Hydrodynamical Hydrodynamical description of first orderdescription of first order

phase transitionsphase transitions

Vladimir Skokov (GSI, Darmstadt)

in collaboration with D. N. Voskresensky

Strongly Interacting Matter under Extreme ConditionsHirschegg 2010

• 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

Phase diagramSchematic phase diagram

CEP

Phase coexistence

RH

ICCBM

FAIRExperimental 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.

Dynamics of order parameterDynamics 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;

Stationary solutionTwo 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.

Non-conserved OPDimensionless form:

Solutionsd=1, ε=0:

d, ε=0:

d, ε<<1: next slide

d, ε<<1, ε>0:

Critical radius:

Non-spherical seedsFor 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 largevalues of ε.

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

Gas-liquid type phase transition

See also L. Csernai, J. Kapusta ‘92; L. Csernai, I. Mishustin ’95;R. Randrup ’08-’09

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

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 rangeorder) is attained rapidly.

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).

Including all fluctuations

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

Equation of motion for density fluctuations in dimensionless form:

fluidity of seeds isControlling parameters for sound wave damping is

Surface tension

Numerical results

Condensed matter physics: Onuki ’07

(Tcr –T)/Tcr =0.15; Tcr=160 MeV; L=5 fm; β =0.2

R<RcrR>Rcr

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

Spinodal instability

see also Randrup ‘09

growing modes k<kc oscillating modes k>kc

ampl

itud

e of

exc

itat

ion

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

Outlook

Joint description of density and thermal transport

Expansion to vacuum; initial conditions

Realistic equation of state

Transport coefficients

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 viscousis 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 sufficienttime to develop. Spinodal instability and formation of droplets couldbe a promising signal of a phase transition.

Hydrodynamic calculations that include stationary 1-order phasetransition are questioned (the expansion time is less than relaxation time of phase separation.