Extended Irreversible Thermodynamics PDF
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ExtendedIrreversibleThermodynamics ONLINE L1BRARY Physics and Astronomy http://www.springer.de/phys/ Springer-Verlag Berlin Heidelberg GmbH D. lou J. Casas-Vazquez G. Lebon Extended Irreversible Thermodynamics Third, Revised andEnlargedEdition With 30 Figuresand15Tables i Springer Professor Dr. David Iou Professor Dr. Iose Casas-Vazquez Departament de Fisica Grup de Fisica Estadistica Universitat Autonoma de Barcelona Edifiei C 08193 Bellaterra, Catalonia, Spain Professor Dr. Georgy Lebon Departement de Physique Universite de Lii~ge Sart Tilman B5 4000 Liege, Belgium Library of Congress Cataloging-in-Publication Data. Jou, D. (David). Extended irreversible thermody namicsl D. Iou, I. Casas-Vazquez, G. Lebon. - 3rd ed. p. cm. Includes bibliographical references and index. ISBN 978-3-642-62505-3 1. Irreversible processes. I. Casas-Vazquez, J. Oose), 1938-II. Lebon, G. (Georgy) III. Title. QC318.I7 J67 2000 536'.7-dc21 00-053784 ISBN 978-3-642-62505-3 ISBN 978-3-642-56565-6 (eBook) DOI 10.1007/978-3-642-56565-6 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, speeifically the rights of translation, reprinting, reuse of illustrations, recitation, broad casting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. http://www.springer.de © Springer-Verlag Berlin Heidelberg 1993,1996,2001 Originally published by Springer-Ve rlag Berlin Heidelberg New York in 2001 Softcover reprint of the hardcover 3rd edition 2001 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant pro tective laws and regulations and therefore free for general use. Typesetting: Camera-ready copy from the authors Cover design: Erich Kirchner, Heidelberg Printed on acid-free paper SPIN 10720979 55/3141lba 5 4 3 2 1 o Prefaceto the ThirdEdition In writing this third edition, weproceeded having in mind two main objectives: peda gogyandphysics.Oneofouressentialconcernsisindeedtoopen thesubject toalarge audience of scientists with adiversity ofinterests.This is thereason we have empha sizedthepedagogical aspects,startingwiththesimplestsituationsbefore treating more complex problems. Our aim is to show that thermodynamics is not restricted to the study of purely thermal phenomena but that it largely covers many different areas, going from continuum mechanics to statistical physics, from nuclear collisions to cosmology, bypassing,forinstance,throughchemistry,rheologyandbiology. Our second objective is tostressthephysicalaspects in their broadest sense,i.e, trying to clarify the foundations and relating the theoretical results with experimental data and applications, rather than introducing formal and lengthy developments. Im portant sections ofthebookaredevoted tothediscussionoffundamental notions,such asthe selection ofbasicvariables, thedefinitionofentropy, temperature, pressure, and chemical potential outside equilibrium and the statement of the second law. We feel that these crucial questions deserve special attention when they cross the borders of classical theories. We haveincluded new original results and opened the door to new extensions andapplications. For the student ortheresearcherit maybestimulatingtogobeyond theclassical theories and todiscover adomain of newideas, new applications, and newproblems. This is what extended irreversible thermodynamics (EIT)offers nowadays:it does not pretendtosolvealltheproblemsraisedincontinuumphysicsandstatistical mechanics, but it can be viewed as an emergent new global framework of non-equilibrium thermodynamics and arapidly advancing frontier with new applications being treated andfundamental questionsbeingaskedandtentativelyclarified. This edition hasbeenextensivelyremodelledcompared tothefirst two. Wehave gone from 12to 18chapters as a result of splitting some of the former chapters into shorter ones andfocusing betteronthefoundationsandontheintroduction tothebasic ideas of the latest developments. Asfor the two preceding editions, we have clearly separatedthe structure ofthe bookintothreeparts, namely thegeneral theory (PartI), microscopic foundations (Part II), and selected applications (Part III). Amongst the VI PrefacetotheThirdEdition most visible changes, wehave split thegeneral presentation ofEIT into two chapters: Chap. 2 devoted to establishing the transport equations and dynamical aspects, and Chap.3in which weintroduce thenon-equilibriumequations ofstateemphasizingthe meaning of entropy and temperature in non-equilibrium states. Comparison with the rational version of extended thermodynamics has beenmademore explicit. Chapter 4 showsanoverview ofHamiltonianformulations,whichhavedeveloped inrecent years intoapromisingdomain. The microscopic foundations arediscussedinChaps.5-9,withtwonewchapters concerning information theory and computer simulations. Both topics were already present intheformereditionsassubsectionsofotherchapters,butrecentdevelopments havejustifiedarenewedandupdatedpresentation. Applications cover half ofthebook:ninechapters from the totaleigthteen.They show that the exploration of new theoretical grounds is both an intellectual challenge and a source of new practical possibilities. Since contemporary technology strives towards higher speed, power, and miniaturization, the transport equations must incorporate memory, non-local, and non-linear effects. Compared to the generalised transportequations incorporating theaforementionedeffects,EITplaysarole similarto that played by classical irreversible thermodynamics with respect to the classical transport equa-tions.Diffusion andelectrical phenomena,treated inasingle chapter in theprevious editions,arethesubjectoftwodifferent, updatedchapters;therelativistic formulation has also been split into two different chapters, one devoted to the foundations and the other to cosmological applications. Finally, let us mention that a wideoverview oftheliteratureonEITaswellasthesolutionstotheproposedproblems canbefoundonthewww.uab.es/dep-fisica/eitwebsite. This book, in its successive editions, has widely benefited from the fruitful sug gestions and comments by colleagues, including M. Anile, J. Camacho, M. Criado Sancho, L.S.Garcfa-Colfn,M.Grmela, P.T.Landsberg,R.Luzzi,W.Muschik, D. Pa v6n, M.Torrisi, A.Valenti and manyothers.Weacknowledgethefinancial support of theDGICyT oftheSpanishMinistryofEducationandCultureunderGrantNos.PB90 0676 and PB94-07l8, of the DGR of the Generalitat of Catalonia, under Grant Nos. 1997SGR 00378 and 2000SGR 00095, of the Belgian Ministry of Scientific Policy under PAl Grant Nos. 21, 29 and IV 6, and a grant from the European Union in the framework of theProgram ofHumanCapitalandMobility(EuropeanThermodynamic NetworkERB-CHRXCT920(07). Bellaterra,Barcelona Liege DavidJou March2001 Jose Casas-Ydzquez GeorgyLebon Preface to the FirstEdition Classical irreversible thermodynamics, asdeveloped by Onsager, Prigogine and many other authors,isbasedonthelocal-equilibriumhypothesis.Outofequilibrium, anysys temisassumed todepend locallyonthesamesetofvariablesaswhenitisinequilibri um. This leads to a formal thermodynamic structure identical to that of equilibrium: intensive parameters such as temperature, pressure and chemical potentials are well defined quantities keeping their usual meaning,thermodynamic potentials are derived as Legendre transformations and all equilibrium thermodynamic relations retain their validity.The theory based on thishypothesis has turned out to bevery useful and has collected anumberofsuccessesinmanypracticalsituations. However, therecentdecadehaswitnessedasurgeofinterestingoingbeyond the classical formulation.There are several reasons for this.One of them is the develop ment of experimental methods able to deal with the response of systems to high frequency andshort-wavelengthperturbations,suchasultrasoundpropagation andlight and neutron scattering.The observedresultshaveledtogeneralisations oftheclassical hydrodynamical theories,byincludingmemoryfunctionsorgeneralised transportcoef ficients depending on the frequency and the wavevector. This field has generated impressive progress innon-equilibriumstatisticalmechanics,butfor themoment ithas not brought about a parallel development in non-equilibrium thermodynamics. An extension of thermodynamics compatible with generalised hydrodynamics therefore appears tobeanaturalsubjectofresearch. Anadditionalreason hasfosteredaninterestingeneralisingtheclassical transport equations, like Fourier's law for heat conduction, Fick's law for diffusion, and Newton'slawforviscousflow.Itiswellknownthatafterintroducing theserelations in thebalanceequations, oneisledtoparabolicpartial-differentialequations which imply that perturbations propagate with infinite speed.This behaviour is incompatible with experimental evidenceanditisalsodisturbingfromatheoreticalpointofview,because collectivemolecular effectsshould beexpectedtopropagateatfinitevelocity, not only in a relativistic framework, but even from a non-relativistic point of view. This un pleasant property can beavoided bytakingintoaccount thefinite nonvanishing relax- VIII PrefacetotheFirstEdition ationtimeoftherespectivefluxes,e.g,heatflux,diffusionflux,momentumflux,some times generically called dissipative fluxes. The subsequentequations are however not compatible withtheclassical non-equilibriumthennodynamics, sincetheyleadinsome circumstancestonegativeentropy production.Thus,athermodynamic theory compati blewiththese phenomena ishighlydesirable,becauseitmayprovidenewinsights into the meaning and definition of fundamental thermodynamic quantities, as entropy and temperature, and may clarify the limitsof validityofthe local-equilibrium hypothesis andoftheusualformulationsofthesecondlawoutofequilibrium. The former problems are not merely academic. It has been observed in several systems that thedissipative fluxes arecharacterized bylong relaxation times.Typical examples are polymeric fluids, heatandelectricconductorsatlowtemperature, super conductors, andsoon.Anaccurateunderstandingofthesesystemsmaythusbeimport ant not only from a theoretical point of view, but also for practical purposes. In real situations,thesesystemsareoutofequilibrium.Accordingly,thereisanurgentneedfor anon-equilibrium thermodynamictheoryable,ontheonehand,tocopewiththeeffects oflongrelaxation timesand,ontheother,tocomplementotherformalismsbasedonthe useofinternalvariables. There areother reasons for thepresentstudy.Oneshould beaware thatclassical irreversible thermodynamics is not the only non-equilibrium thermodynamic theory: othertheories, inparticular theso-calledrationalthermodynamics,haveachieved some valuable results.To reconcile theclassical andtherationalpoints ofview, itwould be of interest to have a theory able to provide a sufficiently wide ground for discussion, thus making their common points evident and their main differences understandable. Extendedirreversiblethermodynamicsisapromisingcandidate. Extended irreversible thermodynamics received a strong impetus in the past decade. Besides the classical thermodynamic variables, this theory introduces as new independent variables the dissipative fluxes and aims to obtain for them evolution equations compatible withthesecondlawofthermodynamics.The centralquantity isa generalised non-equilibrium entropy,depending on both the conserved variables and the fluxes, which sheds new light on the content ofthe second law. This generalised theory is corroborated from a microscopic point of view by the kinetic theory, non equilibrium information theory and other formulations of non-equilibrium statistical mechanics. The purpose of this book is to provide an introduction to the foundations of extended irreversible thermodynamics, todiscussthemainresults andtopresent some of its applications. After more than twenty years of research and several hundreds of papers published by many groups in several countries we feel such a book is sorely needed.Guided bytheaimtobeasillustrativeandpedagogicalaspossible,arelatively PrefacetotheFirstEdition IX simple formulation of the theory is presented, but this is nevertheless more than sufficient for the description of several phenomena not accessible to the classical theory.The various topics treatedin this bookrangefrom thermal waves and phonon hydrodynamics to material and electrical transport, from ultrasound propagation and generalised hydrodynamics to rheology,fromkinetic theory tocosmology.Of course, other formulations of extended thermodynamics and other kinds of applications are possible. They are likely to arise in the near future. We hope that this book will be useful in providing a generalview of present achievementsand in stimulating future research. We areverypleased toacknowledgemanystimulatingdiscussions with ourcol leagues Carlos Perez-Garcfa,Josep-EnricLlebot,DiegoPav6n,Jose-Miguel Rubfand Joseph Lambennontfor morethanfifteenyearsofjoint research, andalso with many other colleagues from the different groups which have devoted their attention to extended irreversible thermodynamics.We also acknowledge the financial support of the Comisi6n Asesora para la Investigaci6nCientfficayTecnica of the Spanish Gov ernment during the years 1979-1986 under grants 3913n9 and 2389/83, and of the Direcci6n General de Investigaci6n Cientffica y Tecnica of the Spanish Ministry of Education, under grants PB86-0287, PB89-0290, and PB90-0676. The collaboration between our groups in Bellaterra and Liege has been made economically possible becauseoftheNATOgrant0355/83. March 1993 DavidJou Jose Casas-Vdzquez GeorgyLebon Contents PartI. General Theory 1 Classicaland Rational Formulations ofNon-equilibriumThermodynamics 3 1.1 TheGeneral BalanceLawsofContinuumPhysics 4 1.2 The Law ofBalanceofEntropy .................................. 13 1.3 Classical Irreversible Thermodynamics ............................ 15 1.4 Rational Thermodynamics ...................................... 24 Problems 33 2 ExtendedIrreversibleThermodynamics: EvolutionEquations 39 2.1 Heat Conduction 40 2.2 One-componentViscousFluid................................... 47 2.3 TheGeneralisedEntropyFluxandEntropyProduction 49 2.4 LinearizedEvolution EquationsoftheFluxes ....................... 52 2.5 Rational ExtendedThermodynamics .. ............................ 54 2.6 Some Comments andPerspectives. ............. .... 65 2.7 Entropy EvolutioninanIsolatedSystem:Anlllustrative Example. ...... 66 Problems 70 3 ExtendedIrreversibleThermodynamics: Non-equilibriumEquations ofState........................... 73 3.1 Physical Interpretation oftheNon-equilibriumEntropy 74 3.2 Non-equilibriumEquationsof State:Temperature ............ 77 3.3 Non-equilibriumEquations ofState:Thermodynamic Pressure.......... 83 3.4 Convexity Requirements andStability 87 Problems 92
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