Hydrodynamical description of first order phase transitions

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. Outline • 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

3. Phase diagram CEP Phase coexistence Schematic phase diagram • Experimental facilities: • SPS (CERN) NA61 • RHIC (BNL) STAR • FAIR (GSI) CBM • NICA (JINR) MPD RHIC CBM FAIR 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: CEP: h=0, v=0 ; First order PT line: h=0, v>0 ; Metastable state: h<>0, v>0; Effective hamiltonian Kinetic coefficient noise term

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. Critical radius: d, ε<<1, ε>0:

9. Non-spherical seeds For non-spherical seeds The coefficients ξ0 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: ←Amplitude ←Radius Noise also affects seed shape Response to the noise

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 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 Critical region Critical dynamics “Conventional” hydrodynamics

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 τψ~ ξ3). 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 » τψ. In opposite case of fast (expansion) dynamics, the system spends short time near CP (τ « τψ), 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. Including all fluctuations • τ » τψ : • critical fluctuations (fluctuations of transverse momentum, fl. of baryon density, etc.) • sound attenuation (disappearance of Mach cone sin(φ)=cs/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 « τψ: 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).

15. Hydrodynamics of 1order PT Surface contribution Shear and bulk viscosities Reference values in vicinity of CEP 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)

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

17. Numerical results Condensed matter physics: Onuki ’07

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

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

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. 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. Further details in: V.S. and D. Voskresensky, arXiv:0811.3868; V.S. and D. Voskresensky, Nucl.Phys.A828:401-438,2009