Microscopic Foundations of Relativistic Fluid Dynamics PDF
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Lecture Notes in Physics Gabriel S. Denicol Dirk H. Rischke Microscopic Foundations of Relativistic Fluid Dynamics Lecture Notes in Physics Founding Editors Wolf Beiglböck, Heidelberg, Germany Jürgen Ehlers, Potsdam, Germany Klaus Hepp, Zürich, Switzerland Hans-Arwed Weidenmüller, Heidelberg, Germany Volume 990 Series Editors Roberta Citro, Salerno, Italy Peter Hänggi, Augsburg, Germany Morten Hjorth-Jensen, Oslo, Norway Maciej Lewenstein, Barcelona, Spain Angel Rubio, Hamburg, Germany Wolfgang Schleich, Ulm, Germany Stefan Theisen, Potsdam, Germany James D. Wells, Ann Arbor, MI, USA Gary P. Zank, Huntsville, AL, USA The series Lecture Notes in Physics (LNP), founded in 1969, reports new developmentsinphysicsresearchandteaching-quicklyandinformally,butwitha highqualityandtheexplicitaimtosummarizeandcommunicatecurrentknowledge in an accessible way. 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Denicol DirkH.Rischke Physics Institute Institute for Theoretical Physics Fluminense Federal University Goethe University Frankfurt Niterói,Brazil Frankfurtam Main,Germany ISSN 0075-8450 ISSN 1616-6361 (electronic) Lecture Notesin Physics ISBN978-3-030-82075-6 ISBN978-3-030-82077-0 (eBook) https://doi.org/10.1007/978-3-030-82077-0 ©SpringerNatureSwitzerlandAG2021 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart 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 orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained hereinorforanyerrorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregard tojurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface Fluiddynamicsisatheorywithwidespreadapplicationsinfundamentalresearchas well as the applied sciences. It relies on the assumption that the details of the microscopicinteractions between particles constituting thefluidplaynoroleinthe dynamicsofthefluidonmacroscopicscales.Thelatterisdeterminedsolelybythe conservation of charge, energy, and momentum. However, the physics on micro- scopic scales does affect that on macroscopic scales at several places. In the absence of dissipation, i.e., for so-called ideal (or perfect) fluids, the solution of the fluid-dynamical equations of motion requires microscopic infor- mation in the form of the thermodynamic equation of state of the fluid. The latter determineshowmuchafluidcanbeacceleratedforagivengradientindensity.For dissipative (or non-perfect) fluids, in addition to the equation of state, also other types of microscopicinformation is required, in the form of the so-called transport coefficients.Themostimportantonesarethebulkandshearviscosity,aswellasthe charge-diffusion coefficient. These determine how much resistance there is for the fluid to reach thermodynamic equilibrium for given gradients in density or fluid velocity. The transport coefficients are functions of space-time and, in a microscopic picture of the system as that of many particles scattering with each other, can be derived by integrating out the momentum-space information contained in the Boltzmann equation, which describes the microscopic motion of particles in phase space. Unless one wants to solve the Boltzmann equation directly, which is a task whosecomplexitygrowswiththenumberofconstituentsofthesystem,andwants to stick with a fluid-dynamical description, the smaller the system the more important the interactions on microscopic scales become and, consequently, the more transport coefficients need to be known for an accurate description of the dynamics of a given system. In the non-relativistic context, the theories of ideal and dissipative fluid dynamics are well-established and have been put to test in a multitude of appli- cations, where they have proven their validity. The most traditional theory of dissipativefluiddynamicsisthewell-knownNavier–Stokestheory.Thesuccessof this theory in non-relativistic applications led to the belief that fluid dynamics should be constructed from a systematic expansion in gradients in density or fluid velocity. The lowest-order truncation of this expansion gives rise to ideal fluid v vi Preface dynamics, while the first-order truncation leads to Navier–Stokes theory itself. However, higher-order truncations of this picture lead to the Burnett and super-Burnett equations, which were proven to be linearly unstable. Whiletherelativisticgeneralizationofidealfluiddynamicsisstraightforward,a relativistic generalization of Navier–Stokes theory encounters severe difficulties, related to the violation of causality and the occurrence of unstable modes. Fol- lowing the procedure developed by Grad in the non-relativistic regime, Israel and Stewart were among the first to derive a relativistic theory of dissipative fluid dynamicsthatiscausalandstable.ThecausalityandstabilityproblemsofNavier– Stokes theory were essentially solved by introducing a delay in time for the gen- eration of the dissipative currents (i.e., bulk-viscous pressure, charge-diffusion current,andshear-stresstensor)from gradientsofthefluid-dynamicalvariables.In this case, these currents become independent dynamical variables that obey equa- tions of motion which need to be solved in addition to the conservation equations. Besides the transport coefficients related to bulk and shear viscosity, as well as chargediffusion,amultitudeofnewtransportcoefficientsoccur,whichdescribethe couplingofthedissipativecurrentsamongeachotherandwithgradientsofdensity and fluid velocity. Nowadays, the main area of application for Israel–Stewart theory is the field of high-energy nuclear physics. The collective behavior of hot and dense nuclear matter created in collisions of heavy atomic nuclei, so-called heavy-ion collisions, atrelativisticenergiescanbeverywelldescribedbyrelativisticfluiddynamics.Due to the smallness of the system and its short lifetime, dissipative effects cannot be neglected,sothatIsrael–Stewarttheoryfindsanaturalapplicationinthedescription of heavy-ion collisions. Nevertheless, despite its successes, this theory remains an ad hoc procedure to derive the equations of motion for the dissipative currents in thesensethat itlacks aparameterwith whichonecanpower-countandthusjudge thevalidityand,ultimately,systematicallyimprovethefluid-dynamicaldescription. Inrecentyears,thisshortcominghastriggeredalotofactivitytoderiveacausaland stabletheoryofrelativisticdissipativefluiddynamicsfromkinetictheoryinamore systematic fashion. The current monograph attempts to summarize these efforts. There are several monographs dealing with the theory of relativistic fluid dynamics. The earliest is Landau’s and Lifshitz’s textbook on “Fluid Dynamics” [1],whichcontainsachapteronrelativisticfluids.However,thediscussionmostly focuses on ideal fluids, and deals with dissipative fluids only very briefly in the unstable and acausal Navier–Stokes framework. A derivation of fluid dynamics from kinetic theory as the underlying microscopic theory can be found in “Rela- tivistic Kinetic Theory—Principles and Applications” by de Groot, van Leeuwen, and van Weert [2]. This monograph contains a detailed discussion of Chapman– Enskog theory, as well as the mathematical tools to derive dissipative fluid dynamics from kinetic theory applied in this monograph, such as irreducible ten- sors. However, the derivation of dissipative fluid dynamics is then only performed in the way suggested by Israel and Stewart, which, as explained above, lacks a systematic power-counting scheme and is not systematically improvable. A text- book with similar content as Ref. [2] is the one by Cercignani and Kremer [3]. Preface vii Adiscussion ofrelativistic fluiddynamics with particular emphasis toapplications ingeneralrelativityiscontainedinthetextbookofRezzollaandZanotti[4],butthe equationsofmotionofdissipativefluidsareonlyderivedphenomenologicallyfrom the second law of thermodynamics. A more recent monograph is the one on “RelativisticFluidDynamicsInandOutofEquilibrium”bytheRomatschke’s[5]. Theviewtakenhereisthatdissipativefluiddynamicscanbeunderstoodintermsof a gradient expansion around local equilibrium. Although it contains a chapter on kinetic theory, the derivation of dissipative fluid dynamics from the underlying kinetictheoryisnotpresented.Inviewofthecurrentsituationoftheliterature,we felt that it would be beneficial, both for researchers new to the field of dissipative fluid dynamics as well as a point of reference for experts, to have a monograph which summarizes the last decade’s progress in our understanding of relativistic dissipative fluid dynamics as it emerges as a causal and stable theory from the underlying microscopic framework of kinetic theory. This book is organized as follows: In Chap. 1, we review relativistic fluid dynamicsfromaphenomenologicalperspective.Westartbyderivingtheequations of motion of an ideal relativistic fluid. Then, we review how dissipation is phe- nomenologically introduced in relativistic fluids. The equations of relativistic Navier–Stokestheoryarederivedviathesecondlawofthermodynamicsand,later, extended using the gradient expansion. Next, we discuss the problems of this traditional theory in the relativistic regime, i.e., the acausality of Navier–Stokes theory, and the phenomenological methods to derive a causal theory of relativistic dissipative fluid dynamics. As an example, we review how Israel and Stewart derived a causal fluid-dynamical theory from the second law of thermodynamics. Finally, the end of this chapter contains a discussion about the existence of non-hydrodynamic modes in relativistic fluid dynamics and how they affect the interpretation of this theory. InChap.2,wediscussthecausalityandstabilityofrelativisticfluiddynamicsin the linear regime. We explicitly demonstrate that relativistic Navier–Stokes theory is linearly unstable when perturbed around global equilibrium: a fundamental flaw that prevents this theory from being applied to describe any relativistic fluid in Nature.Wefurtherconnectthisunphysicalinstabilitywiththeacausalitydisplayed bythetheory.WethendemonstratethatIsrael–Stewarttheorycanbeconstructedto be linearly stable, as long as the transport coefficients satisfy certain constraints. In Chap.3, we expand the analysis performed in the previous chapter by inves- tigating the properties of relativistic fluid dynamics in the nonlinear regime. This can be accomplished by imposing that the fluid expands according to highly symmetricalflowconfigurations,insuchawaythatthesymmetriesoftheproblem aresufficient tosolve for thefluid4-velocity andtheremainingcomponents ofthe conserved currents can be calculated analytically or, at least, semi-analytically. Such solutions are extremely rare and will be derived in this chapter. These rare solutions of relativistic fluid dynamics are then used to study the asymptotic properties of transient fluid dynamics, to a level that is not possible in the linear regime. viii Preface InChap.4,weintroduceanovelderivationoftransientrelativisticfluiddynamics from quantum field theory using linear-response theory. In this derivation, we investigate the conditions for the corresponding retarded Green’s function under whichthelinearizedequationofmotionofadissipativecurrentcanbereducedtoa relaxation-type equation, such as the one appearing in Israel–Stewart theory. The power of this new formalism is demonstrated by applying it to two examples: the linearized Boltzmann equation and the linear-response theory with metric perturbations. InChap.5,wecontinuethediscussionofhowrelativisticfluiddynamicsemerges fromamicroscopictheory,butnowfromtheperspectiveofrelativisticdilutegases. In this case, the derivation of relativistic dissipative fluid dynamics from a micro- scopic theory, here the relativistic Boltzmann equation, can be carried out in great detail, going beyond the linear regime. This issue will be the main topic for the remainder of this book. This chapter starts this discussion by introducing the two most widespread methods usually employed to derive relativistic fluid dynamics from the Boltzmann equation: the Chapman–Enskog expansion and the method of momentsasproposedbyIsraelandStewart.Thesetheorieswillbefurtherimproved and studied inthe followingchapters. InChap.6,arelativisticgeneralizationofthemethodofmomentsisformulated. The main goal of this chapter is to describe the moment expansion of the single-particle momentum distribution function and to derive the equations of motion satisfied by the corresponding expansion coefficients, the so-called irre- ducible moments, which turn out to be moments of the deviation of the single-particle distribution function from a chosen reference state. Such a gener- alization of the method of moments will be essential in understanding how rela- tivistic fluid dynamics actually emerges from the relativistic Boltzmann equation, what its domain of applicability is, and how the equations of motion can be sys- tematically improved. In this chapter, the reference state is assumed to be in local thermodynamicalequilibrium.InChap.9,wewillshowhowtogeneralizethistoa particular non-equilibrium reference state, exhibiting a spatial anisotropy. InChap.7,weinvestigatetheconvergenceofthemethodofmomentsusingthe highly symmetric flow configurations described in Chap. 3: the Bjorken- and Gubser-flow scenarios. We shall demonstrate that these simplified, yet non-trivial, scenarios allow us to systematically derive the equations of motion for all the momentsofthesingle-particledistributionfunction.Thiswillenableustostudythe convergenceofthemomentexpansionandobtainagraspofitsdomainofvalidity. InChap.8,weshowhowrelativisticfluiddynamicscanbederivedfromkinetic theory, using the method of moments. In this case, we derive the equations of motion in the nonlinear regime, including all the possible nonlinear and higher- orderterms.ThereductionofthemicroscopicdegreesoffreedomoftheBoltzmann equationisimplementedbyidentifyingitsmicroscopictimescalesandconsidering only the slowest of these to be relevant in the fluid regime. In addition, the equa- tionsofmotionforthedissipativequantitiesaretruncatedaccordingtoasystematic power-counting scheme in Knudsen and inverse Reynolds numbers. It is mathe- matically proven that the equations of motion of fluid dynamics can be closed in Preface ix termsofonly14dynamicalvariables,asithappensinIsrael–Stewarttheory,aslong asweonlykeeptermsoffirstorderineitherKnudsenorinverseReynoldsnumber, or the product of both. Furthermore, we show how to systematically improve the applicability of relativistic fluid dynamics by going to higher order in Knudsen number, at the same time keeping the equations of motion hyperbolic and causal. We then investigate the applicability of the fluid-dynamical theory derived from kinetictheorybycomparingitssolutionswithnumericalsolutionsoftherelativistic Boltzmann equation. The final chapter of this book, Chap. 9, is devoted to the discussion of how to generalize the moment expansion around the local-equilibrium state, as performed in Chap. 6, to an expansion around a particular non-equilibrium reference state. This state is supposed to exhibit an anisotropy in a single spatial direction in momentum space, in contrast to the local-equilibrium state, which is isotropic in momentumspace intherestframeofthefluid.Thisgeneralization inducesseveral technical complications, as the irreducible tensors used to perform the moment expansion in the local-equilibrium case need to be replaced by new tensors which areirreduciblewithrespecttothesubspaceorthogonaltoboththefluidvelocityand the 4-vector parametrizing the spatial anisotropy. This also requires novel orthog- onality relations for these tensors. It isneedlesstosaythat ourdiscussion ofrelativisticdissipative fluid dynamics as derived from kinetic theory and its stability and causality properties is far from complete. Several more recent developments are not covered, such as causal and stable first-order theories of relativistic dissipative fluid dynamics, relativistic dis- sipative magnetohydrodynamics for unpolarized and polarized fluids, and many more. We reserve a more detailed treatment of these novel and very interesting developments for a future edition of this monograph. Finally, a word of thanks is in order. This book would not have been possible without the unwavering support and countless hours of work of the many col- leagues who co-authored the original papers written by us and referenced in the text: Barbara Betz, Ioannis Bouras, Charles Gale, Carsten Greiner, Ulrich Heinz, Sanyong Jeon, Takeshi Kodama, Tomoi Koide, Etele Molnár, Harri Niemi, Jorge Noronha, Shi Pu, Michael Strickland, Zhe Xu, as well as many others with whom we had fruitful and enlightening discussions over the years. Niterói, Brazil Gabriel S. Denicol Frankfurt am Main, Germany Dirk H. Rischke April 2021 References 1. Landau,L.D.,Lifshitz,E.M.:FluidMechanics.Pergamon,NewYork(1959) 2. De Groot, S.R., Van Leeuwen, W.A., Van Weert, Ch.G.: Relativistic Kinetic Theory— PrinciplesandApplications.North-Holland(1980)
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