Table Of ContentProgress in Mathematical Physics
(cid:86)(cid:111)(cid:108)(cid:117)(cid:109)(cid:101)(cid:54)(cid:51)
Editors-in-Chief
AnneBoutetdeMonvel,UniversitéParisVIIDenis Diderot, France
Gerald Kaiser, Center forSignalsand Waves,Austin,TX, USA
Editorial Board
C. Berenstein, UniversityofMaryland,CollegePark, USA
SirM.Berry, UniversityofBristol, UK
P. Blanchard, University of Bielefeld, Germany
M.Eastwood,UniversityofAdelaide, Australia
A.S. Fokas,University of Cambridge, UK
(cid:70)(cid:46)(cid:87)(cid:46)(cid:32)(cid:72)(cid:101)(cid:104)(cid:108)(cid:44) (cid:85)(cid:110)(cid:105)(cid:118)(cid:101)(cid:114)(cid:115)(cid:105)(cid:116)y(cid:32)(cid:111)(cid:102)(cid:32)(cid:67)(cid:111)(cid:108)(cid:111)(cid:103)(cid:110)(cid:101)(cid:44)(cid:32)(cid:71)(cid:101)(cid:114)(cid:109)(cid:97)(cid:110)(cid:121)
(cid:85)(cid:110)(cid:105)(cid:118)(cid:101)(cid:114)(cid:115)(cid:105)(cid:116)y(cid:32)(cid:111)(cid:102)(cid:32)(cid:77)(cid:105)(cid:115)(cid:115)(cid:111)(cid:117)(cid:114)(cid:105)(cid:44)(cid:32)(cid:67)(cid:111)(cid:108)(cid:117)(cid:109)(cid:98)(cid:105)(cid:97)(cid:44)(cid:32)(cid:85)(cid:83)(cid:65)
D. Sternheimer, Université de Bourgogne, Dijon, France
C. Tracy,UniversityofCalifornia,Davis, USA
Forfurthervolumes:
h ttp://www.birkhauser-science.com/series/4813
Bertrand Duplantier
Editor
Time
Poincaré Seminar 2010
Editor
Bertrand Duplantier
CEA Saclay
Service de Physique Théorique
Gif-sur-Yvette Cedex
France
ISBN 978-3-0348-0358-8 ISBN 978-3-0348-0359-5 (eBook)
DOI 10.1007/978-3-0348-0359-5
Springer Basel Heidelberg New York Dordrecht London
Library of Congress Control Number: 2013932223
Mathematics Subject Classification (2000): 37A35, 37A60, 37N20, 81V80, 82-02, 82C05, 82C35, 82C40, 83F05
© Springer Basel 2013
This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the
material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,
broadcasting, reproduction on microfilms or in any other physical way, and transmission or information
storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now
known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with
reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed
on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts
thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current
version, and permission for use must always be obtained from Springer. Permissions for use may be obtained
through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the
respective Copyright Law.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication
does not imply, even in the absence of a specific statement, that such names are exempt from the relevant
protective laws and regulations and therefore free for general use.
While the advice and information in this book are believed to be true and accurate at the date of publication,
neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or
omissions that may be made. The publisher makes no warranty, express or implied, with respect to the
material contained herein.
Cover design: deblik, Berlin
Printed on acid-free paper
Springer Basel is part of Springer Science+Business Media (www.birkhauser-science.com)
Contents
Foreword ................................................................. ix
Thibault Damour
Time and Relativity
1 Introduction ......................................................... 1
2 The common conception of Time .................................... 3
3 Boltzmann and the first time revolution ............................. 4
4 Einstein, Special Relativity and Time ................................ 8
5 General Relativity and Time ......................................... 9
6 Relativistic Gravity and the Second Law ............................. 11
7 Primordial cosmology and the Second Law ........................... 14
References ........................................................... 16
C´edric Villani
(Ir)reversibility and Entropy
1 Newton’s inaccessible realm .......................................... 20
2 The entropic world .................................................. 25
3 Order and chaos ..................................................... 29
4 Chaotic equations ................................................... 33
5 Boltzmann’s theorem 𝐻 ............................................. 42
6 Entropic convergence: forced march to oblivion ...................... 46
7 Isentropic relaxation: living with ones memories ...................... 55
8 Weak dissipation .................................................... 64
9 Metastatistics ....................................................... 66
10 Paradoxeslost ....................................................... 67
References ........................................................... 73
C´edric Villani
(Ir)r´eversibilit´eet entropie
1 Le royaume inaccessible de Newton .................................. 82
2 Le monde entropique ................................................ 87
3 Ordre et chaos ....................................................... 91
4 E´quations cin´etiques ................................................. 96
vi Contents
5 Th´eor`eme 𝐻 de Boltzmann .......................................... 105
6 Convergence entropique : oubli a` marche forc´ee ...................... 110
7 Relaxation isentropique : s’accommoder de ses souvenirs ............. 119
8 Faible dissipation .................................................... 128
9 M´etastatistiques ..................................................... 130
10 Paradoxesperdus .................................................... 131
R´ef´erences ........................................................... 137
Christopher Jarzynski
Equalities and Inequalities: Irreversibility and the Second Law
of Thermodynamics at the Nanoscale
1 Introduction ......................................................... 145
2 Backgroundand setup ............................................... 148
3 Equilibrium information from nonequilibrium fluctuations ............ 152
4 Macroscopic hysteresis and microscopic symmetry ................... 156
5 Relative entropy and dissipated work ................................ 159
6 Guessing the direction of time’s arrow ............................... 162
7 Entropy production and related quantities ........................... 164
8 Conclusions and outlook ............................................. 166
References ........................................................... 168
Christophe Salomon
Time Measurement in the XXIst Century
The quest for precision: a brief history ............................... 173
Cold atom clocks and the PHARAO Space clock ..................... 179
Fundamental physics tests ........................................... 181
A few applications ................................................... 183
Future ............................................................... 185
References ........................................................... 185
Huw Price
Time’s Arrow and Eddington’s Challenge
1 A head of his time ................................................... 187
2 Introducing ‘time’s arrow’ ........................................... 189
3 The puzzle of the thermodynamic arrow ............................. 191
4 The true puzzle – a first approximationand a popular challenge ..... 192
5 Four things the puzzle is not ......................................... 193
6 What would a solution look like? Two models ........................ 196
7 Which is the right model? ........................................... 198
8 The Boltzmann-Schuetz hypothesis – a no-asymmetry solution? ...... 199
9 The big problem .................................................... 200
10 Initial smoothness ................................................... 202
11 Open questions ...................................................... 204
Contents vii
12 Scepticism about the second law ..................................... 205
13 Where now for the flow of time? ..................................... 207
References ........................................................... 214
Catherine de Mitry
Image of Time’s Irreversibility ............................................ 217
Catherine de Mitry
Image de l’irr´eversibilit´edu Temps ....................................... 219
Foreword
This book is the eleventh in a series of Proceedings for the S´eminaire Poincar´e,
which is directed towards a broad audience of physicists, mathematicians, and
philosophers of science.
The goal of this Seminar is to provide up-to-date information about general
topics of great interest in physics. Both the theoretical and experimental aspects
of the topic are covered, generally with some historical background. Inspired by
the Nicolas Bourbaki Seminar in mathematics, hence nicknamed “Bourbaphy”,
the Poincar´eSeminar is held twice a year at the Institut Henri Poincar´ein Paris,
with written contributions prepared in advance. Particular care is devoted to the
pedagogicalnature ofthe presentations,so asto fulfill the goalof being accessible
to a large audience of scientists.
This new volume of the Poincar´e Seminar Series, “Time” (“Temps”), corre-
spondstothefifteenthsuchseminar,heldonDecember4and18,2010.Itpresents
an interdisciplinary view of the concept of time, which poses some of the most
challengingquestionsinscience, andto the humanmindinits questforanunder-
standingoftheuniverse.Thisvolumedescribesrecentdevelopmentsrelatedtothe
mathematical, physical, experimental, and philosophical facets of this fascinating
concept. Its title could actually be ‘Time’s arrow’, a phrase which seems to have
beenfirstcoinedbySirArthurEddingtoninTheNatureofthePhysicalWorld
(1928), in a challenge to physics recalled by H. Price in his contribution below:
“Time’sArrow.Thegreatthingabouttimeisthatitgoeson.Butthisis
anaspect ofit whichthe physicist sometimes seems inclined to neglect.
In the four-dimensional world ... the events past and future lie spread
out before us as in a map. The events are there in their proper spatial
andtemporalrelation;butthereisnoindicationthattheyundergowhat
has been described as “the formality of taking place” and the question
of their doing or undoing does not arise. We see in the map the path
from past to future or from future to past; but there is no signboard
to indicate that it is a one-way street. Something must be added to
the geometrical conceptions comprised in Minkowski’s world before it
becomes a complete picture of the world as we know it.”
Thefirstsurvey,byThibault Damour,titled“Time and Relativity”, offers
abroaddescriptionofthemanifold fundamentalphysicalissuesatplaywithtime,
