Table Of ContentLecture Notes in Physics
Gabriel S. Denicol
Dirk H. Rischke
Microscopic
Foundations
of Relativistic
Fluid Dynamics
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Gabriel S. Denicol Dirk H. Rischke
(cid:129)
Microscopic Foundations
of Relativistic Fluid
Dynamics
123
GabrielS. 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)
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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)
