PART 2
Formalism	  of	  rela,vis,c	  ideal/viscous	  hydrodynamics	  Adver&sement:	  Lecture	  Notes	  Hydrodynamics• Framework	  to	  describe	  space-­‐&me	  evolu&on	  of	  thermodynamic	  variables	  • Balance	  equa&ons	  (equa&ons	  of	  mo&on,	  conserva&on	  law)	  	  	  	  	  +	  equa&on	  of	  state	  (maEer	  property)	  	  	  	  +	  cons&tu&ve	  equa&ons	  (phenomenology)	  Relativistic Hydrodynamics
Equa&ons	  of	  mo&on	  in	  rela&vis&c	  hydrodynamics	  Energy-­‐momentum	  conserva&on	  Energy-­‐Momentum	  tensor	  Current	  conserva&on	  The	  i-­‐th	  conserved	  current	  In	  H.I.C.,	  Niµ	  =	  NBµ (net	  baryon	  current)	  Tensor/Vector Decomposition
Tensor	  decomposi&on	  with	  a	  given	  &me-­‐like	  and	  normalized	  four-­‐vector	  uµ	

where,	  “Projection” Tensor/Vector•uµ	  is	  local	  four	  ﬂow	  velocity.	  More	  precise	  meaning	  will	  be	  given	  later.	  	  •uµ	  is	  perpendicular	  to	  Δµν.	  
•Local	  rest	  frame	  (LRF):	  •Naively	  speaking,	  uµ	  (Δµν)	  picks	  up	  &me-­‐	  (space-­‐)like	  component(s).	  Intuitive Picture of Projection&me	  like	  ﬂow	  vector	  ﬁeld	  Decomposition of Tµυ	:Energy	  density	  :Energy	  (Heat)	  current	  :Shear	  stress	  tensor	  :(Hydrosta&c+bulk)	  pressure	  	  	  	  P	  =	  Ps	  +	  Π	<…>:	  Symmetric,	  traceless	  and	  transverse	  to	  uµ	  &	  uν	

Decomposition of Nµ	:charge	  density	  :charge	  current	  Q.	  Count	  the	  number	  of	  unknowns	  in	  the	  above	  decomposi&on	  and	  conﬁrm	  that	  it	  is	  10(Tµν)+4k(Niµ).	  
Here	  k	  is	  the	  number	  of	  independent	  currents.	  
Note:	  If	  you	  consider	  uµ	  as	  independent	  variables,	  you	  need	  addi&onal	  constraint	  for	  them.	  	  If	  you	  also	  consider	  Ps	  as	  an	  independent	  variable,	  you	  need	  the	  equa&on	  of	  state	  Ps=Ps(e,n).	  Ideal and Dissipative Parts
Energy	  Momentum	  tensor	  Charge	  current	  Ideal	  part	  Dissipa&ve	  part	  Meaning of uµ	uµ	  is	  four-­‐velocity	  of	  “ﬂow”.	  What	  kind	  of	  ﬂow?	  
Two	  major	  deﬁni&ons	  of	  ﬂow	  are	  1.	  Flow	  of	  energy	  (Landau)	  2.	  Flow	  of	  conserved	  charge	  (Eckart)	  Meaning of uµ (contd.)Landau	  (Wµ=0,	  uLµVµ=0)	  uLµ	Vµ	Eckart	  (Vµ=0,uEµWµ=0)	  Wµ	uEµ	

Just	  a	  choice	  of	  local	  reference	  frame.	  
Landau	  frame	  can	  be	  relevant	  in	  H.I.C.	  Relation btw.
Landau and Eckart

Relation btw.
Landau and Eckart (contd.)Entropy Conservation
in Ideal Hydrodynamics
Neglect	  “dissipa&ve	  part”	  of	  energy	  momentum	  tensor	  to	  obtain	  “ideal	  hydrodynamics”.	  

Therefore,	  Q.	  Derive	  the	  above	  equa&on.	  Entropy Current
Assump&on	  (1st	  order	  theory):	  Non-­‐equilibrium	  entropy	  current	  vector	  has	  linear	  dissipa&ve	  term(s)	  constructed	  from	  (Vµ,	  Π,	  πµν,	  (uµ)).	  
(Prac&cal)	  Assump&on:	  •Landau	  frame	  (omigng	  subscript	  “L”).	  
•No	  charge	  in	  the	  system.	  
Thus,	  α	  =	  0	  since	  Nµ	  =	  0,	  Wµ	  =	  0	  since	  considering	  the	  Landau	  frame,	  and	  γ	  =	  0	  since	  uµ	  Sµ	  should	  be	  maximum	  in	  equilibrium	  (stability	  condi&on).	  The 2nd Law of Thermodynamics
and Constitutive Equations
The	  2nd	  thermodynamic	  law	  tells	  us	  Q.	  Check	  the	  above	  calcula&on.	  

Constitutive Equations (contd.)
Thermodynamic
force

Transport
coefficient

Newton	  tensorshear

Stokes	  scalarbulk“Current”

Equation of Motion:	  Lagrange	  (substan&al)	  	  	  deriva&ve	  :	  Expansion	  scalar	  (Divergence)	  Equation of Motion (contd.)Q.	  Derive	  the	  above	  equa&ons	  of	  mo&on	  	  from	  energy-­‐momentum	  conserva&on.	  	  Note:	  We	  have	  not	  used	  the	  cons&tu&ve	  	  equa&ons	  to	  obtain	  the	  equa&ons	  of	  mo&on.	  Intuitive Interpretation of EoM

Change	  of	  volume	  •Dilu&on	  •Compression	  Work	  done	  by	  pressure	  Produc&on	  of	  entropy	  Conserved Current CaseLessons from (Non-Relativistic)
Navier-Stokes Equation
Assuming	  incompressible	  ﬂuids	  such	  that	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  ,	  Navier-­‐Stokes	  eq.	  becomes	  Diﬀusion	  of	  ﬂow	  Source	  of	  ﬂow	  	  (Kinema&c	  viscosity,	  η/ρ, 	(pressure	  gradient)	  plays	  a	  role	  of	  diﬀusion	  constant.)	  Final	  ﬂow	  velocity	  comes	  from	  interplay	  between	  these	  two	  eﬀects.	  Generation of FlowP	  Pressure	  gradient	  Expand	  Source	  of	  ﬂow	  Expand	  x	  	  Flow	  phenomena	  are	  important	  in	  H.I.C	  to	  understand	  EOS	  Diffusion of FlowHeat	  equa&on	  (κ:	  heat	  conduc&vity	  ~diﬀusion	  constant)	  	  For	  illustra&ve	  purpose,	  one	  discre&zes	  the	  equa&on	  in	  (2+1)D	  space:	  Diffusion ~ Averaging~ Smoothing
R.H.S.	  of	  descre&zed	  heat/diﬀusion	  eq.	  

y	  y	  j	  subtract	  i	  x	  j	  i	  x	  Suppose	  Τi,j	  is	  larger	  (smaller)	  than	  an	  average	  value	  	  around	  the	  site,	  R.H.S.	  becomes	  nega&ve	  (posi&ve).	  
2nd	  deriva&ve	  w.r.t.	  coordinates	  	  Smoothing	  Shear Viscosity Reduces
Flow Difference
Shear	  ﬂow	  Smoothing	  of	  ﬂow	  (gradient	  of	  ﬂow)	  Next	  &me	  step	  Microscopic	  interpreta&on	  can	  be	  made.	  
Net	  momentum	  ﬂow	  in	  space-­‐like	  direc&on.	  
	  Towards	  entropy	  maximum	  state.	  Necessity of Relaxation TimeNon-­‐rela&vis&c	  case	  (CaEaneo(1948))	  Balance	  eq.:	  Cons&tu&ve	  eq.:	  Fourier’s	  law	  τ	  :	  “relaxa&on	  &me”	  Parabolic	  equa&on	  (heat	  equa&on)	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  ACAUSAL!	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  Finite	  τ	Hyperbolic	  equa&on	  (telegraph	  equa&on)	  Heat Kernelx	  x	  perturba&on	  on	  top	  of	  	  background	  Heat	  transporta&on	  causality	  Entropy Current (2nd)Assump&on	  (2nd	  order	  theory):	  Non-­‐equilibrium	  entropy	  current	  vector	  has	  linear	  +	  quadra&c	  dissipa&ve	  term(s)	  constructed	  from	  (Vµ,	  Π,	  πµν,	  (uµ)).	  

Stability	  condi&on	  O.K.	  The 2nd Law of Thermodynamics:
2nd order caseSome&mes	  omiEed,	  	  but	  needed.	  

	  Generaliza&on	  of	  thermodynamic	  force!?	  

Same	  equa&on,	  but	  diﬀerent	  deﬁni&on	  of	  π	  and	  Π.	  Summary:
Constitutive Equations
ω:	  vor&city	  •Relaxa&on	  terms	  appear	  (τπ	  and	  τΠ	  are	  relaxa&on	  &me).	  
•No	  longer	  algebraic	  equa&ons!	  Dissipa&ve	  currents	  become	  dynamical	  quan&&es	  like	  thermodynamic	  variables.	  
•Employed	  in	  recent	  viscous	  ﬂuid	  simula&ons.	  
(Some&mes	  the	  last	  term	  is	  neglected.)	  PART 3
Bjorken’s Scaling Solution
with Viscosity

“Bjorken” Coordinatet	  Boost	  	  parallel	  shis	  Boost	  invariant	  	  Independent	  of	  ηs	  z	  0	  

Bjorken’s Scaling Solution
Assuming	  boost	  invariance	  for	  thermodynamic	  	  variables	  P=P(τ)	  and	  1D	  Hubble-­‐like	  ﬂow	  Hydrodynamic	  equa&on	  for	  perfect	  ﬂuids	  with	  a	  	  simple	  EoS,	  Conserved and Non-Conserved
Quantity in Scaling Solution

expansion	  pdV	  work	  Bjorken’s Equationin the 1st Order Theory(Bjorken’s	  solu&on)	  	  =	  (1D	  Hubble	  ﬂow)	  Q.	  Derive	  the	  above	  equa&on.	  Viscous Correction

Correc&on	  from	  shear	  viscosity	  (in	  compressible	  ﬂuids)	  Correc&on	  from	  bulk	  viscosity	  	  If	  these	  correc&ons	  vanish,	  the	  above	  equa&on	  reduces	  to	  the	  famous	  Bjorken	  equa&on.	  
Expansion	  scalar	  =	  theta	  =	  1/tau	  in	  scaling	  solu&on	  Recent Topics on Transport
Coefficients
Need	  microscopic	  theory	  (e.g.,	  Boltzmann	  eq.)	  to	  obtain	  transport	  coeﬃcients.	  
•	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  is	  obtained	  from	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  Super	  Yang-­‐Mills	  theory.	  
Kovtun,	  Son,	  Starinet,…	  •	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  is	  obtained	  from	  lagce.	  
Nakamura,	  Sakai,…	  •Bulk	  viscosity	  has	  a	  prominent	  peak	  around	  Tc.	  

Kharzeev,	  Tuchin,	  Karsch,	  Meyer…	  Bjorken’s Equationin the 2nd Order Theory

where	  New	  terms	  appear	  in	  the	  2nd	  order	  theory.	  
	  Coupled	  diﬀeren&al	  equa&ons	  Some&mes,	  the	  last	  terms	  are	  neglected.	  
Importance	  of	  these	  terms	  	  Why only π00-πzz?
In	  EoM	  of	  energy	  density,	  appears	  in	  spite	  of	  cons&tu&ve	  equa&ons.	  
According	  to	  the	  Bjorken	  solu&on,	  Relaxation Equation?

Digression:
Full 2nd order equation?
Beyond	  I-­‐S	  equa&on,	  see	  R.Baier	  et	  al.,	  JHEP	  0804,100	  (2008);	  Tsumura-­‐Kunihiro;	  D.	  Rischke,	  talk	  at	  SQM	  2008;	  A.Monnai	  and	  TH.	  According	  to	  Rischke’s	  talk,	  cons&tu&ve	  equa&ons	  with	  vanishing	  heat	  ﬂow	  are	  Digression (contd.): Bjorken’s Equationin the “full” 2nd order theory

See	  also,	  R.Fries	  et	  al.,PRC78,034913(2008).	  	  Note	  that	  the	  equa&on	  for	  shear	  is	  valid	  only	  for	  conformal	  EOS	  and	  that	  no	  2nd	  and	  3rd	  terms	  for	  bulk.	  Model EoS (crossover)
Crossover	  EoS:	  	  Tc	  =	  0.17GeV	  Δ	  =	  Tc/50	  gH	  =	  3,	  gQ	  =	  37	  Relativistic Ideal Gas
Thermodynamic	  poten&al	  for	  rela&vis&c	  ideal	  gases	  Energy-Momentum Tensor
at τ0 in Comoving Frame

In	  what	  follows,	  bulk	  viscosity	  is	  omiEed.	  Numerical Results(Temperature)
T0	  =	  0.22	  GeV	  τ0	  =	  1	  fm/c	  η/s	  =	  1/4π	

τπ	  =	  3η/4p	  Same	  ini&al	  	  condi&on	  (Energy	  momentum	  tensor	  is	  isotropic)	  Numerical	  code	  (C++)	  is	  available	  upon	  request.	  Numerical Results(Temperature)
T0	  =	  0.22	  GeV	  τ0	  =	  1	  fm/c	  η/s	  =	  1/4π	

τπ	  =	  3η/4p	  Same	  ini&al	  	  condi&on	  (Energy	  momentum	  tensor	  is	  anisotropic)	  Numerical	  code	  (C++)	  is	  available	  upon	  request.	  Numerical Results(Temperature)

Numerical	  code	  (C++)	  is	  available	  upon	  request.	  Numerical Results (Entropy)
T0	  =	  0.22	  GeV	  τ0	  =	  1	  fm/c	  η/s	  =	  1/4π	

τπ	  =	  3η/4p	  Same	  ini&al	  	  condi&on	  (Energy	  momentum	  tensor	  is	  isotropic)	  Numerical	  code	  (C++)	  is	  available	  upon	  request.	  Numerical Results (Entropy)
T0	  =	  0.22	  GeV	  τ0	  =	  1	  fm/c	  η/s	  =	  1/4π	

τπ	  =	  3η/4p	  Same	  ini&al	  	  condi&on	  (Energy	  momentum	  tensor	  is	  anisotropic)	  Numerical	  code	  (C++)	  is	  available	  upon	  request.	  

Numerical Results (Entropy)

Numerical	  code	  (C++)	  is	  available	  upon	  request.	  Numerical Results(Shear Viscosity)

Numerical	  code	  (C++)	  is	  available	  upon	  request.	  Numerical Results(Initial Condition Dependence
in the 2nd order theory)

Numerical	  code	  (C++)	  is	  available	  upon	  request.	  Numerical Results(Relaxation Time dependence)

Saturated	  values	  non-­‐trivial	  Relaxa&on	  &me	  larger	  Maximum	  π	  is	  smaller	  Relaxa&on	  &me	  smaller	  Suddenly	  relaxes	  to	  	  1st	  order	  theory	  Remarks• Some&mes	  results	  from	  ideal	  hydro	  are	  compared	  with	  the	  ones	  from	  1st	  order	  theory.	  But	  ini&al	  condi&ons	  must	  be	  diﬀerent.	  
• Be	  careful	  what	  is	  aEributed	  for	  the	  diﬀerence	  between	  two	  results.	  
• Sensi&ve	  to	  ini&al	  condi&ons	  and	  new	  parameters	  (relaxa&on	  &me	  for	  stress	  tensor)	  PART 4
Effect of Viscosity on
Particle SpectraParticle Spectrain Hydrodynamic Model• How	  to	  compare	  with	  experimental	  data	  (par&cle	  spectra)?	  
• Free	  par&cles	  (λ/L>>1)	  eventually	  stream	  to	  detectors.	  
• Need	  prescrip&on	  to	  convert	  hydrodynamic	  (thermodynamic)	  ﬁelds	  (λ/L<<1)	  into	  par&cle	  picture.	  
• Need	  kine&c	  (or	  microscopic)	  interpreta&on	  of	  hydrodynamic	  behavior.	  Cooper-Frye Formula•No	  dynamics	  of	  evapora&on.	  
•Just	  coun&ng	  the	  net	  number	  of	  par&cles	  	  	  	  (out-­‐going	  par&cles)	  -­‐	  	  (in-­‐coming	  par&cles)	  	  through	  hypersurface	  Σ	•Nega&ve	  contribu&on	  can	  appear	  at	  some	  	  space-­‐like	  hyper	  surface	  elements.	  Matter in (Kinetic) Equilibrium
Kine&cally	  equilibrated	  maEer	  at	  rest	  Kine&cally	  equilibrated	  maEer	  at	  ﬁnite	  velocity	  uµ	py	  py	  px	  Isotropic	  distribu&on	  px	  Lorentz-­‐boosted	  distribu&on	  Relativistic Transport Theory
Boltzmann	  equa,on	  Time	  evolu&on	  of	  phase	  space	  dist.	  for	  a	  rareﬁed	  gas:	  gain	  loss	  One-­‐par&cle	  phase	  space	  distribu&on	  Collision	  term	  Lorentz	  invariant	  transi&on	  rate	  Free Streaming, Gain and Lossp	  loss	   gain	  gain	   loss	  x	  Note:	  p	  is	  mass-­‐on-­‐shell.	  H-theorem“Entropy	  current”:	  H-theorem (contd.)H-theorem (cond.2)

H-theorem (contd.3)=0	  H-theorem (contd.4)

1	  0	  1	  One	  can	  iden&fy	  sµ	  with	  entropy	  current	  Collision Invariant
Binary	  collisions	  sa&sfy	  energy-­‐momentum	  conseva&on	  So	  where	  (collision	  invariant)	  Collision Invariant (contd.)Q.	  Check	  this	  iden&ty	  Conserva&on	  law	  5	  conserved	  currents	  (quan&&es)	  Equilibrium Distribution

Thus	  	  	  	  	  	  	  	  	  	  	  should	  be	  collision	  invariant	  α	  and	  βµ	  can	  depend	  on	  posi&on	  x	  and	  are	  to	  be	  µ/T	  and	  uµ/T	  Quantum Statistics
Bose	  enhancement	  or	  Pauli	  blocking	  Collision	  invariant	  Some Remarks• Collision	  term	  vanishes	  if	  f0	  is	  plugged	  in.	  
• If	  α	  and	  β	  is	  constant	  globally,	  f0	  can	  be	  a	  solu&on	  of	  Boltzmann	  equa&on.	  
• Local	  equilibrium	  distribu&on	  is	  NOT	  a	  solu&on	  of	  Boltzmann	  equa&on.	  
• Devia&on	  from	  local	  equilibrium	  distribu&on	  can	  be	  obtained	  from	  Boltzmann	  equa&on.	  Microscopic Interpretation
Single	  par&cle	  phase	  space	  density	  in	  local	  equilibrium	  (no	  entropy	  produc&on	  in	  Boltzmann	  eq.):	  Kine&c	  deﬁni&on	  of	  current	  and	  energy	  momentum	  tensor	  are	  1st Momentuµ	  is	  normalized,	  so	  we	  can	  always	  choose	  aµν	  such	  that	  	  1st Moment (contd.)

Vanishing	  for	  ν=	  i	  due	  to	  odd	  func&on	  in	  integrant.	  

Q.	  Go	  through	  all	  steps	  in	  the	  above	  deriva&on.	  2nd Moment2nd Moment (contd.)where	  Deviation from Equilibrium
Distribution

trace	  part	  	  scalar	  Important	  in	  a	  mul&-­‐component	  gas	  (A.Monnai	  and	  TH,	  (2009))	  Unknowns:	  4	  +	  10	  =	  14	  Taylor Expansion around
Equilibrium Distribution

Taylor Expansion around
Equilibrium Distribution (contd.)14 Conditions
Stability	  condi&ons	  (2)	  Viscosi&es	  (12)	  Epsilons	  can	  be	  expressed	  by	  dissipa&ve	  	  currents	  Stability Condition

Q.	  Check	  the	  above	  deriva&on.	  Stability Condition (contd.)Relation btw. Coefficients andDissipative Currents

Moments	  Bosons	  do	  not	  contribute…	  Solutions

Solutions (contd.)

Epsilons	  are	  expressed	  using	  hydrodynamic	  variables	  Able	  to	  calculate	  par&cle	  spectra	  Cooper-Frye Formulain Viscous Case

Transverse Momentum Spectra
for Pions

Bulk	  pressure	  	  Isotropic	  	  Π	  =	  -­‐ζθ	Shear	  stress	  tensor	  	  Traceless	  	  π	  =	  4ηθ/3	  Summary• Eﬀect	  of	  viscosity	  on	  par&cle	  spectra	  can	  be	  calculated	  using	  hydrodynamic	  variables.	  
• Important	  to	  constrain	  equa&on	  of	  state	  and	  transport	  coeﬃcients	  from	  experimental	  data	  All	  slides	  will	  be	  available	  at	  	  hEp://tkynt2.phys.s.u-­‐tokyo.ac.jp/qgphydro/	  「講義」	 
「2010年度集中講義」