Dynamics of net-baryon density correlations near the QCD critical point

Transcription

1 Dynamics of net-baryon density correlations near the QCD critical point Marcus Bluhm and Marlene Nahrgang The work of M.B. is funded by the European Union s Horizon 22 research and innovation programme under the Marie Skłodowska Curie grant agreement No via the National Science Centre, Poland, under grant Polonez UMO-216/21/P/ST2/435. From Correlation Functions to QCD Phenomenology Bad Honnef, Germany April 4, 218

2 Dynamical effects are very important... At the critical point: ξ fluctuations of the critical mode diverge! Higher moments more sensitive to ξ: σ 2 ξ 2, σ 3 ξ 9/2, σ 4 c ξ 7. Relaxation time τ rel ξ z diverges critical slowing down! B. Berdnikov, K. Rajagopal PRD61 (2) κσ c 4 /c 2 (x1 5 ) net protons.4<p T <.8GeV/c e/e C. Herold, MN, Y. Yan and C. Kobdaj PRC93 (216) no.2 K KA d Tc T T S. Mukherjee, R. Venugopalan, Y. Yin PRC92 (215) Matter created in a heavy-ion collision is small, short-lived and highly dynamical! Study fluctuations within diffusive dynamics of net-baryon number!

3 Goal of this talk Present a numerical implementation of the diffusive dynamics of net-baryon density fluctuations near the QCD critical point: thoroughly tested against equilibrium expectations discuss effects of resolution, system-size and net-baryon charge conservation τ* k vs. aξ z [a.u.] Gauss+surface τ* k Ginzburg Landau τ* k z = 4 ± T/T c deviations in scaling behavior of fluctuation observables from infinite-volume expectations! Stephanov, Phys.Rev.Lett.12 (29) dynamical critical scaling of model B! Hohenberg, Halperin, Rev.Mod.Phys.49 (1977) nonequilibrium and retardation effects in the dynamics! M. Nahrgang, MB, T. Schäfer, S. Bass, in preparation

4 Diffusive dynamics of the net-baryon density µ N µ B = net-baryon charge conservation The diffusive dynamics follows the minimized free energy F: ( ) t n B (t, x) = κ 2 δf[nb ] δn B

5 Diffusive dynamics of the net-baryon density µ N µ B = net-baryon charge conservation The diffusive dynamics follows the minimized free energy F: ( ) t n B (t, x) = κ 2 δf[nb ] + J(t, x) δn B To study intrinsic fluctuations include a stochastic current: J(t, x) = 2T κ ζ(t, x)

6 Diffusive dynamics of the net-baryon density µ N µ B = net-baryon charge conservation The diffusive dynamics follows the minimized free energy F: ( ) t n B (t, x) = κ 2 δf[nb ] + J(t, x) δn B To study intrinsic fluctuations include a stochastic current: J(t, x) = 2T κ ζ(t, x) ζ(t, x) is Gaussian and uncorrelated (white noise): ζ(x, t)ζ(, ) = δ(x)δ(t)

7 Diffusive dynamics of the net-baryon density µ N µ B = net-baryon charge conservation The diffusive dynamics follows the minimized free energy F: ( ) t n B (t, x) = κ 2 δf[nb ] + J(t, x) δn B To study intrinsic fluctuations include a stochastic current: J(t, x) = 2T κ ζ(t, x) ζ(t, x) is Gaussian and uncorrelated (white noise): ζ(x, t)ζ(, ) = δ(x)δ(t) respects the fluctuation-dissipation theorem: P eq [n B ] = 1 Z exp ( F[n B]/T )

8 Couplings motivated by 3-dimensional Ising model ( m F[n B ] = T d 3 2 r 2nc 2 nb 2 + K 2nc 2 ( n B ) 2 + λ 3 3nc 3 nb 3 + λ 4 4nc 4 nb 4 + λ ) 6 6nc 6 nb 6 The couplings depend on temperature via ξ(t ): m 2 = 1/(ξ ξ 2 ) K = K /ξ λ 3 = n c λ 3 (ξ/ξ ) 3/2 λ 4 = n c λ 4 (ξ/ξ ) 1 λ 6 = n c λ 6 M. Tsypin PRL73 (1994); PRB55 (1997) parameter choice: n B = n B n c ξ.5 fm, T c =.15 GeV, n c = 1/3 fm 3 K = 1/ξ (surface tension) λ 3, λ 4, λ 6 (universal, but mapping to QCD) ξ/ξ m 2 fm 3 λ 3 fm 3 λ 4 fm T/T c in this Fig.: λ 3 = 1, λ 4 = 1

9 Different physical scenarios The diffusion equation: t n B = D (m 2 K 2) 2 n n B c ( +D 2 λ3 nc 2 nb 2 + λ 4 nc 3 nb 3 + λ ) 6 nc 5 nb 5 + 2Dn c ζ

10 Different physical scenarios The diffusion equation: Gauss+surface t n B = D (m 2 K 2) 2 n n B c ( Ginzburg-Landau +D 2 λ3 nc 2 nb 2 + λ 4 nc 3 nb 3 + λ ) 6 nc 5 nb 5 + noise 2Dn c ζ consider 3 + 1d system with propagation in 1 spatial dimension diffusion coefficient D = κt /n c for equilibrium calculations D = 1 fm for dynamical calculations T -dependent equilibrium: let system equilibrate for long times dynamics: equilibrate first at high T, then evolve N B perfectly conserved in the numerics!

11 Equilibrium fluctuations

12 Gaussian model - static structure factor.1 T =.5 GeV N x = 64 N x = 128 N x = 256 N x = 512 static structure factor in continuum (ξ 2 = K /m 2 ): S k.1 Gauss+surface λ 3 = λ 4 = λ 6 = S(k) = n2 c m ξ 2 k 2 S k L = 1 fm T = T c N x = 64 N x = 128 N x = 256 N x = κ discretized form (k = 2πκ/L): (semi-implicit predictor-corrector scheme) S k = n2 c m K m 2 x 2 (1 cos(k x)) perfectly reproduced! S k S(k) for x = L/N x! non-zero surface tension suppresses short-wavelength fluctuations!

13 Ginzburg-Landau model - static structure factor.1 Gauss+surface Ginzburg Landau S k.1 T=T c.1 L = 2 fm 1 2 κ nonlinear interactions reduce S k for long-wavelength fluctuations! at this level, results of the Ginzburg-Landau model can effectively be described by a Gauss+surface model with modified m 2 (K essentially unaffected)!

14 Gaussian model - equal-time correlation function numerical correlation length ξ > x! spatial correlations broaden near T c! < n B (r) n B () > L = 5 fm L = 1 fm L = 2 fm L = 4 fm Gauss+surface, λ 3 =λ 4 =λ 6 = T=.5 GeV < n B (r) n B () > L = 5 fm L = 1 fm L = 2 fm L = 4 fm Gauss+surface, λ 3 =λ 4 =λ 6 = T=T c r/ x r/ x

15 Gaussian model - equal-time correlation function < n B (r) n B () > L = 5 fm L = 1 fm L = 2 fm L = 4 fm Gauss+surface, λ 3 =λ 4 =λ 6 = T=T c r/ x impact of exact N B -conservation: local n B -fluctuations need to be balanced within L L dr n B(r) n B () = negative correction term expected perfectly reproduced by the numerics!

16 Ginzburg-Landau model - correlation function < n B (r) n B () > Gauss+surface Ginzburg Landau T=T c L = 2 fm r/ x spatial correlations are significantly smaller! less pronounced finite-size effects!

17 Correlation length and finite-size effects For finite size L, the numerical correlation length is strongly limited due to N B -conservation. 8 7 Gauss+surface 6 Gauss+surface 6 ξ/ξ ξ/ξ 5 4 T=T c T=.5 GeV 2 L = 2 fm L [fm] T/T c very good realization of the relation ξ(t ) 1/m(T )!

18 Correlation length and finite-size effects For finite size L, the numerical correlation length is strongly limited due to N B -conservation. 8 7 Gauss+surface Ginzburg Landau 6 Gauss+surface Ginzburg Landau 6 ξ/ξ ξ/ξ 5 4 T=T c T=.5 GeV 2 L = 2 fm L [fm] T/T c very good realization of the relation ξ(t ) 1/m(T )! nonlinear mode couplings reduce the numerically realized correlation length in accordance with the effective m 2!

19 Volume-integrated variance infinite-volume expectation: σ 2 V = 1 V 2 dx dy nb (x) n B (y) ξ 2 M. Stephanov (et al.) PRL81 (1998); PRD6 (1999); PRL12 (29) exact charge conservation has a significant impact

20 Volume-integrated variance infinite-volume expectation: σ 2 V = 1 V 2 dx dy nb (x) n B (y) ξ 2 M. Stephanov (et al.) PRL81 (1998); PRD6 (1999); PRL12 (29) L = 2 fm Ginzburg-Landau including finite-size and N B -conservation we find for V 2 fm: σ 2 V ξn with n 1.4 ±.1!

21 Volume-integrated variance infinite-volume expectation: σ 2 V = 1 V 2 dx dy nb (x) n B (y) ξ 2 M. Stephanov (et al.) PRL81 (1998); PRD6 (1999); PRL12 (29) Ginzburg-Landau L = 2 fm increasing V increase in σv 2 (n gets closer to 2) before impact of charge conservation visible!

22 Dynamics of fluctuations

23 Dynamic structure factor and relaxation time dynamic structure factor for Gaussian model in continuum: ( S(k, t) = S(k) exp ( t/τ k ) with τ 1 k = Dm2 n c 1 + K k 2) k 2 m 2 τ k [fm] τ k [fm] T=.5 GeV Gauss+surface Ginzburg Landau T=T c Gauss+surface Ginzburg Landau analytic results reproduced by numerics! long-wavelength modes relax slowly! τ k significantly enhanced near T c! nonlinear interactions reduce τ k for modes with small k! κ

24 Dynamical critical scaling analyze ξ-dependence of relaxation time for modes with k = 1/ξ: τ* k vs. aξ z [a.u.] Gauss+surface τ* k Ginzburg Landau τ* k z = 4 ±.1 for both models: τ k = a ξz best fit gives: z = 4 ±.1 a = n cξ D(1 + K ) T/T c Simulations reproduce scaling of model B! Hohenberg, Halperin, Rev.Mod.Phys.49 (1977)

25 Dynamical critical scaling analyze ξ-dependence of relaxation time for modes with k = 1/ξ: τ* k vs. aξ z [a.u.] Gauss+surface τ* k Ginzburg Landau τ* k z = 3 (dashed) z = 5 (dotted) for both models: τ k = a ξz best fit gives: z = 4 ±.1 a = n cξ D(1 + K ) T/T c z = 3, 5 ruled out! Simulations reproduce scaling of model B! Hohenberg, Halperin, Rev.Mod.Phys.49 (1977)

26 Time-dependent temperature Hubble-like T -dynamics: T (τ) = T ( τ τ ) dc 2 s m 2 fm 3 ξ/ξ λ 3 fm 3 λ 4 fm 3 T/T c λ 6 fm τ τ [fm] τ τ [fm] choose c 2 s = 1/3 equilibrate system at T =.5 GeV, D = 1 fm D(T ) = D T /T T c reached at τ τ = 2.33 fm

27 Correlation function and length equal-time correlation function/length during the dynamics: clear deviations from equilibrium expectations rapid modification of T retardation effect equilibrium magnitude of ξ cannot develop nonequilibrium

28 Dynamics of the integrated variance shift of extremum (retardation effect) B. Berdnikov, K. Rajagopal PRD61 (2), S. Mukherjee et al., PRC92(3) (215) maximal value in dynamics smaller than equilibrium value (nonequilibrium effect) expected behavior with varying D

29 Dynamics of the integrated variance shift of extremum (retardation effect) B. Berdnikov, K. Rajagopal PRD61 (2), S. Mukherjee et al., PRC92(3) (215) maximal value in dynamics smaller than equilibrium value (nonequilibrium effect) expected behavior with varying D

30 Conclusions successful test of numerical implementation vs. analytic expectations! significant effect of net-baryon charge conservation! nonlinear interactions significantly reduce ξ finite size affects scaling behavior of fluctuation observables dynamics: critical scaling of model B, nonequilibrium and retardation effects! Thanks to:

31 Conclusions.8 successful test of numerical implementation vs. analytic expectations! significant effect of net-baryon charge conservation! nonlinear interactions significantly reduce ξ finite size affects scaling behavior of fluctuation observables dynamics: critical scaling of model B, nonequilibrium and retardation effects! σ 2 V (Sσ) V (κσ 2 ) V τ τ [fm/c] talk by M. Nahrgang Thanks to:

32 Fluid dynamical fluctuations in heavy-ion collisions The average particle spectra are successfully described using conventional fluid dynamics. What about fluctuations? ( µ T µν = µ T eq µν + T µν visc + Ξµν) = µ N µ = µ (N µ eq + N µ visc + Iµ) = important consequence: nonlinearities cause cutoff dependent corrections to EoS, η... ( e.g. e Λ 3) P. Kovtun et al., JHEP147 (214) box, x = 1 fm, e = 1 GeV/fm 3, η/s = t [fm/c] box_1d_1236_qviol_mitis_test2/outx.dat u 2:1: e [GeV/fm 3 ] < e> [% of e ] box, eta/s=.2 e =1GeV/fm x [fm] based on vhlle code /V [1/fm 3 ] Implementing fluid dynamical fluctuations is important, but requires a sustained and systematic effort! M. Nahrgang et al., CPOD216 I. Karpenko et al., Comput.Phys.Commun.185 (214)

33 T -dependence of local Gaussian fluctuations In finite systems: variance is reduced compared to TD expectations. variance σ Gauss+surface Ginzburg Landau T=T c T=.5 GeV L [fm] variance σ Gauss+surface Ginzburg Landau L = 2 fm T/T c nonlinear couplings in the Ginzburg-Landau model reduce the variance! Could explain why no sign of criticality is observed in second-order moments at NA49 and RHIC BES phase I.

34 T -dependence of local non-gaussian fluctuations Kurtosis vanishes in the absence of nonlinear interactions!.1 T=.5 GeV kurtosis κ T=T c kurtosis κ.1.2 Ginzburg Landau.3 Gauss+surface Ginzburg Landau L = 2 fm L [fm] T/T c negative kurtosis observed for Ginzburg-Landau model with a pronounced signal near T c!

35 Temperature quench and equilibration.5.3 temperature quench: at τ temperature drops from T =.5 GeV to T.2 fast initial relaxation T* = Tc T* =.18 GeV T* =.2 GeV variance σ2.4.1 T quench at τ local variance approaches equilibrium value faster than local kurtosis Ginzburg Landau long relaxation times near Tc kurtosis κ.1 B. Berdnikov, K. Rajagopal PRD61 (2) τ τ [fm]

36 Dynamics of local (non-)gaussian fluctuations variance σ 2 kurtosis κ Gauss+surface Ginzburg Landau T(τ) = T (τ /τ) open symbols: equilibrium values τ τ [fm] shift of extrema for variance and kurtosis (retardation effect) B. Berdnikov, K. Rajagopal PRD61 (2), S. Mukherjee et al., PRC92(3) (215) extremal values in dynamical simulations smaller than equilibrium values (nonequilibrium effect) no dynamical creation of non-gaussianity in Gauss+surface model expected behavior with varying D

37 Volume-integrated kurtosis infinite-volume expectation: (κσ 2 ) V ξ 5 including finite-size and N B -conservation we find for V 2 fm: (κσ 2 ) V ξ n with n 2.9 ±.2!

38 Volume-integrated skewness infinite-volume expectation: (Sσ) V ξ 2.5 including finite-size and N B -conservation we find for V 2 fm: (Sσ) V = aξ n bξ m with n 1.47 ±.5 and m 2.4 ±.5!