Relativistic Ideal and Viscous Hydrodynamics

1
Relativistic Ideal and Viscous Hydrodynamics
Intensive Lecture YITP, December 11th, 2008

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

TH, N. van der Kolk, A. Bilandzic,
arXiv0808.2684nucl-th to be published in
Springer Lecture Note in Physics.
2
Plan of this Lecture

• 1st Day
• Hydrodynamics in Heavy Ion Collisions
• Collective flow
• Dynamical Modeling of heavy ion collisions
  (seminar)
• 2nd Day
• Formalism of relativistic ideal/viscous
  hydrodynamics
• Bjorkens scaling solution with viscosity
• Effect of viscosity on particle spectra
  (discussion)

3
PART 3

• Formalism of
• relativistic ideal/viscous
• hydrodynamics

4
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)
5
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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Intuitive Picture of Projection
time like flow vector field
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Decomposition of Tmu
Energy density
Energy (Heat) current
Shear stress tensor
(Hydrostaticbulk) pressure P Ps P
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.
13
Relation btw. Landau and Eckart
14
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
Assumption (1st order theory) Non-equilibrium
entropy current vector has linear dissipative
term(s) constructed from (Vm, P, pmn, (um)).

• (Practical) Assumption
• Landau frame (omitting subscript L).
• No charge in the system.

Thus, a 0 since Nm 0, Wm 0 since
considering the Landau frame, and g 0 since um
Sm should be maximum in equilibrium (stability
condition).
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The 2nd Law of Thermodynamics and Constitutive
Equations
The 2nd thermodynamic law tells us
Q. Check the above calculation.
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Constitutive Equations (contd.)
Thermodynamic force Transport coefficient Current
tensor shear
scalar bulk
Newton
Stokes
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Equation of Motion
Lagrange (substantial) derivative
Expansion scalar (Divergence)
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Equation of Motion (contd.)
Q. Derive the above equations of motion from
energy-momentum conservation.
Note We have not used the constitutive
equations to obtain the equations of motion.
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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 Averaging
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
Microscopic interpretation can be made. Net
momentum flow in space-like direction. ? Towards
entropy maximum state.
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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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Heat Kernel
x
x
perturbation on top of background
causality
Heat transportation
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Instability

• The 1st order equation is not only acausal but
  also unstable under small perturbation on a
  moving back-ground. W.A.Hiscock and L.Lindblom,
  PRD31,725(1985).
• For particle frame with new EoM, see K.Tsumura
  and T.Kunihiro, PLB668, 425(2008).
• For a possible relation btw. stability and
  causality, see G.S.Danicol et al., J.Phys.G35,
  115102(2008).

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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)).
Stability condition O.K.
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The 2nd Law of Thermodynamics 2nd order case
Sometimes omitted, but needed.
? Generalization of thermodynamic force!?
Same equation, but different definition of p and
P.
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SummaryConstitutive Equations
w vorticity

• 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.
  (Sometimes the last term is neglected.)

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Plan of this Lecture

• 1st Day
• Hydrodynamics in Heavy Ion Collisions
• Collective flow
• Dynamical Modeling of heavy ion collisions
  (seminar)
• 2nd Day
• Formalism of relativistic ideal/viscous
  hydrodynamics
• Bjorkens scaling solution with viscosity
• Effect of viscosity on particle spectra
  (discussion)

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PART 4

• Bjorkens Scaling Solution with Viscosity

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Bjorken Coordinate
t
Boost ? parallel shift Boost invariant ?
Independent of hs
z
0
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Bjorkens Scaling 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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Conserved and Non-Conserved Quantity in Scaling
Solution
expansion
pdV work
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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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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.

Kovtun, Son, Starinet,
Nakamura, Sakai,
Kharzeev, Tuchin, Karsch, Meyer
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Bjorkens Equationin the 2nd Order Theory
where
New terms appear in the 2nd order theory. ?
Coupled differential equations Sometimes, the
last terms are neglected. Importance of these
terms ? see Natsuume and Okamura,
0712.2917hep-th.
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Why only p00-pzz?
In EoM of energy density,
appears in spite of constitutive
equations. According to the Bjorken solution,
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Relaxation Equation?
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Digression Full 2nd order equation?
Beyond I-S equation, see R.Baier et al., JHEP
0804,100 (2008) Tsumura-Kunihiro? D. Rischke,
talk at SQM 2008. According to Rischkes talk,
constitutive equations with vanishing heat flow
are
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Digression (contd.) Bjorkens Equationin the
full 2nd order theory
See also, R.Fries et al.,PRC78,034913(2008).
Note that the equation for shear is valid only
for conformal EOS and that no 2nd and 3er terms
for bulk.
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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 t0 in Comoving Frame
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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Remarks

• Sometimes results from ideal hydro are compared
  with the ones from 1st order theory. But initial
  conditions must be different.
• Be careful what is attributed for the difference
  between two results.
• Sensitive to initial conditions and new
  parameters (relaxation time for stress tensor)

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Plan of this Lecture

• 1st Day
• Hydrodynamics in Heavy Ion Collisions
• Collective flow
• Dynamical Modeling of heavy ion collisions
  (seminar)
• 2nd Day
• Formalism of relativistic ideal/viscous
  hydrodynamics
• Bjorkens scaling solution with viscosity
• Effect of viscosity on particle spectra
  (discussion)

60
PART 5

• Effect of Viscosity on Particle Spectra

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Particle Spectra in Hydrodynamic Model

• How to compare with experimental data (particle
  spectra)?
• Free particles (l/Lgtgt1) eventually stream to
  detectors.
• Need prescription to convert hydrodynamic
  (thermodynamic) fields (l/Lltlt1) into particle
  picture.
• Need kinetic (or microscopic) interpretation of
  hydrodynamic behavior.

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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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Cooper-Frye Formula

• No dynamics of evaporation.
• Just counting the net number of particles
• (out-going particles) - (in-coming particles)
• through hypersurface S
• Negative contribution can appear at some
• space-like hyper surface elements.

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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
Neglecting anti-particles,
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1
4
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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14 Conditions
Landau conditions (2)
Epsilons can be expressed by dissipative currents
Viscosities (12)
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Relation btw. Coefficients and Dissipative
Currents
Finally,
For details, see Sec.6 and Appendix C
in Israel-Stewart paper. (Ann.Phys.118,341(1979))
73
Remarks

• Boltzmann equation gives a microscopic
  interpretation of hydrodynamics.
• However, hydrodynamics may be applied for
  gas/liquid where the Boltzmann equation can not
  be applied.
• Deviation from equilibrium can be incorporated
  into phase space distribution.

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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