Relativistic Hydrodynamics

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Relativistic Hydrodynamics
new solutions with ellipsoidal symmetry
T. Csörgo (KFKI RMKI Budapest)

• Fireball hydrodynamics
• Simple models work well at SPS and RHIC Why?
• Theoretically challenging, difficult problem
• New classes of simple solutions
• Non-relativistic as well as relativistic
• Spherical, cylindrical and ellipsoidal, d1,2,3
• With S. V. Akkelin, L. P. Csernai, P. Csizmadia,
  B. Lörstad, F. Grassi, Y. Hama, T. Kodama, Yu.
  Sinyukov, J. Zimányi,

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In Search of the QGP. Naïve expectations
QGP has more degrees of freedom than pion gas,
hence it has higher entrophy density Entropy
should be conserved during fireball
evolution Hence Look in hadronic phase for
signs of Large spatial size,
Long lifetime, Long
duration of particle emission

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Scaling laws in the observed data

• Slope parameter (effective temperature) mass
• Effective volumes (horizont radii) scale
  mass-0.5

PbPb_at_CERN SPS the effective volumes and
temperatures are independent of particle type,
size, etc, depend only on their mass.
Buda-Lund hydro model gives a natural
explanation to this unexpected observations in
high energy physics. Similar results are
observed also at RHIC -gt see the next fits
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BRAHMS Effective temperature vs mass Hubble
diagram of galaxies and of RHIC
Teff(y0) Tf m ltugt2
Teff(y) Teff(y0) / (1 a y2)
Prediction, Buda-Lund hydro parametr. (Cs. T., B.
Lorstad, 94-96, axial symmetry non-central
collisions with Akkelin, Hama, Sinyukov, Csanad
and Ster (2003)
Edwin Hubble
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Relativistic hydrodynamics

• A baseline for various theoretical aspects of
  RHIC physics / part of our folclore
• Bjorken scaling solution (discovered by R. C.
  Hwa )
• A good example of false illusions
• Phase transitions and observables can be
  calculated relatively easily
• Provides an energy density estimate that is easy
  to measure (good up to a factor of 5
  uncertainty).
• J.D. Bjorken, Phys. Rev. D27, 140 (1983),
• R. C. Hwa, Phys. Rev. D10, 2260 (1974)
• Guess, how many citations these papers have
  received?

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Relativistic hydrodynamics
Charge conservation 4-momentum
conservation 4-velocity field normalization Perf
ect fluid EOS 1 energy density EOS 2
pressure Non-relativistic limit Ultrarelativisti
c gas
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Energy and Euler equations
Decomposing 4-momentum conservation energy
equation rel. Euler equation general
thermodynamic considerations E TS mN - pV -gt
entrophy conservation Energy equation EOS 1
and EOS 2 temperature equation 5 independent
variables 5 equations Euler (3) continuity
temperature
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Self-similar ansatz
Direction dependent Hubble profile coordinates
physical quantities assumption of
self-similarity Trival volume dependence, additio
nal coordinate dependence only through scaling
variable s definition of s why is s a good
scaling variable ?
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New ellipsoidal solutions
Scales depend linearly on time Hence the Hubble
flow becomes spherically symmetric but the
density profile contains an arbitrary scaling
function n(s) that limits the solution to
ellipsoidal symmetry The pressure depends
only on proper time and speed of sound The
temperature is also ellipsoidally symmetric, has
a scaling function related to the density
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Interesting features
Direction independent Hubble flow same as in
the successfull Cracow hydro model of
Broniowski, Florkowski, Baran et al. Simplest
case the density profile and temperature profile
are both constants In non-relativistic limit,
goes back to non-rel hydro new families of
solutions Not yet the final word i) lack of
acceleration ii) directional dependence of
flow? iii) temperature and density cannot yet go
to 0 at the surface continuosly (cut needed)
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Even more generalizations ...
Natural generalizations for more realistic
equations of state Bag model EOS Temperature
dependent speed of sound (similar to non-rel case
with Akkelin, Hama, Lukacs, Sinyukov, PRC
2003) Description for phase transitions, and
sudden timelike deflagrations Accelerating and
smooth surface solutions are related, work in
progress.
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Outlook
Search for hydro solutions behind succesfull
hydro parameterizations (Cracow, Buda-Lund) is
underway Stepping stones are found Generalizat
ion is straightforward?