Non-equilibrium entropy and irreversibility 9789027716408, 9027716404

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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

at?

et?

1G

%)

a

pelt jp Tek!

eal

i

ind.

PS S

298 -

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eve

© a"

oF 7

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ARE

ep ape

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ee IPs,

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det

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iad hy yee gest

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:

Se |

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——

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er

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r

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om

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aed

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ary Sd

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:

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re is. » Soreness i ji. +

eee

Ciboe

‘ ben es"

ae

gs

|

.

maa

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