1. Hydrodynamical description of first order phase transitions Vladimir Skokov (GSI, Darmstadt) in collaboration with D. N. Voskresensky Strongly Interacting Matter under Extreme Conditions Hirschegg 2010

2. Motivation Dynamics of an abstract order parameter non-conserved (CD analogue – model A) conserved (CD analogue – model B) Dynamics of liquid-gas type phase transition Numerical results Conclusions Outline

3. Phase diagram Schematic phase diagram CEP Phase coexistence RHIC CBM FAIR Experimental facilities: SPS (CERN) NA61 RHIC (BNL) STAR FAIR (GSI) CBM NICA (JINR) MPD To map the phase diagram experimentally we have to know consequences of CEP or first order phase transition.

4. Dynamics of order parameter Dynamics at phase transition governs by hydrodynamical modes: fields of order parameters and conserved charges. Conserved order parameter:Non-conserved order parameter: noise term Effective hamiltonian Kinetic coefficient CEP: h=0, v=0 ; First order PT line: h=0, v>0 ; Metastable state: h<>0, v>0;

5. Stationary solution Two stationary homogeneous solutions that are stable to small excitations: Noise term can be considered to be weak if the amplitude of the response to noise, v, is less than solutions of above equation.

6. Non-conserved OP Dimensionless form:

7. Solutions d=1, ε=0: d, ε=0: d, ε<<1: next slide

8. d, ε 0: Critical radius:

9. Non-spherical seeds For non-spherical seeds The coefficients ξ 0 for l>1 are damped. The seed becomes spherical symmetric during the evolution. Numerical results for large deviation from spherical forms and large values of ε.

10. Role of noise The noise term describes the short-distance fluctuations. The correlation radii both in space and time is negligible in comparison to correlation radii of order parameter. Thus the noise can be considered to be delta-correlated: Response to the noise Amplitude Radius Noise also affects seed shape

11. Gas-liquid type phase transition See also L. Csernai, J. Kapusta 92; L. Csernai, I. Mishustin 95; R. Randrup 08-09

12. Critical dynamics vs meanfield Critical region Phase diagram is effectively divided in two parts by the Ginzburg criterion (Gi): 1) region of critical fluctuation 2) region of validity of mean field approximation Conventional hydrodynamics Critical dynamics

13. System inside critical region (Gi »1) development of the critical fluctuations. The relaxation time of long-wave (critical) fluctuations is proportional to the square of the wave-length (in case of H-model the relaxation time τ ψ ~ ξ 3 ). In dynamical processes for successful development of the fluctuation of the system should be inside of the critical region for times much longer than the relaxation time of order parameter τ » τ ψ. In opposite case of fast (expansion) dynamics, the system spends short time near CP (τ « τ ψ ), and the fluctuations are not yet excited. This means that the system is not in full equilibrium, however the equilibrium with the respect to the interaction of neighboring region (short range order) is attained rapidly.

14. τ » τ ψ : critical fluctuations (fluctuations of transverse momentum, fl. of baryon density, etc.) sound attenuation (disappearance of Mach cone sin(φ)=c s /v, see Kunihiro et al 09) some models prredistion: CEP as an attractor of isentropic trajectories (proton/antiproton ration, see Asakawa et al, 09); c.f. Nakano et al. 09 etc… τ « τ ψ : Reestablishment of the mean field dynamics (mean field critical exponents, finite thermal conductivity, shear viscosity, not a Maxwell like construction below CEP, but rather non-monotonous dependence). Including all fluctuations

15. Hydrodynamics of 1order PT 1. Eq. for density fluctuations or sound mode 2. Eq. for specific entropy fluctuations or thermal mode 3. Eq. for longitudinal and transverse momentum (shear mode) current or hydrodynamical velocity. Decouples for fast processes from above two due to absence of mode-mode coupling terms (they are irrelevant for fast processes) Shear and bulk viscosities Reference values in vicinity of CEP Surface contribution

16. Equation of motion for density fluctuations in dimensionless form: fluidity of seeds is Controlling parameters for sound wave damping is Surface tension

17. Numerical results Condensed matter physics: Onuki 07

18. (T cr –T)/T cr =0.15; T cr =160 MeV; L=5 fm; β =0.2 R<R cr R>R cr droplet bubble Parameters are taken to be corresponded quark-hadron phase transition β ~ 0.02-0.2 (effectively viscous fluidity of seeds), even for conjectured lowest limit for ratio of shear viscosity to entropy density c.f. fireball lifetime~ 2L

19. Spinodal instability see also Randrup 09 growing modes k<k c oscillating modes k>k c amplitude of excitation

20. Dynamics in spinodal region. Blue – hadrons, Red – quarks.

21. Outlook Joint description of density and thermal transport Expansion to vacuum; initial conditions Realistic equation of state Transport coefficients

22. Conclusions The controlling parameter of the fluidity of seeds is viscosity-to-surface tension ratio. The larger viscosity and the smaller surface tension the effectively more viscous is the fluidity. Further details in: V.S. and D. Voskresensky, arXiv:0811.3868; V.S. and D. Voskresensky, Nucl.Phys.A828:401-438,2009 Anomalies in thermal fluctuations near CEP may have not sufficient time to develop. Spinodal instability and formation of droplets could be a promising signal of a phase transition. Hydrodynamic calculations that include stationary 1-order phase transition are questioned (the expansion time is less than relaxation time of phase separation.