Nonuniform-temperature effects on the phase transition in an Ising-like model
NNonuniform-temperature e ff ects on the QCD phase transition Jun-Hui Zheng and Lijia Jiang
Center for Quantum Spintronics, Department of Physics,Norwegian University of Science and Technology, NO-7491 Trondheim, Norway
We study the phase transition in a steady temperature-nonuniform system in the frame of the Ising model.We calculate the nonuniform-temperature e ff ects on the phase transition point, the fluctuations, and the corre-lation length of the order parameter. We find the phase transition could happen at a temperature higher thanthe equilibrium phase transition temperature of a temperature-uniform system. Besides, the fluctuations andthe correlation length are enhanced near the phase transition point, and they monotonously increase from thecrossover regime to the first order phase transition regime without exotic behavior at the critical point. Ourstudy is helpful to understand the behaviors of QCD phase transition in the relativistic heavy-ion collision andprovides a method to evaluate the nonuniform-temperature e ff ects for the order parameter. INTRODUCTION
Exploring the QCD phase boundary and critical point (CP)is one of the main goals in relativistic heavy-ion collision(RHIC) experiments [1–5]. In the collider, a fireball quicklyforms and cools down continuously. During the fireball’s ex-pansion, the QCD matter cools from the quark-gluon-plasma(QGP) phase to the hadronic phase. A dynamical phase tran-sition (PT) surface exists in the fireball, which seperates thetwo phases [6–10]. The hadrons and resonances outside thesurface collide with each other and part of them decay. Ahypersurface named chemical freeze-out surface subsequentlycoexists outside the dynamical PT surface, where the inelasticcollision between the hadronic matter ceases [11].Searching the phase boundary and the CP from the dynam-ical process in RHICs, we have to face two fundamental ques-tions. Does the dynamical PT boundary coincide with theequilibrium PT boundary in the QCD phase diagram? Arethe critical behaviors kept to identify the CP?Recent studies show that the chemical freeze-out line fit-ted from experimental data overlaps with the equilibrium PTboundary depicted by the lattice calculations [12–19]. Itstrongly hints the dynamical PT inside the fireball happens ata temperature above the equilibrium PT temperature (see thesketch of an instantaneous fireball in Fig. 1). This is in con-trast to the dynamical delay e ff ects in a temperature-uniformsystem, where the dynamical PT follows and memorizes thebehaviors of equilibrium PT [20–24]. On the other hand, thefluctuations and correlation length of the QCD order parame-ter (i.e., the σ field) have been broadly used to discuss the fluc-tuation behaviors of observables such as net charge, baryonnumber and particle ratios, etc [4, 5, 25, 26]. The correlationlength has been estimated by including the finite size e ff ectand the critical slowing down e ff ect [20, 27]. Yet it is stillunclear what are the behaviors of the fluctuations and correla-tion length near the CP in a finite-size system with nonuniformtemperature.In this article, we pursue to discuss the relation between thedynamical and the equilibrium PT surface in the fireball, bystudying the behaviors of the QCD order parameter field inspatially-nonuniform-temperature profiles near the dynamical FIG. 1. A sketch of an instantaneous fireball. The temperaturedecreases from inner to outer (red to blue). The red solid circlerepresents the dynamical phase transition (PT) surface. The blackdashed line is the chemical freeze-out surface. The green brush linerefers to the isothemal surface of the equilibrium PT temperature intemperature-uniform systems.
PT surface. As both the position and the shape of the dynami-cal PT surface vary with time during the fireball evolution, welook into the σ field in the comoving frame of the PT region.As shown in Fig. 1, we take an instantaneous slender brickcell in the fireball with the dynamical and equilibrium phaseboundary located in the middle. In this brick cell, the temper-ature is spatially nonuniform. We further suppose the relax-ation of the σ field configurations in the brick cell at varioustime approaches to zero. Thus, the σ field reaches the thermalequilibrium distribution instantly. With this Markov assump-tion, the instantaneous dynamical PT surface turns into thestationary PT surface in a nonuniform-temperature system.By mapping the QCD e ff ective potential to the Isingmodel, we illustrate that the PT temperature in a temperature-nonuniform system could be above the equilibrium PT tem-perature T c of a temperature-uniform system. It hints a possi-bility that hadrons form before the QGP cools down to T c dur-ing the fireball evolution. We further deduce and discuss thefluctuation strength and the correlation length of the σ fieldin the PT region, and find that the CP presents no typical crit-ical behaviors compared to the other PT scenarios. At thesame time, the fluctuations and the correlation length in the a r X i v : . [ nu c l - t h ] F e b PT region is significantly increased compared to that in theperiphery of the cell.
TEMPERATURE PROFILE
First, we formalize the temperature profile in the brick cell.For simplicity, we suppose the y-z plane is isothermal, and thetemperature function along the x -axis is spatially dependent, T ( x ) = T c + δ T (cid:18) xw (cid:19) , (1)where δ T is the temperature bias between the two ends of thecell and the width w dominates the range of the region nearthe equilibrium PT surface where a finite temperature gradient( ∼ δ T / w ) presents. Note that the real temperature profile isdetermined by the background matter fields (hadrons, quarksand gluons) [28]. An example with a more realistic tempera-ture profile is present in the appendix, the results qualitativelyagree with that from the tanh-type temperature profile. How-ever, due to the complexities of the RHIC (chemical potential,viscosity etc.), we adopt this simplified tanh-type profile, andfocus our attention mainly on the nonuniform-temperature ef-fects in the PT region. In the following, we will also assumethe baryon chemical potential is homogeneous in the cell. PARTITION FUNCTION
As the local equilibrium assumption in relativistic hydrody-namics is proved to be well-performed [28–33], we carry onthis assumption in our calculation. Thus, the partition func-tion of the system is a product of the local partition functionof the σ field at di ff erent x . In the continuous limit, we have Z [ σ ] = (cid:90) D σ P [ σ ] , (2)where the weight function is P [ σ ] = exp (cid:40) − (cid:90) d r ( ∇ σ ) / + V [ σ ( r )] T ( x ) (cid:41) , (3)with r = ( x , y , z ). The e ff ective potential of the σ field can beobtained from the QCD-inspired models [17, 34–40]. As wemainly focus on the regime near the CP, we parameterize thepotential by analogy to the Ising model. In the simpliest Isingmapping [41–43], we have V [ σ ] = a ( T − T c )( σ − σ ) + b ( µ − µ c )( σ − σ ) + c ( σ − σ ) , (4)where a > b < c > ff ective potential, ( µ c , T c ) marks the position of theequilibrium CP. Note that within this mapping, the PT temper-ature is µ independent. For ∆ µ ≡ µ − µ c > ∆ µ <
0, thee ff ective potential describes the first order PT and crossoverrespectively as the change of temperature. σ (about 45 MeV) is introduced to shift the σ field to the realistic values. Sincethe value of σ will not influence our discussion on fluctu-ations and correlation length, we simply set σ = σ < σ > a = . − , b = − .
25 fm − , and c = . ff ective potentialfrom the linear sigma model [37, 38]. The PT temperature T c =
160 MeV is chosen from the result of lattice simulations[17, 18]. For the temperature profile, the temperature bias isset as δ T =
40 MeV and the width is set to be w = w = . / fm (40MeV / fm), corresponding to the mean temperature gradient ina fireball of radius 10 fm with the central temperature being200 MeV (400 MeV). THE ORDER PARAMETER PROFILE
Since the temperature is spatially nonuniform, the localorder parameter which maximizes the weight function isnever again determined by minimizing the e ff ective potential ∂ V [ σ ] /∂σ =
0, but satisfies the extreme value condition ofthe weight function, δ P [ σ ] /δσ =
0. Explicitly, we have ∇ σ = T ∇ T · ∇ σ + δ V δσ . (5)As we suppose the temperature distribution in the y-z planeis isothermal, the σ ( r ) that maximizes the weight functionmust be flat in this plane. Thus σ ( r ) depends only on x , andEq. (5) reduces to a one-dimensional problem. The boundarycondition is given by the local order parameters at the ends,i.e., σ ( x = − L / = σ L and σ ( x = L / = σ R , where σ L and σ R are the global minimum point of the potential V [ σ ]at x = ∓ L / L is the cell’s length. Note that when L is su ffi cient large, i.e., L (cid:29) w , the magnitude of L will notinfluence the following results.The solution σ c ( x ) to Eq. (5) is presented in Fig. 2, with w = ff erent ∆ µ . A main information from theseorder parameter profiles is that σ c ( x ) changes its sign at x > ∆ µ . It is easy to checkthat, without the temperature gradient term (1 / T ) ∇ T · ∇ σ , thesolution σ c ( x ) is an odd function of x and vanishes at x = / T ) ∇ T · ∇ σ term is always negative ( ∂ x σ < ∂ x T > σ c ( x ), and the sign change of σ c ( x ) will universallyhappens at x >
0. This result can be comprehended directlyfrom the weight function Eq. (3). In the brick cell, the hot partwith high temperature is more easily fluctuated than the coldpart. Therefore, σ c ( x ) will tend to the order parameter valueof the cold part, and σ c (0) becomes positive.Like as the equilibrium PT of the Ising model, we identifythe point of sign change of σ c as the PT point at di ff erent ∆ µ .The PT point always locates at x > FIG. 2. The order parameter profile σ c ( x ) in the brick cell, with thered, green and blue lines represent results in the crossover ( ∆ µ < ∆ µ =
0) and the first order PT ( ∆ µ >
0) scenarios, respectively.FIG. 3. A comparison of the PT temperature in the temperature-nonuniform (red and blue dotted lines) and temperature-uniform sys-tems (black lines). The red dotted line is with w = . w = . temperature T c = T ( x = w = w = . T c .Note that the lifted values of temperature is not universaland depends on the temperature profile. In the RHIC, thetemperature profile usually is not a tanh-type in the space.However, the PT point in a temperature-nonuniform system isgenerally di ff erent from that in a temperature-uniform system.The nonuniform-temperature e ff ects provide a possibility thatthe QCD PT could happen before it cools down to T c . In theappendix we present the result with a more realistic temper-ature profile fitted from the hydrodynamics’ output. The PTtemperature is also lifted, which qualitatively agrees with thatfrom the tanh-type temperature profile. In the following, wekeep our discussion on the tanh-type profile and reveal howthe temperature profile influences the fluctuations and corre-lation length. THERMAL FLUCTUATIONS
The fluctuations of the σ field are also changed by thetemperature profile. We expand the σ field around the vari-ational extremum solution with a small fluctuation, σ ( r ) = σ c ( x ) + δσ ( r ). The weight function P [ σ ] becomes P [ σ ] = exp (cid:34) − (cid:90) d r ( ∇ δσ ) / + m δσ / + c σ c δσ + c δσ T ( x ) (cid:35) , (6)where the mass term m ( x ) = b ∆ µ + c σ c is spatially de-pendent. In the following, we mainly focus on the variance ofthe fluctuations, so we omit the cubic and quartic terms whichare related to the higher order fluctuations [25, 44, 45].Conventionally, we start the discussion from the mass termof the δσ field. In a temperature-uniform system, the massterm relates to the correlation length: ξ = / m . In the nonuni-form case, we follow the old way and define a local corre-lation length: ξ local ( x ) = / (cid:112) m ( x ). We present the resultsof m ( x ) in the brick cell in Fig. 4. In the periphery, m ≈ − corresponds to ξ local ≈ ξ since the temperature becomes flat apart from the center. Inthe central part, m ( x ) presents exotic behaviors for di ff erentPT scenarios. In the crossover regime ( ∆ µ < m ( x ) > ∆ µ = m ( x ) vanishes at σ c =
0, and the local correlation length ξ local diverges. How-ever, in the first order PT regime ( ∆ µ > m ( x ) is negativenear the point σ c =
0, which is in contrast with the positive m in a temperature-uniform system. Therefore, the currentdefinition of the local correlation length is not appropriate inthe PT region with a finite temperature gradient. As we willshow below, the variance of the local fluctuation δσ ( x ) is al-ways positive, and is better-suited to the description of thetemperature-nonuniform system.In the following, we calculate the variance of the fluctu-ation. We presume the size along the y and z direction ismuch smaller than the unknown correlation length. Therefore,we adopt the zero-mode approximation for y and z directionsand thus δσ ( r ) depends only on x . The cross-section of thebrick cell is denoted as S . Discretizing the x -axis with spac-ing length ∆ x , the weight function becomes P [ σ ] (cid:39) exp − S (cid:88) i , j δσ i M i j δσ j , (7)where the nonzero elements of the matrix M are M ii = ∆ x (cid:34) T i − / + T i + / (cid:35) + m i ∆ xT i , (8) M i , i + = M i + , i = − T i + / ∆ x . (9)Here, ‘ i ’ refers to the position x = i ∆ x . The matrix M must bepositive definite so that the solution σ c maximizes the weightfunction. We would like to emphasize the necessity and im-portance of the kinetic energy in P [ σ ] (see Eq. (6)), which is FIG. 4. Panel a) presents the local mass square of the fluctuating σ field, and panel b) presents the variance for di ff erent w and ∆ µ .In both panels, the red, green and blue lines represents results in thecrossover ( ∆ µ < ∆ µ =
0) and the first order PT ( ∆ µ > w = w = . finite and solves the negative m ( x ) problem in the first orderPT scenario. This is because in the brick cell, m ( x ) constructsa potential well as shown in Fig. 4, and the kinetic term has tobe finite due to the uncertainty principle. From the same rea-son, at ∆ µ =
0, the fluctuations on the CP is not divergent dueto a positive ground energy of M . The nonzero kinetic energyrepresents the contribution from the nonzero-mode fluctuationof the σ -field, which plays a crucial role in the temperature-nonuniform system.The variance of the local fluctuations is (cid:104) [ δσ i ] (cid:105) = [ M − ] ii S . (10)In Fig. 4, we plot the results of the variance for di ff erent w and ∆ µ . Note that the maximum point of the variance locates alittle right of the minimum point of m ( x ), because the fluctu-ations in the right of the cell is lifted due to a higher tempera-ture compared to the left. Interestingly, the fluctuations on thePT point monotonically increase from the crossover ( ∆ µ < ∆ µ > ∆ µ = w for all the three scenarios. This can be understood in the ex-treme case that w → ∞ , the temperature becomes flat locallyand the fluctuations near the CP become divergent.We further calculate the correlation length near the PT point FIG. 5. The normalized nonlocal correlation G ( x ) near the PT point.The legends are the same as that in Fig.4. The inset is an enlargementof the cross region marked in the plot. from the normalized nonlocal correlation, G ( x ) = (cid:104) δσ ( x p + x / δσ ( x p − x / (cid:105)(cid:104) δσ ( x p ) δσ ( x p ) (cid:105) , (11)where x p denotes the spatial location of the maximum pointof the variance. We plot the result in Fig. 5. The nor-malized nonlocal correlation doesn’t exactly decay exponen-tially, so we determine the correlation length ξ by requiring G ( ξ ) = exp( − ξ again smoothly increases from thecrossover regime ( ∆ µ <
0) to the first order phase transitionregime ( ∆ µ > ξ ≈ .
74 fm in the central part ofthe brick cell, which is significantly larger than ξ ≈ CONCLUSION AND DISCUSSION
We have developed a method to figure out the stablest orderparameter profile, the fluctuations and the correlation length ina steady temperature-nonuniform system. We find that the PTtemperature is generally ahead for di ff erent temperature gradi-ents in our temperature profile settings. This hints a possibil-ity that, in the Markov approximation, the hadrons may formbefore QGP cools to the equilibrium PT temperature duringthe fireball expansion. In addition, both the fluctuations andthe correlation length near the PT surface are enhanced, anddecrease as the increase of temperature gradient. There is nocritical behavior to identify the CP from the first order PT andcrossover.In the RHIC, the realistic temperature profile as well as thebaryon chemical potential profile vary for di ff erent events atdi ff erent time, and they are a ff ected by dynamical factors likethe viscosity. Even in the Markov approximation, statisticalaverage over fluctuations in di ff erent profiles is needed. Weexpect the statistical average will not change our results qual-itatively.Our study provides a di ff erent perspective to understand theQCD PT in RHIC. Indeed, the nonuniform-temperature ef-fects and the dynamical memory e ff ects [21, 43, 46–52] aretwo extreme cases corresponding to spatial correlation domi-nant and temporal correlation dominant, respectively. Duringthe dynamical evolution of the fireball, the two e ff ects mayinterrelate with each other. In our calculation for a steadynonuniform-temperature system, the correlation length nearthe PT surface increases significantly. Consequently, in thedynamical evolution, the slowing down of dynamics is ex-pected in all the PT regime and the memory e ff ects will beenhanced. Regarding the current measurements on the fluc-tuations of net proton [4, 5], the nonuniform-temperature ef-fects on the high order fluctuations should be studied, and webelieve a combination of the nonuniform-temperature e ff ectsand the dynamical e ff ects will provide a better description tothe PT in RHIC.This work was supported by the Research Council ofNorway through its Centres of Excellence funding scheme(project no. 262633, “QuSpin”). Lijia Jiang thanks XiaofengLuo and Bao-Chi Fu for helpful discussions. APPENDIX
In this Appendix, we show an example with a more realistictemperature profile as show in Fig. 6. Here, T ( x = = T c is the phase transition temperature. The position x = x = − ff ective potential(Eq. (4) in the main text) is valid in the whole temperatureregion. The corresponding extreme solution σ c ( x ) is shownin Fig.7. In this plot, the phase transition happens at x > T ≈
168 MeV is 8 MeV larger than T c .In Fig. 8, we show the results of local mass square andthe variance of the fluctuations. The local correlation lengths(1 / m ) at x = ± ξ local ≈ .
73 fm and 0 . x = − T →
0. InFig. 9, the normalized nonlocal correlation near the PT point isplotted. The correlation length is about 1 .
45 fm. The varianceand correlation length with this temperature profile presentsimilar behaviors as those in the main text.
FIG. 6. The temperature profile in the real space. Here, for x = T = T c . The position x = x = − σ c ( x ), with the red, green andblue lines represent results in the crossover ( ∆ µ < ∆ µ = ∆ µ >
0) scenarios, respectively.FIG. 8. Panel a) and b) present the local mass square and the varianceof the fluctuating σ field, respectively. In both panels, the red, greenand blue lines represents results in the crossover ( ∆ µ < ∆ µ =
0) and the first order PT ( ∆ µ >
0) scenarios, respectively.
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