighbours.
a large
have
should
The levels
levels.
energy
cupied)
have
thus
should
systems
quantum
interval,
a considerable
over
behaviour
the Hamiltonian.
of a full
integrability)
set of
is equivalent
[69].
81
in S demands
conservation
not only
that
the
of the unperturbed
energy
that
implies
problem
as
regards
The estimate
the
Also
transfer.
of nearby
energy
of
inequality
the
energy
levels
poses
no
that
systems
with
very
conservation.
(3.16)
supports
can
be highly
if we estimate
However,
energy
of the order
that
note
{V.}
it must
inside
levels
be at most
may
the
idea
sensitive
to perturbations,
in
by the energy
trans-
bounded
that AS, is not effectively
sense
fer.
which
the coupling
levels
energy
dense
length,
energy
allowed
the total
only couple
can
the coupling
that
in S. Thus,
jumps"
“quantum
parameter
by the operators
provided
generator
of a given
intervals
(8.3)
the
that
be assumed
perturbation
but also
is small,
levels
to observable
lead
energy
the approximate
in (3.11)
not
by the
that
« of the generator does
induced
note
First
perturbations?
external
production
I-entropy
of the
in terms
be defined
property
stochastic
How kan a quantum
is now:
The question
the energy
transfer
by an expression
of the form AE =
we
see
constse at
that
the condition
(8.2) rather than namics basis
of S and
(8.1).
(8.1),
This which
classical
the non-diagonal
must
be done
systems.
the bound
not
in order
been
of the state taken
to obtain
into
(3.16)
account
in
of the form
instability
for
in the form
N(t)
(8.4)
the number of (pure)
is relevant
dy-
(in the eigen-
a bound
of the exponential
smear an initially
(8.1)
is of the form
is that the Hamiltonian
terms
Let us write
where N(t) represents perturbations
have
is characteristic
AS (t) ~ const-In
82
The reason
of the Hamiltonian)
(3.16).
on ¢« for AS) to be small
states
pure state during
if N(t) is of the form
over which the (0,t). Then
N(t) where
const-eexp(ht),
h is a constant.
in the
I-entropy
of states
This
with
smeared
tum version
by the
be described
as well
S.
generator
as
if we
an
an
ithmic,
in time
one.
(2) In order
to have
must
couple
allowed
metries
not
of H.
mix nearby
of “irregular system with eral
(compare
a large
satisfy
This
spectra"
selection
non-interacting
to
not commute
dynamics
of
is clearly
holds
the
for the WCL
which
increase
(B.9)).
I-entropy
to S.
Instead
logar-
just as
in the
in N(t),
The bound
production,
(8.2) is then
the perturbation
of the energy
levels
energy
conservation.
This
any
selection small
picture discussed
rules
to tally well [86].
by Percival
is the following.
subsystems
should
and assume
that
with
sym-
suffice
with
to
the notion
An example
Let S cOnsTst
that
are
which
means
associated
rules
perturbations seems
we expect
is approximately
all,
even
Instead,
levels.
must
intrinsic
production
intrinsic
to a linear
by the approximate
they should
behaviour
of the form
the noise the
commutativity
I-entropy
preferably
many,
for them
of their non-fulfillment.
modelling
property
case
the relevant
for this
not
is possible
conditions
a restriction
the entropy
of the
corresponding
regular classical
as a quanI can
(Appendix A). Obviously. this case can not
instability
increase
from
had put H = 0. This
form of the generator describe
sufficient
describes
then
be seen
instability.
prerequisites
(apart
If they do commute,
same
of giving
H which
thus
increase
in the number
of (8.4) and (8.5)
as the consequences
the Hamiltonian
to a linear increase
can
exponential
necessary
(1) The dissipative with
perturbation
justification
sense
Only three
will
(8.3)),
in the
corresponds
An exponential
of the classical
at present,
exist
form
time.
claim that a complete to hold.
(8.5)
of a
Of sev-
the perturbations
do not couple them either (they are "localized"). The spectrum of S may be dense "by accident" but the perturbations will only couple levels
belonging
to a single
subsystem
with
a more
sparse
Spectrum.
83
to see
chaos
in S either.
in the
AS.
can
=
in-
a linear
Thus
energy.
of the given
state
of the equilibrium
by the entropy
is bounded
I-entropy
in the
increase
(3) The total crease
not expect
we can
then
behaviour,
regular
have
If the subsystems
I-entropy
het
of length
interval
in an
hold
at most
Teoh SegiN
If the
be seen. crease
eee Then
the valid
Lacking some
quantum
duration
of the experiments
and that
the system
note evant
8.d
and the operator
Effects
on the entropy
An instability considerable When
librium).
84
property
external
there
can
the
(8.3) to
It is interesting which
information is the
spectrum
rel-
seems of the
of the perturbations.
functions
for the concepts be no effective
perturbations
The effect
of the form
postulate
for
that
bound
equilibrium.
character
isolation
provided
of the type described
significance
it exists,
tem from
of this
(8.1)
that
chaotic,
some
the only
systems
for the validity
Hamiltonian
satisfy
is far from
for quantum
that
call
we may
which
systems,
(8.2).
on « is still
for effective
of the condition
form
to
proportional
only postulate
can
we
theory,
in-
I-entropy
the
a coefficient
for the restriction
form
a rigorous
is the relevant
but with
be linear
still
can
then
shall
property
the chaotic
equilibrium,
is near
system
far from
remain
must
state
where
interval
in the time
equilibrium
The
the form:
restriction
give this
also
12. We can
of Chapter
(12.2)
the relation
this with
Compare
of the loss
for
long
above
will
introduced
have
in Chapter
way of isolating periods
of information
a
(except
near
due to the
6.
the sysequi-
pertur-
bations
is to destroy
by the fields effect group
F and
hence
of the noise T(F,e)
pative
can
defined
character
of long duration
by the
periods.
cycles
to the definition
work
thus
change
the
entropy functions
Recall
duration.
allowed
S(o3;F,¢)
be the entropy
dynamics
and work
assume
here,
of which of the
evolution
for the optimal
equations
S is far from
where
of the
the values
cycles
that
Going
defined
defined
for the entropy
work cycles
in Chap-
in I(F,D), where D = [0,t] of events,
a bit ahead in Chapter
of arbitrary
10, using
we
let
the perturbed
duration.
S is effectively
(e,t) satisfy an inequality such
per-
conditions
initial
future
and as a result
S(o;F,D)
the notation
is their
relevant)
energy
of the external
the values
cycles
a
change.
ter 6 using the Hamiltonian
First,
penalizes
periods,
re-
from
is not the case
functions,
the variational
long
cycles
be destroyed.
changed
changes
for
dissi-
of the
of lowest
the effects
The noise
in a way which
semi-
(indicating
accessible
will
on the potential
cycles
strongly
as a result
states
depends
equilibrium
that
set of states
state
for a given
This
with work
character
by slightly
of entropy
The
associated
is given in [1] § 1.3). This
(a discussion
system.
that
cycles.
of the mobility
dynamics.
long duration
claimed
of S given
T(F) may have as a result of
of reaching
be replaced
can
turbations
The will
of very
It is sometimes
due
state
possibility
work
any
the unperturbed
non-equilibrium
through
perturbed
destroy
of the work
in terms
group
of long
The
be expressed
will
recurrences
of the motion
the efficiency
maps
which
noise.
the control
of the dynamical
versibility) given
in part
closed
of the form
that AS) is insignificant
during
(8.1)
during
D.
D, i.e.
(or (8.2) when If we can
put
AS} = 0, then
S(o:F.e) < SkosF 50)
(8.6)
85
work
as the set of available
for the LHS.
is larger
cycles
other hand, by (1) of the Proposition
On the
10
in Chapter
S1(o) < S(p3F.e). [f.F
45 so-large
thats
forall
o-¢
Eo
S(osFDjes S1(o) in Chapter 6), then
(3) of the Proposition
(compare
that
it holds
S( pst stone ee For such
F the
the source no
increase
of the
intrinsic
in the
increase
Next, taken
into
we consider account.
induced
by the noise
in the thermodynamic
irreversibility:
system and the mobility
I-entropy S is seen
as a perturbed
semigroup T(F,e) intervals
D such
Let the perturbed
entropy.
is "almost"
There
is
reversible
a group.
that AS, (D) > 0 must
evolution
is
be
in D be given
by
o(t) = T(t) "Lol where
0 = 0(0).
optimal
work
If this evolution
cycle
(in the sense
(1) and (2) of the Proposition
S1(o(t))
|
where
AS; is the
cycle
for
p.
we find
from
10 that
G
n —D
~~
increase
(8.6)
(S.2)
Vv a n
gives
AS, (D") = 0 to sufficient
86
then
S{o(t) she) =1S(esPye)
0
now
that ae = 0 for each
in the sense
need
not hold
to the work
refer
of work
extracted
there
which
defined to the
states
initial
for the
performed from
the
is obviously
heat
energy
from
a given
states.
As the notion are
in general
value:
Py. The work
e.g.
Optimal
cycles
the consideration
by Carnot work
cycles
cycles
are
(on then
(of given origin) which
lead
of S.
the set 2(03;P) of states
p € Eo (without
is well-defined,
96
baths
no bound).
state
voirs) there
Pg and
(as in Chapter 6) to be those lowest
adiabatic
are
definition
is an expectation
by S, and excludes
Using these work cycles, ible
as as
this
that
p = Pp, + (1 - P) Pgs then this
state
1¢ aS® = 0 for the initial
in P which
R. Note
state,
initial
on the chosen
depends
5. We can
of Chapter
processes processes
as those
work cycles
define
(10.1)
for the reversible
equality
with
has
such map
a permanent
change
of S accessin the
reser-
and so is the set E(P) of reproducible of a work
cycle
no corresponding
depends dynamical
on the maps
initial which
state, act
in
an affine only
way
on Eo. A mobility
be defined
tween
S and
This
mobility
T(F,e)
for
the
those
processes
reservoirs
semigroup
semigroup
of dynamical
where
the energy
(and X) is zero
will
for all
be essentially
T(F)
maps
can
exchange
be-
initial
states.
of Chapter
6 or
of Chapter 8.
The observed
quantities
formalism
of the work
transfer
to each R(g), during each process
interval). Chapter
done
in this
averages
are
the ensemble
by S (or each S\)> and of the energy
The observation
in P (i.e.
of fluctuations
will
in each time
be dealt
with
in
11.
The
states
equilibrium
states
for S are
G(F), which are prepared
heat
baths.
This
state
with
brium
states.
means
parameter
that
consider
the
librium
state,
entropy
changes
set of Gibbs using
the
one
where
equal)
states
in the heat
functions,
states
the Gibbs
all S to one of the
are
each S, is in a Gibbs counted
of S to the heat
processes
The thermodynamic entropy
by coupling
By (not all
The coupling
by definition
baths
accessible
from
in P and taking
as non-equilipermits
a given
into
us to non-equi-
account
the
baths.
defined
of S is now
entropy for each
set
P = P(F)
of
as a family
of thermodynamic
pro-
cesses. The
Definition:
P-entropy
of the state
p € E(P)
is
(10.2)
S(psP) = inf, {Sy(olr)) + Jp AS*(osy)} where
the
infimum
is over
all y € P(F)
leading
to final
states
in
o(y) € G(F), and where as*( psy) = a(R) AES is the entropy change R due
to the In order
functions, striction
process
y,
to prove
some
given
initial
on the
in Chapter
state
properties
the desired
restrictions
on F assumed
the
set F are
p of S. of these necessary.
3.b or in (6.12)
entropy The re-
implies
that
97
for every H € F, 9 € E(F)
(10.3)
(0,S(p3P)) < {S(B,H); BE (0.~)}.
cesses
(6.15) and the existence of the Hamiltonian proaida hg equality holds, then all y € in I'(F,H) where eae
r(F,H)
must
from
This follows
be optimal.
Hence
in non-trivial
have
we must
cases
Vim , 9 S(BsH) > S(o5P) to be large
F is assumed
H € F. Furthermore,
for all
(10.4)
A(B) = {S(p.H); H € F}
cessarily
go to zero
as
(0,0), using reversible 10.b
Properties
Proposition:
(1)
B >).
The
From
isothermal
of the entropy P-entropy
this
assumption
over
cycles
Carnot
reversible
perform
we can
(this length will
of length > 0 for all B € (0,)
an interval
to make
enough
follows
the whole
and adiabatic
range
nethat B €
processes.
functions
has the following
properties.
S)(o) < S(psP(F))
for all
p € E(F), equality
holds for p € G(F).
(2) If {p(t)}
is the evolution
thet,” for"all
s*
0 there
is a
that
S,(o) + ASS(osv) + Jp aS*(psy) S,(o). If p € G(F), then, by (10.2), S(psP) < 0 there is a ¥, € P satisfying (10.6) with 02 p(t), y> 4: There is no restriction in taking D(y) = [s,t].
99
by (10.2) and (10.6),
satisfies,
There 7 Fah
$(p(s)sP) < S;(o(s)) + aSS(p(s)sv * 14) + TpAS (0ls)sy * 4) = $,(o(t)) + aSt(e(t)syy) + Jp (as®(p(s)sy) + aS*(0(t) 74) AS*(o(s) sy). +©+ Jp;P) < S(p(t) This
shows
ment,
we
the first that
show
such
state-
second
to prove
the
(10.6)
there
for every y satisfying
y' ~ y,
= y" * Yy> where
In order
statement.
is a M9 tF R. Choose
that as*(osy5) = 0 for all
e, H such that SUP, S(eyy> Then
from
it follows
origin
Sos
transfers
which
cycle Y
Carnot
a reversible
that
(10.4)
be found
H can
Pye c. of
the energies
ac® = g(r)” 'as® (psy) from
R. Then Y, is adiabatic
R to S, for every
and S ends
up with
the entropy
S,(0(¥p)) = Sy(o(v)) + Tp a8*(osy). the infimum
Consequently, processes that izing
where
= S(®,H).
S(o;P) the
the final
infimum
H) by (6.15).
in (10.2) state
corresponds
The third statement
work
follows
from
from
(10.2).
real-
processes
ced of adiabatic
to an optimal
such
o(g3;H)
state
of S is a Gibbs
A sequence
to adiabatic
can be restricted
cycle
(of origin
(3.7) and the equal-
ity
E(B,H) = p(H) - A(p3H). (3) The first
statement
is obvious
or less the same as (3) of the Proposition
100
The
second
is more
in Chapter 6. From
(1)
above follows one
state
leaves can
that it is enough, with the present definition,
in G(F)
is acessible
Sy invariant. of such
short
then
this
duration
is negligible.
(4) Let y satisfy
(10.6).
defined
by a He
different.
From
the
follows
which
of Chapter 8
be accessible
I-entropy
equilibrium
F and a B = B(R)), (10.2)
must
If the initial
gives the final
a P-process
perturbations
state
that
the perturbations
cess y still
p through
If the external
not be neglected,
cycle
from
that
increase
by a due to
state
is Pho then the pro-
state
p(y) for S$ (it is
but the value of As (pysy) is
that R
S(pp3P) < Sy(ely)) + Lp AS*(oy sy). But the energies
Yk AS
R
Weta.)
are
affine
functions
of po. Hence
(py sy) = as®(osy),
and consequently
| R Ya ApS(O,sP)< Syle(y)) + Lp AS”(psy)< S(psP) + €. (5) The elements
of F are,
by assumption,
of the form
H = Hy @ I, + I, ® Ho ; Consequently,
the reproducible
states
between
S, and S,. Note
that
between
the
through
in Gibbs
two
states
from S through
There
systems
of different reversible
there
still
the heat
temperature,
work
is then an equilibrium
can
do not contain
cycles.
state
correlations
be an energy
baths. work
Thus, can
if they are
be extracted
Let B,> Bo be such
o(8,H) which
that
is accessible
p(B, »H,) @ p(B, sH) through S,;-preserving processes,
S(p3P) < S(BsH) = S(By>H,) + S(BysHp).
exchange
i.e.
from
final
into
reservoirs
from S, and So:
contributions
additive
in the
changes
and the energy
entropy
state
the
for any y € F acting on S, we can decompose
On the other hand,
1 ee Si (o(y)) + Lp AS losy) =
= S1(0,()) + Syloplr)) + Felas*oy sr) + AS* (p37)] IV S(0,5P)
fy S(p53P)
the proof of the Proposition.
S(o;P) + e. This concludes
to note
It is important
product
is of the tensor
state
pend on the
initial
the partial
states
S, separately If there
state
and
So» respectively, then
ous
to
if there
ask (3.3).
So» while
still
a way that
of Sos and vice
versa.
In other
descriptions
can
exist
acting
de-
words,
for S, and
case.
no
on S, and
Pas P. < P acting
of processes
interaction
between
a subadditivity in general.
is negative
be used
F can
for
for S, will
involving
The answer
if the fields
entropy
them,
S(0,3P,) and S(p53P.) can be defined.
the entropy functions may
1n=
states
interacting
subsets
are
be no separate
do not not give Markov
in the
for
in fact,
final
set of accessible
S 1 and So» as the
if the
S, and So» even
p = p, ® Po.
form
in general
can
there
systems
teracting
between
(5) will
property
the additivity
that
interaction
an
if we allow
not hold
than
is smaller
expression
the last
then
(10.6),
If y satisfies
the
to control
independently
on the two
P > P, @ Pos then it follows
It is true
that
of S, and
systems,
from property
One
analog-
relation
interaction
then
in such
(3) that
S(osP) < Slo,sPon™ S(p55P5)On the other
hand,
if there
and Sy simultaneously,
P. can not be chosen
102
is no
in such
interaction,
a way
independently,
that
then
while
the elements
F acts
on S,
in P, and
Pc P, ® Po and it may
happen
that
S(p3P) > S(p45P,) + S(po3P5)-
macroscopically
disjoint
quence
of this
is that
fact
A conse-
cycles. that
true
it is not generally
in-
on the
depends
work
the set of allowed
so does
and
state,
jtial
process
name-
blemish,
has a slight
chapter
in this
of an adiabatic
the definition
ly that
bodies.
used
The formalism
into
separated
not
are
if the two subsystems
case
is a likely
This
as one would expect by analogy with(6.5).
fined
disturbing
seems cal
thermodynamics
noted already G(F)
law is assumed).
(if the third
in (1) above that the P(F)-entropy
is uniquely entropy.
classical
of the entropy
of the unicity
in view
the
If we
can
which
I-entropy,
increase
our
this
sight,
At first
set of processes.
for each
one
functions,
of such
but a family
is not one
there
that
is the fact
above
of classi-
However,
it was
of a state
be identified
with
of the system
control
de-
entropy
of the thermodynamic
property
important
The most
in the by
an F' > F, then by (3) the P-entropy of the states in an G(F) does not change. Thus, the dependence of the entropy of p € equilibrium state on P(F) is implicit through the relation choosing
G(F).
The optimal
a universal find
these,
creasing
type,
work
cycles
namely
we do not
P by choosing
associated
the Carnot
have
to solve
a larger
work cycles for the states
with
cycles.
these
Note
the equations
F' > F does
are
of
in order
to
states
that
of motion.
not change
In-
the optimal
in G(F).
not in Now, let S(o3P(F)) = S1(e) for a reproducible state p of S is HamiltonG(F). We assume for the moment that the dynamics the perturbaian (when it is not coupled to the heat baths) i.e. 103
But,
Then S,(e) = S; (pg).
of the fields.
pp) by action
state
equilibrium
initial
an
from
p be prepared
Let
neglected.
8 are
of Chapter
tions
by assumption
S1(o) = S(p;P) = S1(p,)>
in (2) of the Proposition.
cess
S1 (99) S
way leaves
optimal entropy. quently
There
S(o3P) This
is a permanent
another
in this
system,
and conse-
only when
- S1(p) > 0
difference
the world
change
irreversibility,
a genuine
unchanged
system S + bg R(B) with
the total
in the
to equilibrium
back
the system
is the
and
involves
(10.7) smallest
in a process
process
and the whole
states,
equilibrium
and final
p and bringing
of preparing
Consequently,
S1(,)
initial
for the
pro-
adiabatic
by the optimal
defined
state
Py is the final
where
which
total
leads
increase
from
the preparation
one
in the entropy
equilibrium
of
state
of the non-equilibrium
to
state
De
Of all tion tem
the
of being (for all
P-entropy
computable choices
given can
be calculated.
stant
in time
scribe cesses,
the
variant
as
irreversibility
that
soon
the
are
is that
predictive
of an
are
the
intrinsic
It
from
of the Gibbs I-entropy
Note
generally
sys-
is the ori-
is con-
P-entropies
the
a
states
set of available
value.
not
of the
accessible
The non-trivial
due to a limited
P-entropies
This
the distinc-
thermodynamics.
as the entropy
system.
little
for the existence
fields!).
states
to pay
has
the dynamics
for equilibrium
The price
for a closed
I-entropy
solving
equilibrium
state
but they have
of the fact
104
value
us to say which equilibrium
without
the
of time-dependent
gin of its predictive allows
functions,
depro-
importance
unitarily
irreversibility.
in-
10.c
Irreversibility
and approach
two
up in the following
be summed
can
6-10,
in Chapters
in detail
described
formalism,
in the present
of crreversibirity
The causes
to equilibrium
points.
(1) When the mobility semigroup T(F) of S is large enough, that S(o;P(F)) = S1() for all p, then the irreversibility
only as a result of the interaction
(as it is for G(F)), the
rest
number
of degrees
of freedom
of S.
these
equilibrium,
can
be amplified
the
and restrict
and the reversible
ation
of the
result
ultaneously The
fields.
intrinsic
with)
crease
in the
T(F),
in every
Hamiltonian
of the
One objective
which
a way
the P-entropy
that
instability
cycles
to those
P-entropy
this
Here
relaxation
is far from of the dyof short
dur-
increases
as a
of S after
from equilibrium
is just an aspect
states,
takes
(10.7)
evolution
place
(or simby the of the de-
due to the smallness far from
of the present formalism
of approach to equikibrtum. in such
by an
of S away
set of accessible process
the system
The quantity
ones.
the driving
increase
work
useful
by the uncontrolled
When
world.
by the external
perturbations
non-equi-
to the number
compared
reduced
It is also
to reach is limited
The mobility
parameters
of controlled
by a small
a given
from
I-entropy
is significant.
(10.7)
i.e.
state,
in general,
it is not possible,
of the same
state
an equilibrium
namics
of S with
of the world.
(2) When T(F) is limited, librium
content
of the state
I-entropy
of the
set of
to equilibrium,
an approach
in terms
is defined
equilibrium
when
have
that we can
Note
state.
of the
of information
loss
the
from
resulting
states
accessible
in the
as a decrease
be seen
It can
perturbations.
ternal
is es-
by the ex-
induced
I-entropy
in the
increase
to the
due
sentially
such
equilibrium.
is to define
"equilibrium" corresponds
of
a concept
has to be re-defined to an
coming from the intrinsic dynamics
increase
in
of S (defined by
105
it.
ocH) = ECBSH),
hand
of origin
cycles
H. The
(10.8),
On
Proposition. with
property
a significant
follows
passivity
the Gibbs
directly
from
(1) of the relative
(see property
(3.7) and(10.5)
tions
to (1) of the
due
namely that of being passive with respect to P(F)-
o(B,H),
state
itself.
F.
state
it shares
function
P-entropy
p satisfying
a state
‘Some He
not a Gibbs
is certainly the other
that
observe
First we
the
from
derived
is then
to P-equilibrium
in P-
of distance
measure
The necessary
a definition.
for such
processes
to
P-processes
of passivity
the property
to use
it is natural
However,
as
a state a Gibbs
from
apart
by applying
and P-entropy
energy
same
of the
state
not be told
if it can
state
a P-equilibrium
be to define
could
idea
A first
H € F, say).
a constant
the
rela-
P-entropy
below).
The passivity
and put H, - H = AH. ciently
the following
of o implies
If Hy is such
that
property.
Let Hy ar
H + SAH € F for a suffi-
6 > 0, then
small
e(H, ) = p(6,H)CH,]. In order
to show
i. 1%
this,
define
H for t = 0,7;
a work
=H
cycle ae:
Tim,
—>
£ {Ys ot
+ 6AH for t € (0,T).
’
Let S be coupled weakly to R(®) during 0(8,H + SAH).
As described
(0,t) and write
b=
in Chapter 5
Wyse) = 6(p, - e)[AH] . The passivity of p says that Wiyss0) < 0. The substitution 6 > - 6 implies
106
that Wiy_gsp) < 0. The linear term in the expansion
of
Wy 50) in powers
of 5 must then be zero,
i.e.
Tim, , g (es - e) CAH] = 0, gives
p > 0(8,H),
where
relation
and this, combined with the same
(p - p(B3H))(AH] = 0, and the statement
follows.
The condition
F.
true
It is not necessarily performed
the work
two
there
that
know
for the
cycle
values
the corresponding
and that
one),
work
optimal
common
one
is at least
(the reversible
states
not in F). We only
W(y) of Chapter 2 aré generally
work operators
the
that
(recall
in an arbitrary work cycles
for
values
they give the same
that
in
for all elements
give the same expectations
p and p(8,H)
states
the
F. Then
set
of the convex
interior
relative
in the
a point
H, 6 (Fatt Harps
for all
is true
statement
of the
of the work are the same (zero if O(y) =H). There is in general a y € P(F) of origin H such that W(y,9) # W(y,e(B.H)), and then at of tiggpe
one
least
forming
it will
work wh@Pe
In the case one,
be macroscopically
Chapter
is then
9 that
fluctuations
I would define
level like
sense).
driven
far from
on a macroscopic
by an
at this
a thermodynamic
they must
scale.
fluctuations
instability
to draw
(the words
give the same
must
equilibrium
equivalence prediction
can
develop be ar-
to a macrowhich
The conclusion
for cycles
in such
in
claimed
it is unreasonable
of states
used
are
11 it will
be amplified
is that
to
be considered
already
It was
of the motion. point
reversible
micro/macroscopic
In Chapter can
the
irreversibly.
states
intuitive
a system
gued that microscopic scopic
two
if the
different
is uniquely
cycle
apart.
the states
to tell
be lost,
work will
imprecise,
in an
work
by per-
that
means
This
is negative.
be possible
the optimal
part of this
then
The question
here
quantities
a way
performing
to
that
work
on
107
(1)
above.
Proposition
the
from
follow
p and the Gibbs
between
as a distance
P-entropy
(10.8)
- S(osP) + BLe(H) - E(@,H)J.
= S(BH)
S(p|o(B.H);P)
a relative
0(8,H).
state
is
relation
a P-equivalence
Define
way.
in the following
introduced
Instead
fashion.
irreversible
S in an
properties
Its
Only two of them will
be given:
S(plo(B.H);P) > 0 hold if
equality
if
and only
(2) S(o(t)|o(B,H)sP) evolution
defined
is a non-increasing
function
which which
pass-
are
satisfy
of t under the
by H.
(1) follows
The inequality
energy
H, 1.e.
of origin
to P-processes
respect
ive with
of the given
the states
precisely
are
These
(without phase transitions)
systems
For finite
from
(3.7) and (10.5) which enable
us
(10.8) as (P is fixed)
to write
S(p|p(B5H)) = Fidu[e(u,H) - 8] + BA(p3H) E = E(6,H), Both
terms
A(psH)
Q = p(H) - A(o3H).
on the RHS are
= 0
and
E(6,H)
non-negative.
(10.8).
(2) follows
from
the constancy The be used
108
= 0, then
= p(H). unless SP = 0. Consequently
S(o3;P) = S(8,H). from directly
If the LHS
the non-decrease
of the
P-entropy
and
of o(t)(H].
properties to define
of the relative the notion
P-entropy
suggest
of P-equilibrium
states.
that
it can
Definition:
The displacement
of o from
P(F)-equilibrium
is
d(p|G(F)) = inf{S(p|usP(F)); u € G(F)} and p is called a P(F)-equilibrium state if d(o|G(F)) = 0. If the infimum
is achieved
is non-increasing In this defined, can
way a concept
but no general
is doubtful
much
sense
of S.
of S, the
much
shorter
which a rate
than
S is energetically is a problem
An abstract
scheme
which
the formalism for such
it
but
sterile,
a property. rate
If-there
is no such
non-zero
rate,
be defined
to have
rate
the desirable
must
be large
duration
enough
requires
to
in a time (during
the calculation
of such
a detailed
for this
of the
P-processes
of the
Clearly
closed).
property value
to a value near zero
(10.8)
is not enough
by the properties
by the given
is determined
the allowed
to P-equilibrium
a finite
relaxation
entropy
bring the relative
has been
is to find
In order
state
to equilibrium
to look
of S will
state
d(p(t)|G(F))
by H.
significant
to P-equilibrium.
the equilibrium energy
seems
which
defined
of relaxation
if it makes
the equilibrium
total
property
to make
of the environment that
of. relaxation
seem
This
The only problem of relaxation
then the function
the evolution
may
be proved.
then
for py = p(B3H), in t under
and realistic
model.
purpose.
109
the
holds
where
there
the WCL
limit
for an open
This
includes
near
equilibrium,
to the coupling
with
11.a
Observations
mentioned
refers
which text
the
to the dynamics
and
entropy
defined
instant
has the drawback
finite Penrose future
quantum [29]
by an
110
of lacking
introduced
an entropy
on the system, the
same
reason
The two entropy
on the
periodic).
as
concepts
at one
property
(for
and
to all
possible
a non-decrease
in the case are
I-
system
Goldstein
related
and having
con-
of S at a given
a non-decrease
function
systems),
coarse-grained
standard
measurement
it is almost
observations
above.
incomplete
already
In the present
time.
of a state
The
systems
erty for basically defined
of S for all
dependence.
its time
instant
(for classical
entropy
in the entropy
lies
to observations
on the system S. We have
made
Kolmogorov-Sinai
interest
and on the
engine
related
are
concepts
entropy
of observations
the
leads
on the system
Information-theoretic or sequences
problem
systems.
of work for microscopic
definition
about
heat
well-informed
on Szilard's
comments
to some
information This
observations.
through
system
of the
the state
our
with
happens
to see what
increase
we
when
with
of entropy
identification
repeated
entropy
the thermodynamic
is due exclusively
bath.
it is interesting
of information,
is always
system which
irreversibility
the
heat
the
of the often
In view lack
and where
irreversibility.
intrinsic
is no
is in the limit
identity
where
The only case
entropy.
information-theoretic
and
there,
as defined
entropy,
the thermodynamic
between
identity
no
is in general
that there
chapters
in the preceding
We have seen
AND WORK
ENTROPY
MEASUREMENTS,
11.
CHAPTER
prop-
of the P-entropy
not directly
related,
however. ables.
The
P-entropy
Furthermore,
(this
leads
discussed
in Chapter
that
of physical
the
is well
described
ensemble
jour.
scopic
information
been
tive
classical
scopic
by Shaw
instrument
final
states
depending
vation
attempt state,
concept?
to define instead
has
There into
Note
that
is meant
general
extensively
will
in terms in the
micro-
initial
macroscopic The
information. macroscopic
an obserchange
mixture,
there
chapter
in-
process
How will
by a macroscopic
of measurements
treated
different
this
micro-
by the
a sensitive
of resolving
measurements
is forced
to non-conserva-
of different
in this
of behav-
detecting
the question:
There
turbulence
of this
microscopic
initial
capable
what
quite
been
develop
poses
the evolution
The description concepts
can
like
referring
but
state.
system
instrument
the by
the motion
level
description
example.
with
type of motion
to macroscopic
mo-
(described
motion
In this
A measuring
contrasts
initial
the
from
be photon
the two types
when
but mainly
on the
on the system,
the entropy
[72],
functions
of observations
follow
in the
of superpositions
appearance
during
can
likely,
A vivid
is an obvious
of the
states
be
of Brownian
observables
indeterministic
is amplified
would
This
between
by the fields.
state
possible
will
different
of a sequence
process.
distinction
dynamics.
events
which
entropy
examples
not contained
of the dynamics. given
is considered
observations
observations
it is indeed
equilibrium
stability
Typical
of macroscopic
macroscopic
and
dynamics
is completely
the outcome
where
no clear
is possible, far from
work.
information
In fact,
which
by a stochastic
averages),
however,
of the observ-
interpretation
and microscopic
behaviour
no new
fixed
values
information-theoretic
instances
deterministic
give
in this
experiments
In these
mean
12).
sight
considered
tion.
has
only one
to a type of experiment
counting
is,
only with
in [29]
to problems
At first refer
deals
be no
will
observation
or
be allowed.
of thermodynamic literature
([19,59
Wal
given
state
property
The
concept.
will
system
of the
result
of a measurement
Pnby
with
probability
S(a3P)
For the
h(p) =-p inp.
there
bound
re-
on the entropy
(2.2) of [19]
Ty PySplo,) < Hey} -
the
with
coincides
states,
to the equilibrium
be applied
can
inequality
P-entropy
by an amount
an upper
is also
is given by equation
This
Sy(o) This
entropy
leads
the measurement
that
implies
P-entropy
Yk PS (oy 3P) a ne
I-entropy
duction.
is
the measurement
with
of the average
=
replacement
Yk Pr =a
of the
to a reduction
the
where
Pp»
Tor te tall pe The concavity
be
The
here.
by the
be represented
then
can
associated
The information
feature
not be an essential
Yk PL Py ’
pas
of disturbing
measurements
of quantum
can
entropy
to the new
due
the modifications
explaining
here,
outline
Only a bare
selection).
small
is just a very
93-97]
for them.
I-entropy
as the
If o € G(F)
and
P = P(F), then S(p3P)
=
Yk PS (py,5P)
(14.1)
@3, fe) dp BS; ope eto It is important
to realize
0 is a non-equilibrium
state.
is due to the possibility
definition states
are
there
chosen
for the
not
need
of optimizing Pp and
be such
a bound
in the average
The decrease
(10.2) for each Py separately.
that the difference
112
that
the
P-processes
If different
P is a small
may be much larger than
set,
I{p,}.
then
if
P-entropy in the
equilbrium it is clear
11.b
Information
The average implies
and entropy
entropy
decrease
the possibility
librium.
If o = o(8,H),
P-cycles
of origin
we consider
of obtaining then
through work
the state
H, but the
S as an open
then the average
obtained
states
system
out of a system
is passive
p, are
with
not,
in contact
available work
the observation
also
in equi-
respect
in general.
with
a heat
to
If
bath
R(f),
is given by (7.7):
J, PpACO,3H) = B'S, p,S(oy|o(BsH)) < 8'I{p,) where, with the notation
of Chapter 7 (compare
(10.8))
0 < S(p|p(B,H)) = S(B.H) - Sp) +B[(H) - E(B>H)] < S;(e]e(B5H)). The
idea
course
that
originally
tempted
to refute
crease
between
was
given
The validity on the choice Usually ment
will
I must
and
be summed
he hypothesized that
stated
function
above
for an
must
and this
for completeness.
as quantum
ijtially
in uncorrelated
in the product
state
for S + M during
the
introducing heat
idealized
Py and
Pos i.e.
the total
gives
(prime
of the denoting
M =
to be in-
assumed
interaction
follows
apparatus
They are
Py ® Po- The conservation
argu-
standard
systems.
states
depend
instrument.
The notation
that of [19]. The system S = (1) and the measuring described
in-
I [19,
obviously
for the measuring
is considered,
up here
(2) are
of
process
than
less
problem,
information
the
at-
have
by an entropy
is not
of this
what was
of authors
number
which
law is of
[95].
of the claim
I-entropy
the second
be accompanied
treatment
by Szilard
beat
by claiming
instrument
entropy
of entropy
only the
when
conjecture
this
The earliest
the relation
could
A large
demon.
in the measuring
93-97].
engine,
due to Maxwell
information
the
obtaining
an observation
Maxwell's
called
later
59,
such
system
is
I-entropy final
113
of S and M, respectively)
states
partial
1,2 the
states,
S,(e4) + Sylo9) = S;(o4o) = Sz(04) + S1 (5) - Cyo (049). correlation
C10 is the logarithmic Theorem
state.
the
S and M in the final
information
given
by the
by C,0!
is bounded
measurement
that
says
2 of [19]
between
I{pp} < Cyolo49)Let
(k) of probability
reading
strument
state
partial
Pre be the final
to the
of S corresponding
in-
p,. Then
04 = bk PReIK ? reads
(11.1)
and the inequality
0 < S;(p4) - wy P,Sy (Pap) < Tipp}.
S1 (04)
but there
initial
does
this
measurement
measurement
classical
ideal
For an
7
S, (94)
is still
state
not hold
ideal
Pp, = Py. For an
in general,
quantum
and
2 Qs
an average
(equation
entropy
reduction
relative
to the
(7.2) of [19])
0 < Sy(o4) - Ly PySz(O44) = cz S1 (04)
a
Yk Py,Sy (P44)
as S1(p5)
o
S1 (po)
=
Cy5(049)
< I{py}< Sy (p5) - S1(p5). We can
also write
the last
inequality
as
(11.2)
S1(9,) + Syl) < dy PpS7(P4,) + Sylos)Thus
the non-decrease
get the correlations
114
of the total between
entropy
S and M and
remains instead
valid replace
if we forthe final
state
I-entropy
of S by a weighted
average
over
the observed
sub-
ensembles. Consider with
respect
exchange
3.c
the
special
case
where
to the unperturbed
with
Hamiltonian
M is insignificant,
, especially
equations
Py is an equilibrium
then,
H of S.
state
If the energy
by the argument
of Chapter
(3.10),(3.15),
S1(p4)< S04)
If M is also
initially
in an equilibrium
state,
then
S1 (65) =
= S(p5) and there can be no information gain in the interaction a measurement
of S and M. Consequently, with which
exchange
energy
no significant
is in a non-equilibrium
10, which
of Chapter makes
the
alone
state.
an
measure
can
I-entropy-increase
and the
I-entropy
of the preceding
interpretation
Can the
point.
the
state
instrument
a measuring
needs
sensitive
between
of identity
lack
The
on an equilibrium
intrinsic analysis
P-entropy
irreversibility, a debatable
in M
iia3)
S1(05) - Sy(ey) > I{pyts compensating dynamic
quantity,
problem
can
signifying
only be resolved on S and
processes
acting
specially
for preparing the
an
in S, be considered
by extending
similar
interaction
the set of processes
P acting
in M? This
the concepts
of Chapter
will
then
P,-processes
the sensitive
be a set
acting
non-equilibrium
S - M must
as a thermo-
evolution
irreversible
system S + M. There
10 to the combined
In addition,
reduction
the entropy
be taken
into
P, of
on M, esstates
of M.
account
in
on S + M, and evidently
P + P, ® Po.
115
for S + M will
have
all dynamical
maps generated
by P, including
for
property
the non-decrease
The P-entropy
inter-
the measurement
will
in general
not be a sum of contributions
from S and M (as explained
in Chapter
10). Remember
P-entropy
The
action.
them.
The
P,
under
an
an
in general.
This
system
if we can
the system.
As an illustration,
account
tained
in the S - M correlations,
heat baths,
then the P-entropy
(apart from the evolution Consequently,
a picture
mation
contained
an additivity Markov
of separate into
account
descriptions
only
equilibrium
the
states.
M to one
of the
interaction
is simply
and the
not possible.
states
separately,
in M + be R(B). to a P-entropy
contributions
system
con-
Thus there would be an ir-
if the partial
for each
into
to the value S(,3P4)
to the evolution
in the correlations,
is valid
in (uncorrelated) 116
takes
by coupling
e.g.
in S itself).
in S due completely
takes
information
the
of S increases
reversibility
for S + M, which
in a way which
of M obliterates
evolution
of
P-entropies
the
on
of (11.2)
in the fashion
the correlations
If the subsequent
try to define
interaction,
the
after
S and M separately
observations
through
the correlations
exploit
by the
but correlated
for a non-interacting
is true
The same
P-processes.
not be exploited
can
correla-
creates
interaction
if the which
parts,
different
between
seen
is easily
in
to a part of the system,
not be localized
can
system
point
irreversibility
The
way:
in a different
important
is an
P-entropy
of the
be expressed
can
interacting
tions
not be non-decreasing
interaction.
The non-additivity which
it need
10, but
of Chapter
in the Proposition
of (5)
under the assumptions
is additive
-entropy 1 @ P Pa
between
or correlations
interactions
involve
not
if it does
M, even
S and
between
not be divided
can
process
in a specific
P-entropy
in the
increase
the
reason
For this
states.
for uncorrelated
also
that this holds
inforSuch
of S and M give
or if S and M are
An additional
argument
of the difference irreversible in terms
(11.3)
scopic
system
features
of the action
assume
bound
an upper
on
reversible.
of M as a macroto macroscopicthe abstract
In fact,
on the choice
of in-
of Pa. However, not
on S does
of the measurement
the level
an
of S + M
in S and M. The amount
of description
of M, only a observ-
are
S - M correlations
the relevant
that
saying
bound
lower
introduced
in M depends
the description
signifying
and essentially
instrument.
symmetric
irreversibility
trinsic
interpretation
quantity
of M corresponding
of the
readings
is essentially
formalism
has been
the
of the evolution
is microscopic
or of the states
ally different
against
The description
I-entropy
No distinguishing
be made
as a thermodynamic
evolution.
of the
can
able.
always found to be larger than must
make
always
to create
the
must
This
state
non-decrease be taken
of a Markov
the measurement.
argu-
the preceding
description
for the appli-
of the measurement
can
be obtained
if we
state
account
The
S and M interact
In the case
of S and leave
of the properly
a term at least equal
also
concept.
maps
induced
defined
total
in a formal
way
can
no longer
ourof
as an operation
by the entropy
through
to I{p,} to the average information
restrict
out a description
considered
is then
of the dynamical
into
from
to be drawn
of M. The measurement level
must
for a measurement.
necessary
description
to the partial
on how to assign
assumption
in which
of the evolution
conclusion
a Markov
on S at the The
systems.
of the entropy
cability
selves
to the two
the necessity
is again
process
implicit)
the correlations
general
The ments
(perhaps
the reversal
prohibit
that such models
I{pyt. It seems
an assumption
changes
the entropy
is
increase
entropy
The average
[96,97].
level
a phenomenological
in M on
production
the entropy
and calculated
for the apparatus
mechanisms
realistic
or less
more
described
have
Many authors
P-processes. of S + M
the addition
of
entropy of S after disappear
through
117
Instead
in M.
the evolution
after
The entropy
per definition.
data,
such
permanent
are
results
the measurement
then
can
a measurement
be defined as (P = P,)
FLp,S(ojsP) + h(p,)]. This
PM
M. A new
combination
available
measure-
(at t = 0, say), the PM entropy of the state at t = O- can be as
defined
expression
case
of repeated
will
clearly
with
equality
11.c
for equilibrium
Exchange
works
of view
above.
a completely
taken that
of muscle
may
the
without
in spite
context.
of the transfer
be formulated.
on the with
at variance
This
of work
difficulty
level of work
was
of treating [100]. between
state the
described experiments
in the gedanken
the problem
on a molecular
information
the notion
engine"
heat
of the difficulties
point
is to define with
any
and
entropy
between
"well-informed
is obviously
the crucial
microscopic
how a definition systems
here,
in connection
by McClare
property and S,(e) 0 If there
of work from S to M, and Q the heat transfer.
the transfer
Q = 0, i.e.
there
W = ae” : Work the
recieving
is no total
is energy system.
entropy
the asymmetry
If
in M + R(Bg)» then
increase
without
transfer
Note
W
p4 = Po, then W < 0. We call
i.e.
is no interaction,
an entropy between
in
increase
S and M in the
definition.
tion
At first
sight
of work.
If S is in an equilibrium
one would expect, cycles
of origin
to M. However,
120
there
is a possible
objection state
to this
defini-
Py = p(B.H,).
then
from the passivity of Py with respect to P,Hy that
there
could
with the suggested
be no work
definition
transfer
from S
this does not hold.
In the limit where S(p53P.) (3.15)
= S1 (95) it follows
from
(3.10) and
that
M
M_
BAE
is AS}
we
ByQ.
We aeM(1 - BB)»
the
when
happens the
of the
ing case
by the
provided
the
case
heat limit
ideal
engine. of an
the correlations
interaction
is seen expect
we must
remain
in the energy to find
M acts
while
is
section. as an
insignificant
process,
reversible
slow,
infinitely
where
processes,
of the Carnot
I-
case
of the preceding
process
limit-
in the
special
Another
of the world.
for
is excluded
no changes
are
there
where
of
stage
of F is a special
F. The action
oc-
that
S and M. This
on M, a thing which
measurement
Then
interaction
general
system
interaction
consists
of the
effect
back
of S and the rest
entropies
ideal
react
classical
strictly
A third
S can
at some
between
of correlations
the creation
the process,
than
process
It involves,
act on S.
the fields
when
that
in S is due to the fact
general
of S and M is a more
interaction
curring
of work
By: The availability
B
and this
of irreversibility
for a closed
plausibly
function
on
bounds does
be not
system.
123
[101].
It seems
to restore
the
on thermodynamics disposal
at our
ceedingly
time-homogenous
dynamics.
by the Brussels
group
increase
that
the decay
system
is irreversible,
as the
matched
by the entropy
increase
that
below sented tems
such
here
does
showing
compensating the treatment First, work.
a picture
a perfect entropy
a general
is obtained
sets
of thermalized
changed
124
with
remark
through
state
in the apparatus.
It will
be argued
and
the heat
baths
given
(heat
between
entropy
between
baths)
parameters.
is called
heat,
that
with
in [107].
of the environment
of freedom
sys-
for a
is consistent
the distinction
by a set of macroscopic
to spin
is no need
view
on the relation
a partition
degrees
there
phenomenon
pre-
The formalism
irreversibility
in F. This
thermodynamics
work
initial
effect,
increase
is
of the
restoration
a genuine
of the spin echo
In classical
F described
echo
leads spin
in the
of the polarization
is not well-founded.
not ascribe
F and performs
This assumption
14.1-3 of [102]).
the spin flips (see figures to the conclusion
the fields
generates
which
in the apparatus
entropy
a compensating
by introducing
be circumvented
must
(12.1)
experiments,
spin echo
e.g.
with
In dealing
an approach.
in such
has to be paid
price
A certain
[102-106].
pursued
been
has
which
is a program
This
specific
one
only from
S* starting
a function
to construct
tempt
facing any at-
the difficulty
underlines
(12.1)
The condition
not
will
much.
very
of the entropy
the value
change
necessarily
of magnitude
by orders
this
Increasing
small.
to be ex-
by F is likely
given
of the mobility
the control
systems
large
for
that
however,
be noted,
It should
technology.
in our
of the art
state
on the
depend
must
of irreversibility
the concept
that
namely
is different,
here
taken
of view
The point
by nature.
given
something
as
state
initial
the means
considered
Planck
that
text
in his well-known
given by Planck
of irreversibility
nition
the defi-
with
above
the argument
to compare
It is interesting
and
heat
and
of S into
and a system
The energy exchanged
exwith
F
entropy ous
with
One
thermodynamics. erty
are
world
No S - F correlations
allowed.
tion
of F on S is to perform
echo
example
The
and many
group
Boltzmann,
having have
point others, not
does
of reversing
the
different
been
unitary
in principle,
is that
include
accessible thing
but
evolution ago,
going
inverted.
back
to
we may have
way the means
and the possibility when
the evolution
it is hardly
spin-
by the Brussels
chosen
equilibrium
a century
In the
be instantaneously
the approach
final
and the ac-
created
are
transformations.
in a natural
of influencing
agents,
S and the external
has a long tradition
which
"irreversible"
a natural
no possibility by external
can,
these
crucial
between
correlations
when
no sense
makes
states
described
of the equilibrium
the passivity
that
there
We saw
11.
in Chapter
to distin-
of the type
of F on S from measurements
the action
guish
discuss-
When
necessary
in S, it is of course
changes
ing the entropy
by the
allowed
limit.
any obvious
without
increase
law would
second
second
the
from
follows
set of processes
the
to S + F. In fact,
prop-
the passivity
be that
longer
no
states
of the equilibrium
law applied
would
consequence
of classical
the foundation
destroying
without
importance
mental
has any funda-
increase
an entropy
such
that
to accept
impossible
It seems
experiment.
in the spin-echo
of the fields
the action
increase
an entropy
if we associate
is abandoned
the assumption
However,
it.
without
to get anywhere
be impossible
it would
Indeed,
entropy.
of the thermodynamic
for the definition
it is the basis
and
formalism,
in the present
over
taken
was
assumption
This
the work cycles.
which define
in the parameters
with the changes
in F associated
change
is no entropy
there
that
assumption
but obvi-
is a tacit
there
Thus,
baths.
in the heat
change
by the
in S is compensated
change
the entropy
processes
reversible
that for
the condition
for S through
function
equilibrium entropy
the
defines
and heat
into work
decomposition
This
work.
is called
states. there
This was
Even
may
essentially
of relaxation
so today.
of
processes
the character-
responding
information-theoretic
entropy.
The most
rigorous
result
(for classical
systems)
is that
of Penrose
and
Their
construction
direction
in this
This
coarsegrained tion
continuous
measures
are the same asymptotically
laxation
to equilibrium
equilibrium sponding
of infinite
states
K-mixing
property,
recurrences,
would
assumption
of the
associated
with
formalism,
if it could
The
Prigogine
namics
of K-systems
or other
seems
to disagree
ever,
I will
ical
systems
The problem some
126
more
have
that
involves is then
clearly
also
[103-106].
intrinsic
with
that
the application
an approximation transferred
understanding
the
two
is reof
is a corre-
The
strong
problems
the present
simplify
for the
very
all
removes
to Hamiltonian
approximation
completely
argue
detailed
an
there
[108]).
which
is basic
and coworkers
Misra,
coarse-graining
be applied
property
K-mixing
where
to equilibrium
approach
spatial
systems,
To-
perturbations
local
the
state.
there
i.e.
(in time),
with
(compare
on the
that
implies
this
of K-mixing,
the assumption
gether with
assump-
is absolutely
state
by the equilibrium
defined
to that
relative
set of all
by the
the measure
initial
by a non-equilibrium
generated
algebra
is that
of the entropy
in the definition
process.
A crucial
on the system.
observations
future
generated
an algebra
by introducing
is done
be seen
can
of the observational
description
a Markov
as providing
[29].
mentioned
already
Goldstein
and a cor-
system
on the
observations
repeated
involving
periments
ex-
consider
can
one
dynamics,
time-homogenous
Using a fixed
Hamiltonian.
a time-variable
demands
which
interpretation,
dynamic
thermo-
its obvious
with
of passivity,
condition
natural
the very
having
a problem without
becomes
states
of the equilibrium
ization
systems. recent
They claim that
This
of the present of K-mixing
the dy-
not due
irreversibility procedures.
work.
models
to that of justifying
to
philosophy
of a coarse-graining
of the physics.
by
work
Howto physnature.
the model
by
For abstract
value:
With the notation a
- S = Inu(Q)
dis-
the dynamics equilibrium
a microcanonical
can
distribution
the equilibrium
from
of the entropy
the deviation
represent
then
is not
probability
representing
by the probability
defined
I-entropy
The
state.
of energy
stationary
of as with
energy
of constant
on a surface
the concept
a unique
be thought
instead
can
tribution
systems
with
systems
Ergodic
defined.
dynamical
of [29]
- |du olno
where fdu = u(2), fdu p = 1. Without the introduction of phase cells
space
which
systems,
mical
value
the physical scribed
in a way which
K-shift
T (for a K-flow
(h(T) > 0), then of ergodic
sense
assigned
KS entropy
value
in the
irreversible
but the corresponding
theory,
be pre-
T = T,) has a positive
is certainly
this dynamics
physical
of
rate
h(T)Sog. and hence on the
of Seq’
Due to the finite amount
Ty » take
must
If a given
must depend on the quantity
relaxation
entropy
to the physics.
the model
relates
Consequently,
dimension.
Seq of the equilibrium
size
of undefined
systems
of physical
of constants
to the lack
dyna-
of abstract
true
more
physical
represent
1tonHami
of a classical
entropy
is even
This
is undefined.
jan system
due
thermodynamic
the equilibrium
dom)
of free-
of degrees
N is the number
fh, where
(volume
of information
value that
of Seq of a physical
can
system,
by observing
be obtained
the the system
is limited:
provided
that
perturbations follows
system
from
contained
to be closed
is considered
due to quantum
directly
information
sponding
the
measurements
the definition
of the
in n repetitions
to a generating
can
and that
be neglected.
KS entropy
of an observation
partition must satisfy:
that
the It the
corre-
n-h(T) Ae
hs
127
The
Eats Segh(T)
at
(1222)
to a manageable
If h(T) corresponds system,
span.
with
the rate that
showed
such
of the Goldstein-
order
as Seq? in or-
of magnitude
increase
of the entropy.
the use of the asymptotic
K-mixing
property according
of Seq is so enormous tween
information
the
thermodynamic of thermal
that
fluctuations,
is due to the fact
that
and the
change
due to
gain by (11.1).
to the information
the a priori
be-
by the observation
entropy
the average
where
is related
on a system
is provided
An exception
values
no relation
in general,
in measurements
obtained
entropy.
the measurement
is,
there
to (izetee
to macroscopic
corresponding
of information
The amount
contradicts
This
a significant
der to have
[109]).
enough
is fine
should be valid for a time t
the model
is of the same
t-h(T)
that
(Goldstein
entropy
of increase
partition
the chosen
when
interpretation
With this
rate
it is the asymptotic
h(T)
to identify
attempt
of the thermodynamic
of increase
entropy
Penrose
may
one
hand,
On the other
time
of a
an enormous
provides
(12.2)
the limit
then
from
amount of information
if Seq is the entropy
and
system,
of the
observations
macroscopic
of quantum
not be neglected.
can
measurements
real
of the RHS the effects
of t of the order
For values
asymp-
K-mixing
of genuine
by a h(T) > 0) only if
(measured
behaviour
totic
the lack
not see
We will
model:
K-mixing
of the
on the validity
a limit
gives
then
I, Ae
inequality
distribution
This
for the outcomes
of the measurement is given by the equilibrium state. When the system is forced far from equilibrium (and the a priori distribution is different), served. value
In fact, far from
is considerable, ment
128
the deviation
from
if a macroscopic
the equilibrium
even
is insignificant
equilibrium quantity
one,
we will
if the information
can
is observed conclude
content
on the thermodynamic
easily
scale.
be ob-
to have
that
a
Seq - S
of the measureThe
increase
of
the entropy with
after
no obvious
It should
such
relation
be noted
also a transient From models
an observation to the
that
KS entropy
entropy
production arguments
be justified
for all
this
limit
corresponds
interpretation
values
amount This
to a kind
is that
of information
restriction
equilibrium lines
in Chapter
subjective
function
on
dependence are
was
of description,
ditions
tion
physical
give an
scale
of Seq°
only departures system
the
advocated
from
on the
and idea
each
to the
point
corresponding
by Grad
[111].
sensitive
of a authors.
performed entropy
"The
and
of operations
of this
nature
The precise The
intrin-
by many
[110]:
by Bridgman
on the universe
measuring
that
of view
He ascribed on
there
level
to a certain
dependence
the ob-
initial
con-
The maximum
of statistical
of an ensemble
containing
inference. the
setting
degree
entropy
developed
consistently
most
theory
the experimenters
system.
as a form
idea was
An information
[7,112].
of the
sented
was
however.
taken
entropy
subjective
by Jaynes entropy
only
(see Chapter 8).
The
state
also
on the
states,
changes".
universe
irreversibility
served
depend
must
functions, was
The
system
on the type of experiments
not described,
entropy
many
entropy
expressed
clearly the
when
change
must
K-mixing
Obviously
a non-trivial
for non-equilibrium
therefore
in general
on the
of defining
of the entropy
system was
the
the
7.
or anthropomorphic
The dependence
describe
parameter
of freedom of a large
Due to the difficulties sic entropy
that
of coarse-graining.
if we consider
in a few degrees
can
(h > 0) to be taken.
is infinitesimal
is fulfilled
described
clear
of the time
the observations which
entropy
model.
much larger than h(T). it seems
if we can allow the limit Seq +o
phenomenon,
of a K-mixing
the Goldstein-Penrose
the preceding
can
is a transient
is used,
of ignorance
formalism It allows
information
that
with
the
of the
is thus
pre-
the constructhe expecta-
129
tem, time
system
as
relevant
less
becomes
passes.
procedure
preparation
procedure
preparation
thus
as we
like,
hand,
we can
a given
entropy
(10.8)
to p with
the
state.
vanishes. to all
respect
in principle, I-entropy
the predictions
state
to an equilibrium
I-entropy.
the
we can,
obtaining
consider
from
can make
is uniquely
which
an entropy
state,
of a reproducible
Let
0(p,H) Assume
sense
in the
there
that
P-processes,
then
this P(F)
choose
the future which
that
defines
for any
In fact,
p be a state
in the
contained
in the limit.
about
I-
as maximum
information
the
If we consider
functions.
entropy
the
that
It seems
not be interpreted
12 can
of Chapter
P-entropies
approach
This
processes.
to pro-
be used
it can
thermodynamics.
with
contact
a direct
also makes
set of work
a given
through
duce work
is that
relevance
This
system.
of the
in the
contained
information
a
of having
has the advantage
for the
of relevance
notion
definite
here
presented
The formalism
state
of the
in the past history
tained
sys-
con-
information
the
that
statement
for the general
except
any
in a closed
of the entropy
increase
for the
mechanism
specific
to
provide
not
does
formalism
entropy
the maximum
However,
(7.3).
of the I-
similar
is quite
which
experiments
in reproducible
entropy
fact.
this
with
on the increase
a statement
law then becomes
The second
in-
at certain
values
consistent
I-entropy
the largest
and having
stants
given
have
of observables
of a number
tions
as
given
large
On the other which
we
is P-equivalent
the relative
is a state
P-
o' equivalent
i.e.
Wyse") -=-Wly.e), 5 al yc PE, and furthermore
that
o'(H) = p(H),
Sp") = Steck):
Then we find that S,(0') = §(8,H), follows
130
that
and from (3.4) and (3.5)
it
p' = p(B,H). But
it was
give
the
obtain same
indicated
same
in Chapter
values
of the work
a contradiction.
predictions
10.c
as
This
o will
that
for all
means
p and
p(®,H)
P-processes,
that
a state
need
not
and thus
o' which
we
gives
the
satisfy
S1(p') < S(p3P), in general.
we may
methods
are
mentioned
in Chapter
On a more can
may
reader
set out
in Chapter
and that
ter
level,
the concept
noticed
have
forces
to external
those
of Meixner
[50]
[114].
a similarity
quite
The entropy
closely
to the
processes
of thermodynamic
10 and’ the thermodynamics
of Day
10 correspond
(page 40).
like
in the contextof rational
erudite
[113]
of the system
6.
formal
be developed
the pro-
into account
take
models,
on macroscopic
based
those
response
by the
generated
The only treatments
the motions.
which
thermodynamics
of irreversible cesses
of reversing
have
the
not consider
does
formalism
entropy
the maximum
Again,
thermodynamics. between
the formalism
of Coleman
functions
and Owen
defined
entropy"
"largest
The
in Chap-
of [113]
If we write
s(o,¥) = - lp as® (oy) m(p sq) ="Supistp.r)s r€ PIs where
the supremum
a reference
equilibrium
to that of equations is given
is over all processes state
Pg» then
mapping this
the state
notation
(5.6) and (8.6) of [113],
into
corresponds
and the P-entropy
by
S(p3P) = S(pq) - m(9509)-
U3
rational
In the the
Instead,
scheme
irreversibility
has to be postulated.
entropy
have
functions
idea
to the
to give a sense
dynamics
no proper
there
approach
thermodynamics of each
of irreversibility.
thermodynamic
For the same reason,
to be prescribed
is, of course,
in order
in the
process
the energy and the
to define
is then of a purely phenomenological nature. In cononian trast to this situation, the statistical mechanics of Hamilt
model, which systems
a direct
provides
functions
for equilibrium
formalism
this
relation
of the available of Hamiltonian
work, systems
roles
concepts
of energy
through
is extended
which
makes
and work
and entropy
In the present
(3.7). states
to all
leading to (10.5).
of the Hamiltonian
multiple
132
states
the energy
between
relation
through
The fundamental
thermodynamics in defining
and the entropy
the use
property
possible
the dynamics,
of equilibrium
is the the states.
APPENDIX
A.
A short
QUANTUM MARKOV
introduction
jan dynamics
is given
in open
a thermodynamic
PROCESSES to the topic
quantum
systems.
interpretation
of irreversible
The problems
Markov-
associated
of this type of dynamics
with
are de-
scribed. A.1
Reduced
Consider
a small
The dynamics family
dynamics system
of S + R defined
of dynamical
itial
state
S interacting
maps
with
a large
by a Hamiltonian
for S, all
depending
reservoir
Hou generates on the assumed
a in-
of R:
T(t)[X] = Trp epU(t)” LX STUCEY IP ey where
R.
(A.1)
X € B(H.), U(t) = exp[ -F tHe pl. These maps have a strong
positivity
property
called
complete
positivity
(CP for short)
+ bag Yq+ TEXGXs] Ys 20>
for all lated
{X;5Y; € B(Ho) }4 all n. This property can also be formu-
in the following way:
of nxn
(A.2)
matrices
T defines
with
In the tensor product space B(H) ® My
operator-valued
a map LPs = Te
entries
X =
j © BUMg))
(X,
Id,
(TUX) 43 = TIX, 4]
which can
is positive
be written
(and in fact CP) for all n. Every normal
in the form
iXInsbasVguXailers» where
V. € B(H
CP map
3)
Without
(A.3) the CP property
the dynamical
maps
would
133
containing
system
way to a larger
and physical
on the properties
ground
the reduced
In general
there
will
condition
in S. One necessary not change
significantly
The simplest namical
type
Only with some additional
as a result of reduced
time-homogenous
evolution
the state
of R does
that
of the
dynamics
interaction
T(t){1]
of dy-
of norm continuity
of probability
= 1,
a complete
description
S
be proved
of the generator
L of the semigroup:
= dt 1’) |t-9 : L hid
Tit) = exp(tl), that
L has the form
if i L{X] = L ytx] + -LH,X],
a
(A.5)
+
V. E B(H.), H* = H [13-15].
dissipative However,
134
S.
(A.4)
(A.4) and an assumption
(A.2),
and of conservation
where
with
is a semigroup
OS T(t eT sha
The conditions
It can
is clearly
state
initial
on the chosen
an
provide
maps
Tesuere
allows
be an autonomous
back-
of CP maps.
for S do not
maps
in the past.
instant
of R at a particular assumptions
dynamical
detailed
interpretations
as they depend
description,
interesting
for a more
[13]
preserve
not necessarily
S would
See
of the states.
the positivity
in a trivial
lifted
the dynamics
that
in the sense
be inconsistent,
The decomposition
part and a Hamiltonian
the dissipation,
described
part
of L into
is not unique
by the bilinear
a
in general.
expression
D(X,Y)
= LEXY]
is unique.
- LIX]Y -.X LLY],
The CP property of T implies
that
Gi Yi D(X; 5X5) 520. +
(A.6)
+
L is Hamiltonian
if and only
The evolution
of the
if D = 0.
state
of S expressed
by the dual
maps
T(t)*[o] = p-T(t) is defined
by what is often called a Matkovian master equation
Sed a(t) aed = L*felt)l ,
(A.7)
where L* has the explicit *
TS |
The detailed
do so.
of
in such
A2>0,
The most
tion
t-2,
thorough
= van
between
a way
which
Hove
review
limit)
while
clear
physi-
the
limiting
to a dissipative involves
where
the
S and R and the time
evolution
the weak
strength
parameter
i
t is
that xt
=
treatment of the WCL
been given by Spohn
lead
which,
system S + R is
in time,
procedure
constant.
of the derivation
in the WCL limit was performed
extensive
a very
of the total
a direction
limiting
used
( = WCL
interaction
the
scaled
The most
limit
coupling
not define
and approximations
procedures must
does
and
do not have
As the evolution
is
type of evolution
procedures,
limiting
defined,
well
i
a simple
of such
involves
and
interpretation.
Hamiltonian
+
justification
mathematically
being cal
+
difficult
extremely
form
by E.B.
and similar
of the master
Davies
limiting
[37-39].
procedures
equa-
An has
[40].
135
to the unperturbed
corresponds such
o(B,H) (and so does L*, of course)
(A.8) detacled
called
For all X € B(H)
[24,43,44]:
balance
the relation
Ly satisfies
by L*. Furthermore,
generated
the semigroup
under
is invariant
state
Gibbs
this
state
[24]
Ljto(B.H)] = L*{o(B.H)] = 0. Hence
part,
the Gibbs
Li annihilates
and
commute
parts
the two
that
(which
part
and a dissipative
evolution)
of
the structure
on
a Hamiltonian
into
be decomposed
It can
the generator:
to be given
(para-
state
of R is a KMS
state
reference
information
more
B) allows
meter
the
where
case
The
Lqlx o(BsH)] = LyX] p(B.H). is of a special
the generator
temperature)
When g = 0 (infinite
In (A.5) all the V. can be chosen to be self-adjoint (Vv; = V.) [64,65]. This means that the dissipative part is of the form
form:
(3.11),
which
a stochastic
from
obtained
that
is also
Hamiltonian
perturbation. A.2
Markov
processes
The semigroup
evolution
and the corresponding evolution
equations
equation.
It is well
(Chapman
- Kolmogorov
responding [115]. must
Markov
In order
in classical known,
process
to qualify
to repeated
variable
136
physics, that
as a Markov order
dynamics
process,
conditional
observations
probability
the
n
- Planck
property
to define - Uhlenbeck a random
a corprocess
evolution
probabilities all
derived
(which corresponds
are the values
at t, < ty A halidesSG the then
similar
Fokker
of the system)
{p(x|y)}
in A.1
recalls
the semigroup
the Ornstein
map T): If XporeeoX
X observed
e.g.
does not suffice
as e.g.
higher
reduced
immediately
however,
equation)
from the basic transition to the dynamical
idealized
master equation
have well-defined
(corresponding
of the
of a random
P(X5o++2 9X, [X4) = P(X, |X, _ 4 PUK, 4 [Xpg) + + P(%p 1X4).
is necessary
to introduce
the
system.
A general
was
defined
in [56].
are
maps
where
the P. are
ution
of some
maps there ding
of unitary
Thus, can
there
as describing
The natural
dynamical
Limits
of
operations
on
the CP
of all
Especially,
out as a set of operations.
separation
between
and the unitary
way of generalizing map
the operations
operations
which
of time-dependent
the action
the dynamical
no subset
the classical
resulting
on the
system
as a simple
maps
of the
semigroup
from
time-ordered
evolution
limit-
they are
operations.
act as unitary
is in general
be separated
to measurements
can
maps
hand
com-
under
not closed
are
above
of the
that
from
is different
combinations
of projection-type
on a
the set of operations
which
The expectations
is no absolute
vations
that
to realize
of convex
is to define
resol-
of the spectral
projections
On the other
which
sidered
expectations
are the von Neumann
combination.
states.
positive
completely
are
the operations
is not re-
formalism
the full
and convex
ing cases
some
context
called
often
are
which
maps
of
on S. The observations
of observations
has a structure
case.
commutative
products
the probabilities
observable.
system
position
defines
the orthogonal
It is important quantum
processes
a process
that
E). Examples
(called
on
stochastic
sequence
recall
We just
quired.
instruments
quantum
In the present
operations.
of measuring
by a set of dynamical
represented
it
called
scheme Such
of any
the outcomes
the action
systems,
for quantum
property
an analogous
to define
In order
corresponbe con-
can
fields.
external
property
Markov
a sequence
of obser=
composition
of the
and the operations
TE |e aE.) = T(ty-ty)sEy*T(to-ty)e.--T(ty-ty_4)°E,
(A.9)
137
The associated
=
3)
p(Ey.+++sE,
If we now
of S at
(A.9) has recently been given
for the WCL and a singular
by Diimcke
state
initial
p(T(E,,.--sE,){1]).
of the expression
A justification
the
POS)
hdd
ty < t, {So pes
that
given
probability,
consider
coupling
S in a time-dependent
limit
[116].
field,
then
re-
the
can be regarded as a limiting case of (A.9), where the operations are now unitary. The dynamical maps Til Sat} X(s) + X(t) are defined by the solutions of sulting evolution
ta c+
-
(A.10)
X = L(t)[X]
(t)
iT]
Lg ty (N(t).+1, generator of the form given in (A.5).
where L, is a dissipative A direct have
consequence
the semigroup
Tés gu} e23T
for all choices scribed
7.a:
is that
equation
maps
the dynamical
property
(A.11)
Gog) s
Polt) = Tis star teevedl,”The equivalence proved
of (A.9) and (A.10)
in [57].
let the external
In order
fields
to see
in a well-defined
this
be 6-functions
relation
in an
sense intuitive
is way,
in time
H(t) = Hy + Dy, 6(t - ty) hy Then the solution ations
138
given by
of (A.10)
is of the form (A.9) with the oper-
_
yt
Es
i
T(t) = exp(Ly + #lHy.J)t If we let H(t) be a step function are
intervals
union
on R, then
of such
the evolution
intervals
is again
T(U AL) = T(A,)-T(Ap)*... T(A,)
The
=
exp(L4
defined
a composition
by (A.10)
in a
of maps
TCA ) [A |
[61],
formula
- Trotter
by the Lie
be written
can
case
in this
which
che ATH, 91)
is recovered
case
general
H(t) = H, for t € AY, where A,
T(s4t) = lim, | 9 T(dy)+T(A)e... Tld)s where U A, = cSt).
aie
of the corresponding
(A.11)
property
of the semigroup
A consequence solutions
sup, |A,| ae=
equation
master
the
is that
for the states
(A.12)
o(t) = Lalo(t)] = pIH(t) .o(t)]
or
satisfy relative time:
For any two
a type of H-theorem: entropy
For all
of the time-evolved
initial
states
states
is non-decreasing
directly from is that
interpretation states otonic.
apart
in
s where Lo annihilates
that
it will
but
general,
through
other
invariant
state
is most
which
state,
in
p(p,H),
is
fact
in Chapter
4. A consequence
of this
defined
for an arbitrary
time-dependent
system
the dynamical
Hamiltonian
some
have
as argued
unique
Ly (in line with
(A.8) with the new Gibbs
not satisfy
(A.12)) will often
to H + Ho but keeping
the Hamiltonian
changing
by
obtained
p(BsHy)> by (A.8). The generator
will
(A.12)
not
have
the passivity
property.
that if the work performed by S + R is defined by (2.7), where po(t) js the solution of (A.12), then W(y,0) > 0 for suitable choices of H and y with O(y) = H, when po is the invariant state This means
for the value show
this
in the general
the invariant the
statement
calculation. tions,
140
H of the Hamiltonian.
which
case,
as
there
is no explicit
state for H + Ho: It is sufficient, but typical
in a special The simplest are
example
equivalent
difficult
quite
It seems
case
is provided
to a Markovian
formula
however,
through
to for
to prove
an explicit
by the Bloch
equa-
equation
in a
master
two-dimensional The
metric
Bloch
Hilbert
space.
equations
with
dissipative
homogenous
part) are as follows
relaxation
(rotation
[59]
o) + gMxH(t)
=-)(M-—
a/o. |=!
sym-
(A.14)
where H is the applied magnetic field, M the magnetic polarization of the sample,
A the
magnetic
and Mo the equilibrium
ternal
ratio
inverse
of the relaxation
magnetization
field Ho: Mo is given by Curie's
where
N is the number
responding
master
of spins,
equation
can
M(t) = Tr(p(t)u), where
(9, 505503) are
pressed
which
formula
be found
from
in the following
Pauli
matrices.
g the gyroin a fixed
ex-
Mo = Ta*nh’BA»
we put equal
H(t) =- weA(t), the
time,
to 1. The cor-
the relations
w= z gho = ro|
The work
can
now
be ex-
way
Wyse) = fat A(t) SHE, Partial
integration
gives,
over
the
interval
(0,t)
Wo= (M(t) - M(0))H(0) + a}nat - iy)+A A(t) = A(0) has been used.
where
Note that H(0) + Ho: Consider
the
type of cycle where
special
ft) = NC) TOY Lola bere iT)
Then
we find
H
Por Oe
kts
that
[ee(mce) ~My) A(t) = Ae] t
=
=
és
=
141
= Fi(0)«(M(x) - (0)) + (1 = eT) Ae(A(0) - Mig). ted The relation between (0) and the stationary value M(0) predic from
from
is obtained
(A.14)
(A.15)
- (M - M,)0? + gMxH(0) = 0, and the
solution
is
where we have used that M(0)> from (A.15). Also note that H
t +0, in such a way that gt|H| = © and
Now let || +, es
Then we can neglect the dissipation
\al {a are constant.
in
the calculation of M(t): M, (0) + M,(0) cosp + M(0)xe sing
M(t)
M, (0) = M(0) - M, (0); and in the limit
where M, (0) = (M(0)-e)e, we find
H(0)-M)(1
=-
- cosp) + (M(0)+e)(A(0)-e)(1 “~
- cosy)
+ F(0)-M(0)xé sing + g” 'rpe-(M(0) - fig) This
not negative
is clearly
expression
definite.
In fact,
choose
A(O) = cMy (c < 0), é-Mp = 0, m= 7, to find that M(0) = Mo and Consequently,
W = 2\H(0)-Mp|. As the
rate
of relaxation
is determined which
tude
142
can
to the stationary
by \, it is evident
be extracted
of \W.
W can be made as-large
This
from
is also
that
the system
the order
state
the average
as we please.
defined
of the
rate
of work
rate
is of the order
by M(0)
of magni-
of energy
dissipa-
pation
ancy
in (A.14).
is small
Hence,
for work
sible
fails
cycles
cycle with
non-passivity
fail
if the absolute
size
of this
discrep-
in this system of a single spin -5 particle,
true that the model that
even
one
always
to be passive
in a maximal
of arbitrary
origin
of the form
described
above
holds.
note
the preceding
Also
we can
way.
that
combine
1h41s Note
a rever-
to show that
the
arguments
if B = 0 (or fh = 0), as it then follows that Mo ="0, For
8 > 0 the conclusion
H(t), and clearly
is that
(A.15)
(A.14)
is not valid
for arbitrary
does not give the correct
stationary
states for H + Ho: It may be of some a different
type
thermodynamics
Non-KMS
The
property
above
refers
[117].
is seen the
from
exact
functions
that
fact
reduced
the
dynamics,
here.
is zero.
but
Redfield
the Bloch
discovered
equations
an experimental
deviation
of quantum of the
Therefore
system
that even
the
the time-homogenous
of thermal
unless
result which
was was
a proof of this
function
must
p is assumed
to be invariant.
Making
dynamics
shown
This
hold
the dissipative
for
part
by Talkner
quite result
defined
forces.
by the correlation
for the unitary
system S + R is conventionally
treated
equilibrium. which
not be satisfied
dynamics
dynamics
to time-dependent
KMS property,
can
This
Markovian
different
in his from
is included
here.
evolution
of the
as follows
r(X,¥st) = o(U(t)*XU(t)Y) = p(XU(t)YU(t)*), where
and
processes
in a formalism
The autocorrelation closed
found
properties
that
of the Markovian
[118],
used
to observe
with
of the generator thesis
of Markov
to the response
the
between
He actually
of non-passivity
It is interesting
that
based on (A.14).
property
is inconsistent
to recall
of inconsistency
from the predictions A.4
interest
(A.16)
the replacements
143
the reduced
dynamics
for p through
the autocorrelation
function
for S should
and defining that
bp
pI
YsY@I,
X>Xe@l,
following
in the
it is again
for t < 0,
that
assumed
(A.17)
for t > 0
= p(T(t)[X]Y),
o(XT(-t)[Y]),
reduced
be written
way
r(X,Y;t)
where
(A.1), we find
the
state
js invariant
under
the
dynamics
Tit) Lol. =-0°1(t)s=-00.s0al The function
Yk
(A.16)
has the
Lt? 0.
positive
semi-definite
property
(A.18)
r(Xy 4X45 ty - t,) > 0 +
for all {X, € B(H), ty € R};> all n. Note that + \+ ) r(X,Y¥3t)” a r(Y> jk gt).
up If the reduced dynamics given by T(t) has the CP and semigro The properties, then the property (A.18) holds also for (A.17).
proof is rather straightforward. As a matrix
Order the t,: t, >to > «+2 the
in B(H.) eM,
y where X l,m = T(t, - t,UXy], 1
co follows
when
quantity
of this
behaviour
The asymptotic
rectly from the expression for r(t): lim where
a) tT! AS(t)
to
S
This
of «.
distribution.
tributions
will
A closely systems
namical
[121].
Kifer
give
the same
related
asymptotic
sensitivity
He compared
hyperbolicity
condition
is
which
is due to
I-entropy
Note the the assumption as all
initial
dis-
general
dy-
behaviour. of more
manifolds
the deterministic
random paths obtained when a small some
of the
property
Riemannian
on compact
production
is not essential
state
initial
growth
is thus
of A. There
eigenvalues
of I-entropy
unlimited
the lack of an equilibrium of a Gaussian
>
rate
constant
an asymptotically
r;
positive
all
is over
the sum
independent
=
was
trajectories
diffusion
the probability
by
discussed
and the Under
term is added.
of the diffusion
path
remaining within a small 6 of the unperturbed trajectory during >. The exponent is the sum of (0,t) is exponentially small as t the positive Liapounov exponents is again independent
of e. It is also interesting
Pesin's
formula
relating
in this
context
[74].
152
(when these are defined)
and it
to recall
the KS entropy and the Liapounov exponents
We see
that
the hyperbolic
systems
have
a
sensitivity
to perturbations
sic properties
(Liapounov
which
exponents
in terms
or KS entropy)
of intrin-
and which
is
(in an asymptotic
of the perturbations
of the amplitude
independent
is expressed
sense). The
limit
the
during
by the noise I-entropy:
For some
interval
the perturbation
The opposite
system
the
that
The condition
mitted.
is that where
system
and a closed
be neglected
can
diffusion
above
of the system.
the behaviour
nates
the
treated
(0,t)
is not can
domi-
is where the
limit
description
significantly
is peraffected
in terms
be written
of
6 > 0
Bolt).< 6 » From
(B.5)
follows
2 eTrig
-|
that
it is sufficient
(B.6)
6 > 0 this
is also
the spectral
resolution
of i:
-
=
that
26.
-r(t)]
(1 - exp( 4 2t|51))] h is a weighted average
of the positive eigenvalues
of the type
153
t and
of
function
is a non-decreasing
k(t)
formula.
in Pesin's
occuring
it satisfies
, lim,
9 (inet)
=]
big | k(t) ay sae
fall
: Vim, peg KKt) =c9 If (B.6)
5 dy p;In(2|A,|) Kec
Ne
weaker
the following
then
is satisfied,
ms
bound
hold
must
TOYS
eo