Relativistic Ideal and Viscous Hydrodynamics

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Relativistic Ideal and Viscous Hydrodynamics
The 8th Heavy Ion Café April 12th, 2008

• Tetsufumi Hirano
• Department of Physics
• The University of Tokyo

TH, N. van der Kolk, A. Bilandzic, to be
published in Springer Lecture Note in Physics.
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Disclaimer

• I am not an expert of viscous hydro in the sense
  that I have never published papers on it. So far,
  I have played with an ideal hydro model for gt10
  years.
• I do not have enough time to polish up todays
  slides.
• Keep intuitive picture, sometimes not
  mathematically/numerically correct.

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Plan of Lecture

• PART 1 (INTRODUCTION)
• Introduction to hydrodynamics in H.I.C.
• PART 2 (FUNDAMENTALS)
• Formalism of relativistic ideal/viscous
  hydrodynamics
• PART 3 (PHENOMENOLOGY)
• Basic Checks
• Elliptic flow
• Ideal hydrodynamic model
• Application of ideal hydrodynamic model to H.I.C.
  and comparison with data

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PART 1

• Introduction to hydrodynamics in relativistic
  heavy ion collisions

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(No Transcript)
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Why Hydrodynamics?

• Static
• Quark gluon plasma
• under equilibrium
• Equation of states
• Transport coefficients
• etc

Energy-momentum
Conserved number

• Dynamics
• Expansion, Flow
• Space-time evolution of
• thermodynamic variables

• Local thermalization
• Equation of states

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Longitudinal Expansion in Heavy Ion Collisions
Freezeout Re-confinement Expansion,
cooling Thermalization First contact (two
bunches of gluons)
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Bottom-Up Approach to Heavy Ion Physics

• The first principle (QuantumChromo Dynamics)

• Inputs to phenomenology (lattice QCD)

• Complexity
• Non-linear interactions of gluons
• Strong coupling
• Dynamical many body system
• Color confinement

• Phenomenology (hydrodynamics)

• Experimental data
• _at_ Relativistic Heavy Ion Collider
• 200 papers from 4 collaborations
• since 2000

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PART 2 (FUNDAMENTALS)

• Formalism of
• relativistic ideal/viscous
• hydrodynamics

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Relativistic Hydrodynamics
Equations of motion in relativistic hydrodynamics
Energy-momentum conservation
Energy-Momentum tensor
Current conservation
The i-th conserved current
In H.I.C., Nim NBm (net baryon current)
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Tensor/Vector Decomposition
Tensor decomposition with a given time-like and
normalized four-vector um
where,
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Projection Tensor/Vector

• um is local four flow velocity. More precise
• meaning will be given later.

• um is perpendicular to Dmn.

• Local rest frame (LRF)

• Naively speaking, um (Dmn) picks up time-
• (space-)like component(s).

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time like flow vector field
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Decomposition of Tmu
Energy density
(Hydrostaticbulk) pressure P Ps P
Energy (Heat) current
Shear stress tensor
ltgt Symmetric, traceless and transverse to um
un
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Decomposition of Nm
charge density
charge current
Q. Count the number of unknowns in the above
decomposition and confirm that it is
10(Tmn)4k(Nim). Here k is the number of
independent currents. Note If you consider um as
independent variables, you need additional
constraint for them. If you also consider Ps as
an independent variable, you need the equation of
state PsPs(e,n).
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Ideal and Dissipative Parts
Energy Momentum tensor
Charge current
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Meaning of um
um is four-velocity of flow. What kind of
flow? Two major definitions of flow are
1. Flow of energy (Landau)
2. Flow of conserved charge (Eckart)
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Meaning of um (contd.)
Landau (Wm0, uLmVm0)
Eckart (Vm0,uEmWm0)
Wm
Vm
uEm
uLm
Just a choice of local reference frame. Landau
frame might be relevant in H.I.C.
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Relation btw. Landau and Eckart
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Relation btw. Landau and Eckart (contd.)
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Entropy Conservationin Ideal Hydrodynamics
Neglect dissipative part of energy
momentum tensor to obtain ideal hydrodynamics.
Therefore,
Q. Derive the above equation.
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Entropy Current
In what follows, we consider the Landau frame
only and omit subscript L. For simplicity, we
further assume there is no charge in the system.
Assumption (1st order theory) Non-equilibrium
entropy current vector has linear dissipative
term(s) constructed from (Vm, P, pmn, um).
We have assumed Nm 0, so the second term does
not appear in this case.
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The 2nd Law of Thermodynamics
The 2nd thermodynamic law tells us
Q. Check the above calculation.
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Constitutive Equations
Thermodynamic force Transport coefficient Current
tensor shear
scalar bulk
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Equation of Motion
Expansion scalar (Divergence)
Lagrange derivative
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Equation of Motion
Q. Derive the above equations of motion from
energy-momentum conservation.
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Intuitive Interpretation of EoM
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Conserved Current Case
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Lessons from (Non-Relativistic) Navier-Stokes
Equation
Assuming incompressible fluids such that
, Navier-Stokes eq. becomes
Final flow velocity comes from interplay
between these two effects.
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Generation of Flow
P
Pressure gradient
Expand
Expand
Source of flow
? Flow phenomena are important in H.I.C to
understand EOS
x
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Diffusion of Flow
Heat equation (k heat conductivity diffusion
constant)
For illustrative purpose, one discretizes the
equation in (21)D space
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Diffusion Smoothing
R.H.S. of descretized heat/diffusion eq.
y
y
subtract
j
j
i
i
x
x
Suppose Ti,j is larger (smaller) than an average
value around the site, R.H.S. becomes negative
(positive). 2nd derivative w.r.t. coordinates ?
Smoothing
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Shear Viscosity Reduces Flow Difference
Shear flow (gradient of flow)
Smoothing of flow
Next time step
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Bjorkens Equation in the 1st Order Theory
(Bjorkens solution) (1D Hubble flow)
Q. Derive the above equation.
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Viscous Correction
Correction from shear viscosity (in compressible
fluids)
Correction from bulk viscosity
? If these corrections vanish, the above
equation reduces to the famous Bjorken
equation. Expansion scalar theta 1/tau in
scaling solution
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A Few Remarks
1. and are dimensionless
quantities in natural units and intrinsic
properties of fluids. 2. One often says,
viscosity is small. However, one has to say
this in the context of heavy ion collisions in a
more precise way Viscous coefficients
are small in comparison with entropy
density.
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Recent Topics on Transport Coefficients
Need microscopic theory (e.g., Boltzmann eq.) to
obtain transport coefficients.

• is
  obtained from
• Super Yang-Mills theory.
• is obtained from
  lattice.
• Bulk viscosity has a prominent peak around Tc.
• Systematic study of pi-N system based on
• Chapman-Enskog method

Kovtun,Son,Starinet,
Nakamura,Sakai,
Kharzeev,Tuchin,Karsch,Meyer
Otomo, Itakura, Morimatsu
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Digression
Pa N/m2
hwater gt hair BUT nwater lt nair
(Dynamical) Viscosity h 1.0x10-3 Pa s
(Water 20?) 1.8x10-5 Pa s (Air 20?)
Kinetic Viscosity nh/r 1.0x10-6 m2/s
(Water 20?) 1.5x10-5 m2/s (Air 20?)
Non-relativistic Navier-Stokes eq.
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Necessity of Relaxation Time
Non-relativistic case (Cattaneo(1948))
Balance eq.
Constitutive eq.
Fouriers law
t relaxation time
Parabolic equation (heat equation)
ACAUSAL! Finite
t Hyperbolic equation (telegraph equation)
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Entropy Current (2nd)
Assumption (2nd order theory) Non-equilibrium
entropy current vector has linear quadratic
dissipative term(s) constructed from (Vm, P, pmn,
um).
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Constitutive Equations

• Relaxation terms appear (tp and tP are relaxation
  time).
• No longer algebraic equations! Dissipative
  currents become dynamical quantities like
  thermodynamic variables.
• Employed in recent viscous fluid simulations.

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Bjorkens Equationin the 2nd Order Theory
where
New terms (written in red) appear in the
2nd order theory. ? Coupled equations Usually,
the last terms are neglected. Their importance ?
Natsuume-Okamura
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Relaxation Equation?
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More on Bjorkens Solution
Assuming boost invariance for thermodynamic
variables PP(t) and 1D Hubble-like flow
Hydrodynamic equation for perfect fluids with a
simple EoS,
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Bjorken Coordinate
t
Boost ? parallel shift Boost invariant ?
Independent of hs
z
0
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What is conserved, what is NOT conserved in
Ideal Hydrodynamics
expansion
pdV work
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Model EoS (crossover)
Crossover EoS Tc 0.17GeV D Tc/50 dH 3, dQ
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Energy-Momentum Tensorat Initial Time
In what follows, bulk viscosity is neglected.
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Numerical Results (Temperature)
T0 0.22 GeV t0 1 fm/c
h/s 1/4p tp 3h/4p
Same initial condition (Energy momentum tensor
is isotropic)
Numerical code (C) is available upon request.
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Numerical Results (Temperature)
T0 0.22 GeV t0 1 fm/c
h/s 1/4p tp 3h/4p
Same initial condition (Energy momentum tensor
is anisotropic)
Numerical code (C) is available upon request.
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Numerical Results (Temperature)
Numerical code (C) is available upon request.
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Numerical Results (Entropy)
T0 0.22 GeV t0 1 fm/c
h/s 1/4p tp 3h/4p
Same initial condition (Energy momentum tensor
is isotropic)
Numerical code (C) is available upon request.
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Numerical Results (Entropy)
T0 0.22 GeV t0 1 fm/c
h/s 1/4p tp 3h/4p
Same initial condition (Energy momentum tensor
is anisotropic)
Numerical code (C) is available upon request.
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Numerical Results (Entropy)
Numerical code (C) is available upon request.
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Numerical Results(Shear Viscosity)
Numerical code (C) is available upon request.
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Numerical Results(Initial Condition
Dependencein the 2nd order theory)
Numerical code (C) is available upon request.
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Numerical Results(Relaxation Time dependence)
Relaxation time larger ?Maximum p is
smaller Relaxation time smaller ?Suddenly relaxes
to 1st order theory
Saturated values non-trivial
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Boltzmann Equation
Liouville eq. (f 6N1 D)
Reduction of phase space distribution
BBGKY hierarchy
Stosszahl ansatz
Closed equation for f(1) ?Boltzmann eq.
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Microscopic Interpretation
Single particle phase space density in
local thermal equilibrium
Kinetic definition of current and energy
momentum tensor are
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Matter in (Kinetic) Equilibrium
Kinetically equilibrated matter at rest
Kinetically equilibrated matter at finite velocity
um
py
py
px
px
Lorentz-boosted distribution
Isotropic distribution
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1st Moment
um is normalized, so we can always choose amn
such that
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1st Moment (contd.)
Vanishing for n i due to odd function in
integrant.
Q. Go through all steps in the above derivation.
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2nd Moment
where,
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Deviation from Equilibrium Distribution
(Just neglecting anti-particles)
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1
Grads 14 moments EoM for epsilons can be also
obtained from BE.
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Taylor Expansion around Equilibrium Distribution
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Taylor Expansion around Equilibrium Distribution
(contd.)
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Relation btw. Coefficients and Dissipative
Currents
Again assuming no net charge,
For explicit expression of B,D, and E, see
Israel-Stewart. Roughly speaking, these are
obtained from the 2nd, 3rd , or 4th moments of
f0(1f0)
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Modified Cooper-Frye Formula

• Change of flow as a consequence of dissipative
  fluids
• Distortion of distribution

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Summary (part 2)

• Relativistic Ideal/Viscous Hydrodynamics is
  introductory reviewed.
• Naïve extension of Navier-Stokes equation to its
  relativistic version has a problem on causality.?
  Need the 2nd order corrections in entropy
  current.
• Microscopic interpretation based on Boltzmann
  equation is made.

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ReferencesFar from Complete List

• General
• L.D.Landau, E.M.Lifshitz, Fluid Mechanics,
  Section 133-136
• L.P.Csernai, Introduction to Relativistic Heavy
  Ion Collisions
• D.H.Rischke, nucl-th/9809044.
• J.-Y.Ollitrault, 0708.2433nucl-th.
• Viscous hydro transport coefficient
• C.Eckart, Phys.Rev.15,919(1940).
• M.Namiki, C.Iso,Prog.Theor.Phys.18,591(1957)
• C.Iso, K.Mori,M.Namiki, Prog.Theor.Phys.22,403(195
  9)
• I.Mueller, Z. Phys. 198, 329 (1967)
• W.Israel, Ann.Phys.100,310(1976)
• W.Israel. J.M.Stewart, Ann.Phys.118,341(1979)
• A.Hosoya, K.Kajantie, Nucl.Phys.B250, 666(1985)
• P.Danielewicz,M.Gyulassy, Phys.Rev.D31,53(1985)
• I.Muller, Liv.Rev.Rel 1999-1.

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References for Astrophysics/Cosmology

• N.Andersson, G.L.Comer, Liv.Rev.Rel.,2007-1
• R.Maartens, astro-ph/9609119.

Disclaimer Im not familiar with these kinds of
review