Dynamical fluctuations in critical regime and across the 1st order phase transition
aa r X i v : . [ nu c l - t h ] A p r Nuclear Physics A 00 (2018) 1–4
NuclearPhysics A / locate / procedia XXVIth International Conference on Ultrarelativistic Nucleus-Nucleus Collisions(Quark Matter 2017)
Dynamical fluctuations in critical regime and across the 1storder phase transition
Lijia Jiang a,b , Shanjin Wu b , Huichao Song b,c,d a Frankfurt Institute for Advanced Studies,Ruth Moufang Strasse 1, D-60438, Frankfurt am Main, Germany b Department of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, China c Collaborative Innovation Center of Quantum Matter, Beijing 100871, China d Center for High Energy Physics, Peking University, Beijing 100871, China
Abstract
In this proceeding, we study the dynamical evolution of the sigma field within the framework of Langevin dynamics.We find that, as the system evolves in the critical regime, the magnitudes and signs of the cumulants of sigma field, C and C , can be dramatically di ff erent from the equilibrated ones due to the memory e ff ects near T c . For the dynamicalevolution across the 1st order phase transition boundary, the supercooling e ff ect leads the sigma field to be widelydistributed in the thermodynamical potential, which largely enhances the cumulants C , C , correspondingly. Keywords:
Dynamical critical phenomena, correlations and fluctuations, critical point, first order phase transition
1. Introduction
The STAR collaboration has measured the higher order cumulants of net protons in Au + Au collisionswith collision energy ranging from 7 . κσ (cid:16) κσ = C / C (cid:17) shows a large deviation from the poisson baseline, and presents an obvious non-monotonic behavior at lowercollision energies, indicating the potential of discovery the QCD critical point in experiment [3].Within the framework of equilibrium critical fluctuations, we calculated the fluctuations of net protonsthrough coupling the fluctuating sigma field with particles emitted from the freeze-out surface of hydro-dynamics [4]. Our calculations can fit the C and κσ data by tuning the related parameters, as well asqualitatively describing the acceptance dependence of the cumulants of net protons. However, our calcula-tions over-predicted both C and C data due to the positive critical fluctuations, which are in fact intrinsicfor the traditional equilibrium critical fluctuations [5, 6, 7].Recently, Mukherjee and his collaborators have studied the non-equilibrium evolution for the cumulantsof sigma field in the critical regime, based on the Fokker-Plank equation [8]. The numerical results showedthat, as the system evolves near the critical points, the memory e ff ects keep the signs of the Skewness andKurtosis at the early time, which are opposite to the signs of the equilibrium ones at the freeze-out pointsbelow T c . However, their calculations focus on the zero mode of the sigma field, which has averaged out / Nuclear Physics A 00 (2018) 1–4 the spatial information at the beginning and can not directly couples with particles to compare with themeasured experimental data.To solve this problem, one could directly trace the whole space-time evolution of the sigma field withinthe framework of Langevin dynamics. In this proceeding, we will present the main results from our recentnumerically simulations of the Langevin equation of the sigma field, using an e ff ective potential of the lin-ear sigma model with constituent quarks. As discovered in early work [8], we also observe clearly memorye ff ects as the system evolves in the critical regime, which largely influence the signs and values of the cumu-lants C and C . In addition, we find that for the dynamical evolution across the 1st order phase transitionboundary, the supercooling e ff ect leads the sigma field to be widely distributed in the thermodynamicalpotential, which largely enhances the corresponding cumulants C − C at the freeze-out points.
2. The formalism and set-ups
In this proceeding we focus on the dynamical evolution of the order parameter field within the frame-work of the linear sigma model with constituent quarks. According the the classification of the dynamicaluniversality classes [9], our approach belongs to model A, which is not in the same dynamical universal-ity class of the full QCD matter evolution [10], but easy for numerical implementations. The linear sigmamodel is an e ff ective model to describe the chiral phase transition, which presents a complete phase diagramon the ( T , µ ) plane with di ff erent phase transition scenarios, including a critical point [11, 12]. As the massof the sigma field vanishes at the critical point, the related thermodynamical quantities become divergentdue to the critical long wavelength fluctuations of the sigma field. In the critical regime, the semi-classicalevolution of the long wavelength mode of the sigma field can be described by a Langevin equation [13]: ∂ µ ∂ µ σ ( t , x ) + η∂ t σ ( t , x ) + δ V e f f ( σ ) δσ = ξ ( t , x ) , (1)where η is the damping coe ffi cient and ξ ( t , x ) is the noise term. Both of these two terms come from theinteraction between the sigma field and quarks, and satisfy the fluctuation-dissipation theorem [13]. Herewe take η as a free parameter, and input white noise in the calculation. The e ff ective potential of the sigmafield is written as: V e f f ( σ ) = U ( σ ) + Ω ¯ qq ( σ ) = λ (cid:16) σ − v (cid:17) − h q σ − U + Ω ¯ qq ( σ ) (2)where U ( σ ) is the vacuum potential of the chiral field, and the related values of λ , σ , h q and U are set bythe vacuum properties of hadrons. Note that here we have neglected the fluctuations of ~π , since its mass isfinite in the critical regime. Ω ¯ qq represents the contributions from thermal quarks, which has the form: Ω q ¯ q ( σ ; T , µ ) = − d q Z d p (2 π ) { E + T ln[1 + e − ( E − µ ) / T ] + T ln[1 + e − ( E + µ ) / T ] } (3)where d q is the degeneracy factor of quarks, and the energy of the quark is E = q p + M ( σ ) . Here weintroduce an e ff ective mass for the quark, M ( σ ) = m + g σ [4, 6]. After the chiral phase transition, quarksobtain e ff ective mass and turn to constituent quarks. With g = .
3, the e ff ective mass of the constituentquark at T = m p ∼
930 MeV.Based on the e ff ective potential Eq. (2), one can obtain the corresponding phase diagram in the ( T , µ ) plane,which is plotted in the left panel of Fig. 1.For the numerical implementations, we first construct the profiles of the initial sigma field accordingto the probability function P [ σ ( x )] ∼ exp ( − ε ( σ ) / T ) (where ε ( σ ) = R d x h ( ∇ σ ( x )) + V e f f ( σ ( x )) i ),then evolve the sigma field event by event through solving the Langevin equation Eq.(1). With the obtainedspace-time configurations of the sigma fields, the moments of the sigma field at a certain evolution time canbe calculated as: µ ′ n = h σ n i = R d σσ n P [ σ ] R d σ P [ σ ] , (4) Nuclear Physics A 00 (2018) 1–4
120 160 200 240 280 32004080120160 traj. II [ M e V ] [MeV] crossover 1st order traj. I 0 20 40 60 80-2.0x10 -1.5x10 T < T c = 240 MeVT = T c T > T c [] Fig. 1. Left panel: the phase diagram on the ( T , µ ) plane, obtained from the linear signa model with constituent quarks. Right panel:the thermodynamic potentials with di ff erent temperatures ( T < T c , T = T c and T > T c ), but with the same chemical potential µ = where σ = R d x σ ( x ). The cumulants of sigma field can be further obtained from the values of these abovemoments.Note that numerically solving Eq.(1) also needs to input the space-time information of the local temper-ature and chemical potential, T ( t , x , y , z ) and µ ( t , x , y , z ), for the e ff ective potential, which are in principleprovided by the evolution of a back-ground heat bath. For simplicity, we assume that the heat bath evolvesalong simple trajectories with constant chemical potential (traj. I and traj. II in Fig. 1), and the temperaturedrops down in a Hubble-like way [8]: T ( t ) T = tt ! − . , (5)where T is the initial temperature and t is the initial time. Considering that the dynamical evolution of the σ field belongs to the universality class of model A, we set the damping coe ffi cient η to be a constant value.
3. Numerical results
Fig. 2 presents the time evolution for the cumulants of sigma fields. The left and right panels show theresults of evolution on the crossover phase transition side (along traj. I with µ =
200 MeV, which is also closeto the critical point) and on the 1st order phase transition side (along traj. II with µ =
240 MeV), respectively.For each case, we choose three constant damping coe ffi cients for the dynamical evolution, which are shownas three colored solid lines. We also plot the equilibrated values of the sigma field (dotted lines) fromthe equilibrium critical fluctuations along traj. I and traj. II, using the mapping between temperature andevolution time in Eq. (5).For the case with traj. I, the evolution of the cumulants for critical fluctuations presents clear memorye ff ects. For C and C , the signs and values are di ff erent from the equilibrated ones at later evolutiontime. For example, at t =
12 fm / c, both C and C show a positive sign, which is opposite to the sign ofthe equilibrated one. In dynamical evolution scenario, the increase of cumulants is also delayed due to thecritical slowing down. With the increase of the damping coe ffi cient η , the dynamical evolution becomesslower, and behaves like di ff usion process. In the early paper [4], it was found that the equilibrium criticalfluctuations always over-predict C and C due to the intrinsic positive contributions. The calculationspresented in Fig. 2 (left) show that the dynamical evolution of the sigma field near the critical point couldchange the sign of C and largely delay the increase of C , which has the potential to qualitatively describedi ff erent cumulant data with a properly chosen freeze-out scheme and well tuned parameters.The right panel presents the dynamical evolution along traj. II, which is across the first order phasetransition boundary. For the equilibrium values, C − C show discontinuity at the phase transition temper-ature. Note that the thermodynamical potential has two minima around T c (Fig. 1, right), which leads to thediscontinuity of C − C . For the dynamical evolution scenario, C − C continuously change during theevolution and the values of C − C at late time are much larger than the maximum values of the equilibrated / Nuclear Physics A 00 (2018) 1–4 = 1 fm -1 = 3 fm -1 = 7 fm -1 eq C C traj. I (crossover side) t [fm] C t [fm] C traj. II (1st order side) = 1 fm -1 = 3 fm -1 = 7 fm -1 eq C C t [fm] C t [fm] C Fig. 2. Dynamical evolution for the cumulants of sigma fields. The left and right panels are the results evolving on the crossover sidewith µ =
200 MeV (along traj. I) and on the 1st order phase transition side with µ =
240 MeV (along traj. II). ones. As shown in Fig. 1 (right), there exists a barrier between two minima of the thermodynamical poten-tial, which prevents part of the sigma’s configurations evolve to the real minimum at certain temperaturesclose to the phase transition. Such supercooling e ff ect leads the sigma field to be widely distributed in thethermodynamical potential, which also largely enhances the cumulants C and C at the first order transitionside.
4. Summary
Using Langevin dynamics, we simulate the dynamical evolution of the sigma field with the e ff ectivepotential from the linear sigma model. We found, as the system evolves in the critical regime, the memorye ff ects keep the signs of C and C from the early evolution, which are di ff erent from the equilibrated onesat the possible freeze-out points below T c . For the dynamical evolution across the 1st order phase transitionboundary, the supercooling e ff ect leads the sigma field to be widely distributed in the thermodynamicalpotential, which largely enhances the cumulants C and C , correspondingly. Acknowledgements
We thanks the discussions from U. Heinz, Y. X. Liu, S. Mukherjee, M. Stephanov, D. Teaney , H.Stoecker, Y. Yin and J. Zheng. This work is supported by the NSFC and the MOST under grant Nos.11435001,11675004 and 2015CB856900.
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