Second-order Relativistic Hydrodynamic Equations Compatible with Boltzmann Equation

Second-order Relativistic Hydrodynamic Equations Compatible with Boltzmann Equation and Critical Opalescence around the QCD Critical Point Teiji Kunihiro (Kyoto) Yuki Minami (Kyoto) and Kyosuke Tsumura (Fuji Film co. ) Quark Matter 2009 Knoxville, March 30 --- April 4, 2009

Critical Opalescence around the QCD Critical Point and Second-order Relativistic Hydrodynamic Equations Compatible with Boltzmann Equation Teiji Kunihiro (Kyoto) Yuki Minami (Kyoto) and Kyosuke Tsumura (Fuji Film co. ) Quark Matter 2009 Knoxville, March 30 --- April 4, 2009

Contents I Density fluctuations around QCD critical point with rel. dissipative hydrodynamics; new possible signal for identifying the QCD critical point* II Derivation of Israel-Stewart type hydrodynamic equations on the basis of the (dynamical) renormalization group method** ; only brief exposition of the result * Yuki Minami and T. K. , in preparation, ** Kyosuke Tsumura and T. K. , in preparation

Classical Liq. -Gas P Critical point Solid Liq. gas Triple. P T The same universality class; Z 2 H. Fujii, PRD 67 (03) 094018; H. Fujii and M. Ohtani, Phys. Rev. D 70(2004) Dam. T. Son and M. A. Stephanov, PRD 70 (’ 04) 056001 Density fluctuation is the soft mode of QCD critical point! The sigma mode is a slaving mode of the density. c. f. The coupling of the density fluctuation with the scalar mode was discussed in, T. K. Phys. Lett. B 271 (1991), 395

Spectral function of density fluctuations The density fluctuation depends on the transport as well as thermodynamic quantities which show an anomalous behavior around the critical point. Especially, the existence of the density-temperature coupling. Missing in the previous analyses. For non-relativistic case with use of Navier-Stokes eq. L. D. Landau and G. Placzek(1934), L. P. Kadanoff and P. C. Martin(1963), R. D. Mountain, Rev. Mod. Phys. 38 (1966), 38 H. E. Stanley, `Intro. To Phase transitions and critical phenomena’ (Clarendon, 1971) We apply for the first time relativistic hydrodynamic equations to analyze the spectral properties of density fluctuations, and examine possible critical phenomena.

Relativistic Hydrodynamics dissipative terms (1) Energy-frame (2) Particle frame ; Eckart(1940), unstable Tsumura-Kunihiro-Ohnishi, Phys. Lett. B 646(2007) (3) Israel-Stewart

Linear approximation around thermal equilibrium; etc In the rest frame of the fluid, Inserting them into , and taking the linear approx. ・Linearized Landau equation (Lin. Hydro in the energy frame); Rel. effects with Solving as an initial value problem using Laplace transformation, we obtain , in terms of the initial correlation.

Spectral function of density fluctuations in the Landau frame In the long-wave length limit, k→ 0 sound modes thermal mode Rel. effects appear only in the width of the peaks. Rel. effects rate of isothermal expansion rate：：sound velocity Notice: ：specific heat Long. Dynamical ： ratio enthalpy As approaching the critical point, the ratio of specific heats diverges! The strength of the sound modes vanishes out at the critical point.

Eq. of State of ideal Massless particles thermal mode Rayleigh peak sound mode Brillouin peak [1/fm] Brillouin peak [Me. V] [1/fm] Rel. Non-rel. In the energy(Landau) frame, relativistic effects appear only in the peakheight and width of the Rayleigh peak.

Spectral function from I-S eq. IS Non-rel. (N-S) [1/fm] For No contribution in the long-wave length limit k→ 0. Conversely speaking, the first-order hydro. Equations have no problem to describe the hydrodynamic modes with long wave length, as it should.

Particle frame; the new equation TKO= K. Tsumura, T. K. K. Ohnishi; Phys. Lett. B 646 (2007) 134 Rel. case (TKO) Non-rel. (Navier-Stokes) [1/fm] Rel. effects appear in the Brillouin peaks (sound mode) but not in The Rayleigh peak.

Critical behavior of the density-fluctuation spectral functions In the vicinity of CP, only the Rayleigh peak stay out, while the sound modes (Brillouin peaks) die out. Critical opalescence c. f. So, the divergence of and the viscocities therein can not be observed, unfortunately.

Spectral function of density fluctuation at CP 0. 4 The sound mode (Brillouin) disappears Only an enhanced thermal mode remains. Spectral function at CP The soft mode around QCD CP is thermally induced density fluctuations, but not the usual sound mode. Suggesting interesting critical phenomena related to sound mode. Eg. Disappearance of Mach cone at CP! A hint for detecting CP! Needs explicit examination.

Cf. STAR, ar. Xiv: 0805/0622, to be published in PRL R. B. Neufeld, B. Muller, and J. Ruppert, ar. Xiv: 0802. 2254 g. T (z t) u g. L

Why at all do sound modes die out at the Critical Point ? The correlation length The wave length of sound mode The hydrodynamic regimel << However, around the critical point as t 0 << So the hydrodynamic sound modes can not be developed around CP!

II An RG derivation of 2 nd order hydrodynamic equations K. Tsumura and T. K. , preliminary results was presented at JPS meeting Sep. 23, 2008, in preparation

Relativistic Boltzmann equation Conservation law of the particle number and the energy-momentum H-theorem. The collision invariants, the system is local equilibrium Maxwell distribution (N. R. ) Juettner distribution (Rel. )

Geometrical image of reduction of dynamics X Invariant and attractive manifold eg. O M ; distribution function in the phase space (infinite dimensions) ; the hydrodinamic quantities (5 dimensions), conserved quantities.

T. K. Prog. Theor. Phys. 94 (’ 95), 503; 95(’ 97), 179 S. -I. Ei, K. Fujii and T. K. , Ann. Phys. 280 (2000), 236 RG/E equation Y. Hatta and T. Kunihiro, Ann. Phys. 298 (2002), 24 T. K. and K. Tsumura, J. Phys. A: Math. Gen. 39 (2006), Slow dynamics (Hydro dynamics + relaxation equations） (I) Energy frame Particle frame (II) Relaxation equations (very long)

The viscocities are frame-independent, in accordance with Lin. Res. Theory. However, the relaxation times and legths are frame-dependent. The form is totally different from the previous ones like I-S’s, And contains many additional terms. contains a zero mode of the linearized collision operator. Conformal non-inv. gives the ambiguity.

Summary • Density fluctuations is analyzed using rel. hydro. • The second-order terms in IS theory do not affect the (longwave length) hydrodynamic modes. • As a dynamical critical phenomena, Brillouin peaks due to sound modes disappear. The disappearance of the sound mode suggests the suppresion or even disappearance of Mach cone, which can be a signal of the created matter hitting CP. Need further explicit calculation for confirmation, • The (dynamical) RG method is applied to derive generic second-order hydrodynamic equations. • There are so many terms in the relaxation terms which are absent in the previous works, especially due to the conformal non-invariance, which gives rise to an ambiguity in the separation in the first order and the second order terms (matching condition). • A practical use of I-S level equations, however, may be problematic.

Critical Opalescence is a general phenomenon for the matter with 1 st order transition. © University of Cambridge Do. ITPo. MS, Department of Materials Science and Metallurgy, University of Cambridge Information provided by doitpoms@msm. cam. ac. uk. Experiment at lab. HBT? http: //www. msm. cam. ac. uk/doitpoms/tlplib/solid-solutions/videos/laser 1. mov

Back Ups

for Definitions of critical exponents

Israel-Stewart eq. in particle frame ： relaxation times tends to coincide with the Eckart equation.

Stability of I-S eq. Eckart eq. (1 st order); → diverges; unstable I-S eq. Rel. time of thermal conductivity → stable. → unstable! even with finite rel. time. W. A. Hiscock and L. Lindblom Phys. Rev. D 35(1987)

B. Muller@NFQCD 08 STAR data Away side shape modification Central Au+Au 0 -12% (STAR) STAR 2. 5 < p. Ttrig< 4 Ge. V/c 1< p. Tassoc < 2. 5 Ge. V/c Technique: Measure 2 - and 3 particle correlations on the awayside triggered by “high” p. T hadron in central coll’s. Cone-shaped emission should show up in 3 particle correlations as signal on both sides of backward direction. ( 1 - 2)/2

B. Muller@NFQCD 08 The Mach cone R. B. Neufeld, B. Muller, and J. Ruppert, ar. Xiv: 0802. 2254 Unscreened source with min/max cutoff g. T Energy density (z ut ) - Momentum density g. L

References on the RG/E method: • • T. K. Prog. Theor. Phys. 94 (’ 95), 503; 95(’ 97), 179 T. K. , Jpn. J. Ind. Appl. Math. 14 (’ 97), 51 T. K. , Phys. Rev. D 57 (’ 98), R 2035 T. K. and J. Matsukidaira, Phys. Rev. E 57 (’ 98), 4817 S. -I. Ei, K. Fujii and T. K. , Ann. Phys. 280 (2000), 236 Y. Hatta and T. Kunihiro, Ann. Phys. 298 (2002), 24 T. K. and K. Tsumura, J. Phys. A: Math. Gen. 39 (2006), 8089 (hep-th/0512108) • K. Tsumura, K. Ohnishi and T. K. , Phys. Lett. B 646 (2007), 134 C. f. L. Y. Chen, N. Goldenfeld and Y. Oono, PRL. 72(’ 95), 376; Phys. Rev. E 54 (’ 96), 376.