new exact solutions of rel hydro - indico.cern.ch · zimanyi’19@budapest, 2019/12/05 csörgő, t....
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Zimanyi’19@Budapest, 2019/12/05 Csörgő, T.
A new family of exact solutionsof relativistic viscous hydrodynamics
T. Csörgő 1,2 and G. Kasza 1,2
1Wigner Research Center for Physics, Budapest, Hungary2EKE GYKRC, Gyöngyös, Hungary
Introduction and motivation
Viscous Relativistic Hydro
Multipole solutions with
temperature dependent speed of sound
Family of solutions with temperature profile
Outlook
Summary
Partially supported by
NKTIH FK 123842, FK123959
K 133046 and EFOP 3.6.1-16-2016-00001
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Zimanyi’19@Budapest, 2019/12/05 Csörgő, T.
Context
Renowned exact solutions, reviewed in arXiv:1805.01427
Landau-Khalatnikov solution: dn/dy ~ Gaussian
Hwa solution (1974) – Bjorken: same solution + e0 (1983)
Chiu, Sudarshan and Wang: plateaux, Wong: Landau revisited
Revival of interest: Zimányi, Bondorf, Garpman (1978)
Buda-Lund model + exact solutions (1994-96)
Biró, Karpenko, Sinyukov, Pratt (2007)
Bialas, Janik, Peschanski, Borsch+Zhdanov (2007)
CsT, Csanád, Nagy (2007-2008)
CsT, Csernai, Grassi, Hama, Kodama (2004)
Gubser (2010-11)
Hatta, Noronha, Xiao (2014-16)
CsT, Kasza, Csanad, Jiang (2017-18)
New simple solutions Evaluation of measurables
Rapidity distribution Advanced initial energy density (Kasza)
Viscous solutions Theorems
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Zimanyi’19@Budapest, 2019/12/05 Csörgő, T.
Goal
Need for solutions that are:
explicit
simple
accelerating
relativistic
viscous
realistic / compatible with the data:
lattice QCD EoS
ellipsoidal symmetry
multipole solution (spectra, v2, v3, v4 …, HBT)
finite dn/dy
Some of our recent new exact hydro solutions:
finite in y: CsT, Kasza, Csanád, Jiang (CKCJ): arXiv.org:1805.01427
viscous: Csanád, Nagy, Jiang, CsT: arxiv.org:1909.02498
This has been generalized to T-dependent speed of sound
to multipole solutions and to spatially dependent T profiles
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Zimanyi’19@Budapest, 2019/12/05 Csörgő, T.
Self-similar solutions for perfect fluid
Publication (for example):
T. Cs, L.P.Csernai, Y. Hama, T. Kodama, Heavy Ion Phys. A 21 (2004) 73
3D spherically symmetric HUBBLE flow:
No acceleration:
Define a scaling variable for self-similarly expanding ellipsoids:
EoS: (massive)
ideal gas
Scaling function n(s) can be chosen freely.
Shear and bulk viscous corrections in NR limit: known analytically.
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Zimanyi’19@Budapest, 2019/12/05 Csörgő, T.
New, exact solutions with viscosity
Csanád, Nagy, Jiang, CsT: arxiv.org:1909.02498
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Zimanyi’19@Budapest, 2019/12/05 Csörgő, T.
Exact solution with viscosity
Csanád, Nagy, Jiang, CsT: arxiv.org:1909.02498
Case A: Case C: Case D:
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Zimanyi’19@Budapest, 2019/12/05 Csörgő, T.
CNJC viscous solutions in 1+3 dim
Bulk viscosity important at late stage, heats upShear viscosity effects cancel for asymptotically Hubble flows
Csanád, Nagy, Jiang, CsT: arxiv.org:1909.02498
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Zimanyi’19@Budapest, 2019/12/05 Csörgő, T.
New viscous solutions in 1+3 dim
T-dependent speed of sound
CsT and G. Kasza
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Zimanyi’19@Budapest, 2019/12/05 Csörgő, T.
New viscous solutions in 1+3 dim
T-dependent speed of sound
Bulk viscosity effect at late time:same as rescaling Tinit
for a perfect fluid
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Zimanyi’19@Budapest, 2019/12/05 Csörgő, T.
New viscous solutions in 1+3 dim
T-dependent speed of sound
Bulk viscosity effect at late time:same as rescaling Tinit
for a perfect fluid
![Page 11: New exact solutions of rel hydro - indico.cern.ch · Zimanyi’19@Budapest, 2019/12/05 Csörgő, T. A new family of exact solutions of relativistic viscous hydrodynamics T. Csörgő1,2](https://reader033.vdocuments.mx/reader033/viewer/2022041712/5e489774df24f85a80378148/html5/thumbnails/11.jpg)
Zimanyi’19@Budapest, 2019/12/05 Csörgő, T.
New viscous solutions in 1+3 dim
T-dependent speed of soundNow for mB = 0 case
Bulk viscosity effect at late time:same as rescaling Tinit
for a perfect fluid
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Zimanyi’19@Budapest, 2019/12/05 Csörgő, T.
Conclusions
Explicit solutions of a very difficult problem
New exact solutions
for Hubble flows
for arbitrary EOS with const e/p
with T dependent e/p
for temperature profiles
for multipole solutions*
shear effects cancel at late time
bulk viscosity heats up matter
Late time evolution ~ perfect fluid, with rescaled Tinit
A lot to do …E.g. rotating viscous solutions
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Zimanyi’19@Budapest, 2019/12/05 Csörgő, T.
Thank you for your attention
Questions and Comments ?
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Zimanyi’19@Budapest, 2019/12/05 Csörgő, T.
Auxiliary variables: hx, t, W , hp, y
Consider a 1+1 dimensional, finite, expanding fireballAssume: W=W(hx)
Notation T. Cs., G. Kasza, M. Csanád, Z. Jiang, arXiv.org:1805.01427
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Zimanyi’19@Budapest, 2019/12/05 Csörgő, T.
Hydro in Rindler coordinates, new sol
Assumptions of TCs, Kasza, Csanád and Jiang, arXiv.org:1805.01427 :
From energy-momentum conservation, theEuler and temperature equations are obtained:
For the entropy density, the continuity equation is solved.
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Zimanyi’19@Budapest, 2019/12/05 Csörgő, T.
A New Family of Exact Solutions of Hydro
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Zimanyi’19@Budapest, 2019/12/05 Csörgő, T.
A New Family of Exact Solutions of Hydro
New: not discovered before,as far as we know …
Family:For each positive scaling
function t(s), a different
solution, with same T0, s0, k, l
Not self-similar: Coordinate dependenc NOT on
scaling variable s ONLY, butadditional dependence on
H = H(hx) too.
Explicit and Exact:Fluid rapidity, temperature,
entropy density explicitly givenby formulas
New feature:Solution is given
as parametric curves of H in hx :
(hx(H),W(H,t))Simlification, for now:
limit the solution in hx
where parametric curvescorrespond to functions
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Zimanyi’19@Budapest, 2019/12/05 Csörgő, T.
Illustration: results for T
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Zimanyi’19@Budapest, 2019/12/05 Csörgő, T.
Illustration: results for fluid rapidity W
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Zimanyi’19@Budapest, 2019/12/05 Csörgő, T.
Perfect fluid hydrodynamics
Energy-momentum
tensor:
Relativistic
Euler equation:
Energy conservation:
Charge conservation:
Consequence is entropy conservation:
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Zimanyi’19@Budapest, 2019/12/05 Csörgő, T.
New, exact solution with bulk viscosity