# Presentation: 190577

## Exact results in analytic hydrodynamics 

**UTILIZING THE FLUID NATURE OF QGP**
**M. Csanád, ****T. Csörgő****, M. I. Nagy**** **
**ELTE**
**MTA KFKI RMKI **
**Budapest,  Hungary**
**Quark Matter 2008, Jaipur, Rajastan, India**** **
** ****February 8, ****200****8 **
** **

## High temperature superfluidity at RHIC!

**High temperature superfluidity**** at RHIC!**

**All “realistic” hydrodynamic calculations for RHIC fluids to date have assumed zero viscosity**
******= 0****  ****→********perfect fluid**
**a ****conjectured**** quantum limit:**_**P. **__**Kovtun**_**, **_**D.T. Son**_**, **_**A.O. **__**Starinets**_**, **_**hep-**__**th**__**/0405231**_** ****How **** “ordinary” ****fluids ****compare to**** ****this limit?****(4 ********) **η**/s**** > 10 **
**RHIC****’s perfect**** ****fluid **
**	****(4 ********) **η**/s****  ****~1 **** ****!**
**    ****T**** > ****2 Terakelvin**
**The hottest **
**& most perfect fluid **
**ever made...**

**(**4

- R. Lacey *et al*., Phys.Rev.Lett.98:092301,2007

## Relativistic hydrodynamics

**Energy-momentum tensor:**
**	****	****Relativistic **
**	****Euler equation:**
**	****Energy conservation:**
**	****Charge conservation:**
**	****	****	****Consequence is entropy conservation:**

## Context

**Renowned ****exact**** ****solutions**
**Landau-Khalatnikov solution****:  ****dn/dy ~ Gaussian**
**Hwa**** ****solution **** ****(**PRD 10, 2260 (**1974**)) **-**** ****Bjorken **** ********0 ****estimate (1983)**
**Chiu, Sudarshan and Wang: plateaux **
**Baym, Friman, Blaizot, Soyeur and Czyz: finite size parameter ******
**Srivastava, Alam, Chakrabarty, Raha and Sinha: dn/dy ~ Gaussian**
**	****	****Revival of interest:       **** ****Buda-Lund model + exact solutions, **
**	****	****	****	****        ****Biró,  Karpenko+Sinyukov, Pratt (2007),**
**	****	****	****	****        ****Bialas+Janik+Peschanski, Borsch+Zhdanov (2007)**
**New simple solutions**
**	****  ****Evaluation of measurables**
**	****	****Rapidity distribution****	****	**** Advanced initial energy density **
**	****	****HBT radii ****	****	****	**** ****Advanced life-time estimation**

## Goal

**Need for solutions that are:**
**explicit**
**simple**
**accelerating**
**relativistic**
**realistic / compatible with the data****:**
**	****	****	****lattice QCD EoS**
**	****	****	****ellipsoidal symmetry (spectra, v****2****, v****4****, HBT)**
**	****	****	****finite dn/dy**
**Report on a new class that satisfies these criteria**
**	****	****	****but not simultaneously**
**	****	****	**arXiv:0709.3677v1 [nucl-th] PRC(2008) in press

## Self-similar, ellipsoidal solutions

**Publication (for example):**
**T. Csörgő, 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**** ********(s)**** ****can be chosen ****freely.**
**	****	**** ****Viscous corrections in NR limit****:  ****known analytically.**

## New, simple, exact solutions

- If   = d = 1 , general solution is obtained, for
- ARBITRARY initial conditions. It is STABLE !

- Possible cases (one row of the table is one solution):

- Hwa-Bjorken, Buda-Lund type

- New, accelerating, d dimension

- d dimensional with p=p(,)  (thanks T. S. Biró)

- Special EoS, but general velocity

**Nagy,CsT, Csanád:**** ****arXiv:0709.3677v1**

## New simple solutions

**Different**** final states from ****similar**** initial states are reached by varying**** ******

## New simple solutions

**Similar**** final states from ****different**** initial states are reached by varying**** ******

## Rapidity distribution

**Rapidity distribution from the 1+1 dimensional solution, for  ********> 1.**
**T****f****: slope parameter.**

## Pseudorapidity distributions

**BRAHMS data fitted with the analytic formula of**
**Additionally: y********η  transformation**

## BRAHMS rapidity distribution

**BRAHMS data fitted with the analytic formula of**

## Advanced energy density estimate

- Fit result:  > 1
- Flows accelerate:     	do work
- 			initial energy density is higher than Bjorken’s

- Corrections due to work and  acceleration. Ref:

- For   > 1 (accelerating) flows, both factors > 1

## Advanced energy density estimate

- Correction depends on timescales, dependence is:

- With a tipical f/0 of ~8-10, one gets a correction factor of 2!

## Conjecture: EoS dependence of 0

- Four constraints
- 	1) Bj is independent of EoS (= 1 case)
-  	2) cs2= 1 case is solved for any > 0.5

- Corrections due to respect these limits.
- 3) cs2  dependence of  is known
- 4) Numerical hydro results

- Conjectured formula – given by the principle of Occam’s razor:

- Using  = 1.18, cs = 0.35,  f/0 = 10, we get  ecs/eBj = 2.9
-  0 = 14.5 GeV/fm3 in 200 GeV, 0-5 %  Au+Au at RHIC

## Advanced life-time estimate

**Life-time estimation: for Hwa-Bjorken type of flows**
** ****	****Makhlin ****& ****Sinyukov, Z. Phys. C 39, 69 (1988)**
**Underestimates lifetime (Renk, CsT, Wiedemann, Pratt****, ...****)**
**Advanced life-time estimate: **
**	****       ****width**** ****of**** ****dn/dy related to  acceleration and work**
**At RHIC energies: correction is about +20%**

## Conclusions

**Explicit simple accelerating relativistic hydrodynamics**
**Analytic (approximate) calculation of observables **
**Realistic rapidity distributions; BRAHMS data well described**
**No go theorem: same final states, different initial states**
**New estimate of initial energy density: **
**	****	********c****/********Bj****  ****~ 2 ****@ ****RHIC**
**	****	****dependence on  c****s**** estimated****,  ********c****/********Bj****  ****~  3 for c****s**** = 0.35**
**Estimated work effects on lifetime: **
**	****	****~**** 20%**** ****increase  @ RHIC**
**A lot to do ...**
**more general EoS**
**less symmetry, ellipsoidal solutions**
**asymptotically Hubble-like flows**

## New simple solutions in 1+D dim

**Fluid trajectories of the 1+D dimenisonal new solution**

## 
 Back-up Slides

** ****Back-up Slides**

## How Perfect is “Perfect” ? Measure η/s !

**Damping  (flow, fluctuations, heavy quark motion)  ~ **η**/s**
- FLOW: *Has the QCD Critical Point Been **Signaled by Observations at RHIC?,* R. Lacey *et al*., Phys.Rev.Lett.98:092301,2007 (_nucl__-ex/0609025_)
*The Centrality dependence of Elliptic flow, **the Hydrodynamic Limit, and the Viscosity **of Hot QCD*, H.-J. Drescher *et al*., (_arXiv__:0704.3553_)
- FLUCTUATIONS:* Measuring Shear Viscosity **Using Transverse Momentum Correlations **in Relativistic Nuclear Collisions*, S. Gavin and M. Abdel-Aziz, Phys.Rev.Lett.97:162302,2006 (_nucl__-__th__/0606061_)
- DRAG, FLOW:* Energy Loss and Flow of **Heavy Quarks in Au+Au Collisions **at √s**NN** = 200 GeV* (PHENIX Collaboration), A. Adare *et al*., Phys.Rev.Lett.98:172301,2007 (_nucl__-ex/0611018_)

**C****H****A****R****M****!**

## Landau-Khalatnikov solution

**Publications:**
**L.D. Landau, Izv. Acad. Nauk SSSR 81 (1953) 51**
**I.M. Khalatnikov, Zhur. Eksp.Teor.Fiz. 27 (1954) 529**
**L.D.Landau ****and**** S.Z.Belenkij, Usp. Fiz. Nauk 56 (1955) 309**
**Implicit 1D solution with approx. Gaussian rapidity distribution**
**Basic relations:**
**Unknown variables:**
**Auxiliary function:**
**Expression of                    is a true „tour de force”**

## Landau-Khalatnikov solution

**Temperature distribution (animation courtesy of ****T. Kodama****)**
**„****Tour de force” implicit solution:****	**** t=t(T,v), r=r(T,v)**

## Hwa-Bjorken solution

**The Hwa-Bjorken solution / Rindler coordinates**

## Hwa-Bjorken solution

**The Hwa-Bjorken solution / Temperature evolution**

## Bialas-Janik-Peschanski solution

**Publications:**
**A. Bialas, R. Janik, R. Peschanski,  ****arXiv:0706.2108v1**
**Accelerating, expanding **1D**  solution**
**	****interpolates between Landau and Bjorken**
**Generalized Rindler coordinates:**
**                         **

## Hwa-Bjorken solution

**Hwa-Bjorken solution**

**Publications:**
**R.C. Hwa****, Phys. Rev. D****10, 2260 (1974)**
**J.D. Bjorken, Phys. Rev. D27, 40(1983)**
**Accelerationless, expanding **1D** simple boost-invariant solution**
**Rindler coordinates:**
**Boost-invariance (valid for asymptotically high energies):**
**                         **

- depends on EoS, e.g.

## New simple solutions in 1+d dim

**Th****e fluid**** lines (****red****) and the  ****pseudo-orthogonal freeze-out surface (black)**** **

## Rapidity distribution

**Rapidity distribution**

**Rapidity distribution from the 1+1 dimensional solution, for            .**

## 1st milestone: new phenomena

**1****st**** milestone: new phenomena**

## 2nd milestone: new form of matter

## 3rd milestone: Top Physics Story 2005

**3****rd**** milestone: Top Physics Story 2005**

**http://arxiv.org/abs/nucl-ex/0410003**
**PHENIX White Paper: second most cited in nucl-ex during 2006 **

## 4th Milestone: A fluid of quarks

**4****th**** Milestone: ****A fluid of quarks**