Thermalization of dense hadronic matter in Au + Au collisions at the energies available at FAIR
aa r X i v : . [ nu c l - t h ] O c t Thermalization of dense hadronic matter in Au + Au collisions at energiesavailable at the Facility for Antiproton and Ion Research
Somnath De , Sudipan De , and Subhasis Chattopadhyay Institute of Physics, Bhubaneswar, Odisha, India Universidade de S˜ao Paulo, S˜ao Paulo, Brasil and Variable Energy Cyclotron Centre, 1/AF, Bidhan Nagar, Kolkata, India
The conditions of local thermodynamic equilibrium of baryons (non-strange, strange) and mesons(strange) are presented for central Au + Au collisions at FAIR energies using the microscopictransport model UrQMD. The net particle density, longitudinal-to-transverse pressure anisotropyand inverse slope parameters of the energy spectra of non-strange and strange hadrons are calculatedinside a cell in the central region within rapidity window | y | < . PACS numbers: 25.75.-q, 25.75.Dw, 51.30.+i, 12.40.Ee
I. INTRODUCTION
The motivation of the relativistic heavy ion col-lider experiments is to explore the properties ofstrongly interacting matter (partonic or hadronic)at the finite temperature and/or density. The cur-rent heavy ion research facilities e.g; RelativisticHeavy Ion Collider (RHIC) and Large Hadron Col-lider (LHC) are focused in unveiling the propertiesof deconfined quark-gluon matter created at the ex-treme temperature and almost vanishing net baryondensity [1, 2]. At this regime the lattice quantumchromodynamics (lQCD) simulations have reporteda crossover from hadronic to partonic phase andthe existence of critical point where the first or-der phase transition line terminates [3]. Thus RHIChas initiated the beam energy scan program to findthe location of critical point in the QCD phase di-agram (temperature (T)- baryo-chemical potential( µ B ) plane) [4].In contrary to the above experiments, the futureCompressed Baryonic Matter (CBM) experiment atFAIR / GSI laboratory is aimed to explore anotherfacet of QCD phase diagram; at high baryon density( ∼ − ρ , ω ) decaying to dilepton pairs, (ii) Pro-duction of multi-strange hyperons (Ξ, Ω) (iii) Disso-ciation of Charmonium (J/Ψ) and charmed hadron ( D , Λ c ) states, etc. The existence of first orderphase transition from hadronic to partonic matterand restoration of chiral symmetry at the large µ B is expected to be found from the FAIR energy scanprogram [7]. Earlier experiments such as; RHIC-AGS and CERN-SPS were aimed to explore abovefeatures through the measurement of bulk observ-ables like; flow and momentum spectra of hadrons.However their efforts were constrained due to limitedbeam luminosity. In recent years a similar researchprogram (NICA) at JINR-Dubna has been proposedto explore phases of nuclear matter at high baryondensity [8]. But the CBM experiment would be moreefficient for the detection of bulk and rare probes,with the availability of high intensity ion beams [9].In order to compute the dynamic evolution of thematter created in such collisions, we need macro-scopic/microscopic models. The macroscopic mod-els like; hydrodynamics rely upon the assumption oflocal thermal equilibrium of the created matter on acertain time scale. The actual thermalization crite-rion has seldom been tested. There are a few works,addressed the issue at higher collision energies inthe framework of perturbative QCD [10] or color-glass condensate theory [11]. On the other handmicroscopic Monte Carlo models like: UrQMD [12],HSD [13], AMPT [14] work on the postulated in-teraction among it’s constituents (parton, hadron,or string) and does not require any assumption oflocal thermal equilibrium. Therefore, it is very im-portant to test weather the dense baryonic mattercreated in these collisions achieve a local thermalequilibrium or not. In particular, we have inves-tigated the time-scale of local thermal equilibrationof non-strange and strange baryons in an elementaryvolume in phase-space from the time evolution oflongitudinal-to-transverse pressure anisotropy andslope of the energy spectrum. For this purpose,we have employed the microscopic, N-body trans-port model called Ultra-relativistic Quantum Molec-ular Dynamics (UrQMD). A comparison betweenthe model and the data for central Pb+Pb collisionsat different energies at CERN-SPS can be found inRef. [15]. We considered the most central collisionsof gold (Au) nuclei at four beam energies associ-ated with the CBM experiment. The incident beamenergy has obvious implication on the time-scale ofequilibration, which can be found in the subsequentsection.The organization of the paper goes as the fol-lowing. In the next section, we briefly recapitulateabout the microscopic transport model UrQMD andthen discuss about the methodology of our analysis.In section 3, we show the results for the time evo-lution of density, ratio of longitudinal-to-transversepressure and inverse slope parameter of the energyspectra for non-strange baryons, strange baryonsand strange mesons. In section 4, we have utilizedthe statistical thermal model to extract the post-equilibrium thermodynamic parameters e.g., tem-perature, chemical potentials and calculated the en-tropy density of the system. Finally we have sum-marized the findings in section 5. II. METHODOLOGY OF THE ANALYSIS
The model UrQMD has been extensively usedin recent years for describing heavy ion collisionsof center of mass energy ranging from few GeV/nucleon to few TeV/ nucleon [12]. We used theUrQMD-version:3.3p2 in default cascade mode with-out invoking any hydrodynamic evolution for theinitial state. It includes 55 baryon species (up tomass 2.25 GeV) and 32 meson species (up to mass1.9 GeV) and their corresponding anti-particles andiso-spin projected states. Particle production inUrQMD occurs through inelastic collisions, decayof meson, baryon resonances, and string fragmen-tation mechanism. At low energies (E lab < lab ∼ lab ) 10A, 20A, 30A, 40A GeV. For eachenergy we ran the simulation at different time stepsranging from 1fm/c to 15 fm/c. 6 × events havebeen analyzed for each time step. The center of massframe is chosen as the computational frame in ouranalysis. We have considered a cell of dimension2 × × about the origin of Au + Au system.The test volume has been chosen such that the effectof collective flow of the system on the observableswill be minimum and at the same time the particlenumber should be large enough for reasonably smallfluctuation in the observables. Additionally a mo-mentum rapidity cut | y cm | < . p ) and Neutron ( n ), the strange baryons in-clude Lambda (Λ), Sigma (Σ), Cascade (Ξ), Omega(Ω) and the strange mesons include Kaons: K + , K .All the higher mass resonances (baryon and meson)are allowed to decay. We did not include Ω in thepressure calculation at E lab = 10A and 20A GeV dueto its limited statistics at lower energies. However weexpect that inclusion of Ω does not modify any con-clusion drawn in this work. We have also calculatedthe energy spectra (EdN/d p vs. E) of Protons andLambdas inside the cell. Lastly the above quantitiesare statistically averaged over the number of eventsfor each time step. III. RESULTSA. Time evolution of net particle density
The time (t) is the elapsed time in center of massframe. Time t = 0 fm/c corresponds to the mo-ment when two nuclei touch each other. The netparticle density ρ ( t ) is defined as the difference ofparticle density and anti-particle density. The evo-lution of net non-strange baryon density ( ρ NSB ), netstrange baryon density ( ρ SB ), net kaon density ( ρ SM ),and net strange baryon to kaon ratio ( ρ SB / ρ SM ) aredepicted in Fig. 1 at E lab = 10A, 20A, 30A, 40AGeV. The net particle density starts from a smallvalue, reaches a maximum around t = 2R/( γ cm v cm )when the two nuclei pass through each other and t (fm/c) ) - ( f m B N S ρ = 10A GeV lab E = 20A GeV lab
E = 30A GeV lab
E = 40A GeV lab
E(a)Au+Au, b = 2 fm|y| < 1.0 t (fm/c) ) - ( f m BS ρ = 10A GeV lab E = 20A GeV lab
E = 30A GeV lab
E = 40A GeV lab
E(b)Au+Au, b = 2 fm|y| < 1.0 t (fm/c) ) - ( f m M S ρ = 10A GeV lab E = 20A GeV lab
E = 30A GeV lab
E = 40A GeV lab
E(c)Au+Au, b = 2 fm|y| < 1.0 t (fm/c) s M ρ / s B ρ = 10A GeV lab E = 30A GeV lab
E = 40A GeV lab
E(d)Au+Au, b = 2 fm|y| < 1.0
FIG. 1: (Color online)(Upper panel) Time evolution of net density of (a) non-strange baryons ( ρ NSB ), (b) strangebaryons ( ρ SB ), (Lower panel) (c) kaons ( ρ SM ) and (d) net strange baryon to kaon ratio ( ρ SB / ρ SM ) inside the central cellfor Au + Au collisions (b = 2 fm) at the laboratory energies 10A, 20A, 30A, 40A GeV. The error bars are statisticalonly. then falls down as the system expands. The genericfeature has been found in agreement with earlierworks [17, 18]. Here R is the radius of Au nucleus, γ cm and v cm are the Lorentz boost and velocity incenter of mass frame. Thus we found the maximummatter density near 6 fm/c at 10A GeV and 3 fm/cat 40A GeV for all species. The production of non-strange baryons has been found almost similar for allthe beam energies, but the strange baryon and me-son production becomes larger with increasing beamenergy. This is probably because of the fact thatstring excitation mechanism has major contributionto strangeness production at higher energies. Thepeak of baryonic (non-strange and strange) matterdensity has been found at 40A GeV, which is about7–8 times the ground state nuclear matter density.The time evolution of strange baryon to meson ra-tio has clearly shown the net strangeness contentof the created matter is dominated by baryons forall the beam energies. The ratio has been foundto grow with time because the kaons and Lambdasare produced through same strong interaction. How-ever the kaons have suffered less scatterings in themedium due to its small interaction cross-section with other hadrons [19], thus escapes the reactionvolume quickly. The production of kaons is largerat higher beam energies can be seen from the non-monotonus behaviour of the ratio at smaller times. B. Isotropization of pressure components ofbaryons and mesons
We have studied the isotropization of differentcomponents of microscopic pressure of non-strangebaryons, strange baryons and kaons for an expand-ing system. The pressure components are highlyanisotropic immediately after the collision. Ther-mal equilibrium is established in the cell when theyhave become nearly isotropic. Different componentsof microscopic pressure are calculated in UrQMDusing ideal gas ansatz [20]: P ( x,y,z ) = X i p i ( x,y,z ) V ( p i + m i ) , (1)where, p i is the momentum, m i is the mass of i’thhadron and V is the volume of the cell under con- t (fm/c) T / P L P Non-strange baryonStrange baryon) , K + Kaon (K(a) = 10A GeV lab
Au+Au, Eb = 2 fm, |y| < 1.0 t (fm/c) T / P L P Non-strange baryonStrange baryon) , K + Kaon (K(b) = 20A GeV lab
Au+Au, Eb = 2 fm, |y| < 1.0 t (fm/c) T / P L P Non-strange baryonStrange baryon) , K + Kaon (K(c) = 30A GeV lab
Au+Au, Eb = 2 fm, |y| < 1.0 t (fm/c) T / P L P Non-strange baryonStrange baryon) , K + Kaon (K(d) = 40A GeV lab
Au+Au, Eb = 2 fm, |y| < 1.0
FIG. 2: (Color online) Time evolution of longitudinal-to-transverse pressure ratio ( P L /P T ) of non-strange baryons,strange baryons, kaons inside the central cell for Au + Au collisions (b = 2 fm) at the laboratory energies (a) 10A,(b) 20A, (c) 30A, (d) 40A GeV. The error bars are statistical only. sideration. The longitudinal and transverse compo-nents of pressure for an ensemble of hadrons are de-fined as: P L = h P z i ; P T = 12 ( h P x i + h P y i ) , (2)here the h .. i corresponds to the statistical aver-age over the number of events. The time evolu-tion of the longitudinal-to-transverse pressure ratio( P L /P T ) for the above mentioned hadron species areshown in Fig. 2 at the four beam energies.The P L /P T ratio of baryons (non-strange andstrange) starts from a large value at initial times,ultimately settles down to a value close to 1.0.This reflects the longitudinal( z ) and transverse( x, y ) momentum distribution of baryons are highlyanisotropic at initial times. Successive elastic scat-terings in the medium have made their momentumdistribution nearly isotropic. We found that theratio P L /P T becomes 1.0 around 6.5 fm for non-strange baryons and 7 fm for strange baryons atE lab = 10A GeV. However the system further evolvesand the ratio reaches a constant value ∼ ≥ P L /P T from unity after-equilibrium, is possibly arising due to finite shearviscosity of the hadronic matter [21]. The ratio ismore closer to unity as the system approaches to-wards ideal fluid limit. This has been shown by arecent study on the pressure isotropization in quark-gluon plasma for Au + Au coliisions at top RHICenergy [22]. For other beam energies the P L /P T ra-tio of baryons has become unity much earlier, andit achieves a constant value ∼ ≥ ≥ ≥ t ∼ P L /P T ratio of kaons approaches to 1.0at early times t ∼ ∼ t (fm/c) ( G e V ) s l ope T p Λ -1/3 ~ t(a) = 10A GeV lab Au+Au, Eb = 2 fm, |y| < 1.0 t (fm/c) ( G e V ) s l ope T p Λ -1/3 ~ t(b) = 20A GeV lab Au+Au, Eb = 2 fm, |y| < 1.0 t (fm/c) ( G e V ) s l ope T p Λ -1/3 ~ t(c) = 30A GeV lab Au+Au, Eb = 2 fm, |y| < 1.0 t (fm/c) ( G e V ) s l ope T p Λ -1/3 ~ t(d) = 40A GeV lab Au+Au, Eb = 2 fm, |y| < 1.0
FIG. 3: (Color online) Time evolution of the inverse slope parameter ( T slope ) of the energy spectra of Proton andLambda inside the central cell for Au+Au collisions (b = 2 fm) at the laboratory energies (a) 10A, (b) 20A, (c) 30A,(d) 40A GeV. The error bars are statistical only. inal excited hadron and suffer less resonant scatter-ings than baryons in medium. Thus we found thepressure isotropization time of baryons and mesonsreduces by about 3 fm/c from E lab = 10A GeV toE lab = 40A GeV. C. Thermalization of energy spectra ofbaryons
In this section, we adopted an alternate ap-proach to equilibrium which would reinforce thefindings of earlier section. We investigated thetime scale of local thermalization of baryonic mat-ter from the time evolution of inverse slope param-eter of the energy spectra (
EdN/d p vs E). Forthis purpose we have parameterized the energy spec-tra of Proton and Lambda inside the cell by Tsal-lis distribution [24]. An important criticism oftenarises that systems obeying non-extensive statisticsachieve thermal equilibrium or not. Here we refer tothe work of B´ır´o and Purcsel [25] which has shownthat two non-extensive subsystems do achieve a com-mon equilibrium distribution within the frameworkof non-extensive Boltzmann equation. The Tsal- lis distribution has extensively been used in recentyears for describing the transverse momentum ( p T )distribution of produced hadrons at RHIC and theLHC energies [26, 27]. The special merit of the dis-tribution is: at low energy limit it reduces to anexponential distribution and at high energy limit itreduces to a power-law distribution [28]. Thus it canaccommodate both equilibrium and non-equilibriumphenomena. A recent work has found that Tsallisdistribution fits reasonably good all particle spectrafor p T <
10 GeV at midrapidity in d +Au, Cu + Cu,Au + Au collisions at RHIC [29]. Keeping the factsin mind, we write the energy spectra of Proton andLambda inside the cell of dimension 2 × × about the origin of Au + Au system as; E d Nd p = C (1 + EbT ) − b , (3)where E is the energy of baryon in the unit of GeVand b = 1/(q-1) is dimensionless. C has the unit ofGev − and T is in GeV. q is called the non-extensiveparameter of Tsallis distribution. The values of C,b, T are obtained through fitting the energy spectraup to E = 3 GeV. The inverse slope parameter of t (fm/c) ( G e V ) s l ope T Λ -1/3 ~ t -1 ~ t(a) = 20A GeV lab Au+Au, Eb = 2 fm, |y| < 1.0 t (fm/c) ( G e V ) s l ope T Λ -1/3 ~ t -1 ~ t(b) = 40A GeV lab Au+Au, Eb = 2 fm, |y| < 1.0
FIG. 4: (Color online) The scaling behaviour of inverse slope parameter ( T slope ) of the energy spectra of Lambdainside the central cell for Au+Au collisions (b = 2 fm) at (a) 20A, (b) 40A GeV laboratory energies. The blackdotted line denotes the scaling due to longitudinal expansion and the red dashed line denotes scaling due to threedimensional expansion. The error bars are statistical only. this distribution is given by: T slope = T + ( q − E. (4)In the asymptotic limit E →
0, the inverse slope pa-rameter ( T slope ) gives the thermodynamic temper-ature of the system [25]. We have calculated the T slope of proton and Lambda energy spectra at E =0.1 GeV (nearly pion mass) and studied its time evo-lution at the four beam energies. The error in T slope arises from the errors in the fitting parameters T and b . The results are depicted in Fig. 3.We have found T slope (for Proton and Lambdaboth) falls sharply with time and then almost scalesas ∼ t − / for t & T slope corresponds to the local tempera-ture of system, then we can infer an isentropic lon-gitudinal expansion sets in inside the above men-tioned cell analogous to Bjorken ideal hydrodynam-ics. The temperature follows the Bjorken scalingsolution. We consider the time as the local thermalequilibration time scale of the system at which thescaling behaviour of the slope parameter has initi-ated. Similarly we have found the t − / scaling holdsgood for t & & & >
10 fm/c at E lab = 40A GeV),the T slope is seen to scale as ∼ t − owing to threedimensional spherical expansion of the system (seeFig. 4). The assumption of Bjorken hydrodynamicregime with the above mentioned scaling solution,namely, initial one-dimensional flow and ideal gasequation of state, could be dubious at lower colli-sion energies. Although earlier works at AGS andSPS energies [30] have found phenomenological suc-cess based on it. However it may be noted that wedo not study thermalization of whole reaction vol-ume, rather concentrate at the very central part of the system only. And for this region the above as-sumptions may be relevant, at least we can identifyclearly the Bjorken scaling regime of T slope for allenergies (see Fig. 3). Thus we have found time scaleof thermalization of energy spectra roughly in agreeswith the pressure isotropization time of baryons anddecreases with the increase in laboratory energy forthe above mentioned cell.A natural question about the analysis involving T slope could arise that how better does the Tsallisdistribution fit the spectrum compared to any clas-sical distribution. In order to see that, we fit theenergy spectra of baryons with Maxwell-Boltzmann(MB) distribution; f ( E ) = C ′ exp ( − ( E − µ ) /T ). C ′ is a constant, T is the temperature and µ is thechemical potential in usual notation. We have fittedthe spectra for Au +Au collisions at E lab = 30A GeVfor the same range of E and calculated the chi-squareper degrees of freedom ( χ /ndf ) for different times.The result is depicted in Fig. 5. It has been observedthat Tsallis distribution mostly gives lower value of χ /ndf which is close to unity in comparison to MBdistribution. The inverse slope parameters of bothdistributions at different times are listed in Table I.Several facts emerge upon close inspection. First,the two parameters are very similar at early times(say up to 3 fm/c). This might be due to numericalequivalance of the two distributions at these times.However it can be noted that χ /ndf comes out verylarge at those times for both distributions; thus theparameters may not be describing a good fit.Now in the thermal regime, say for t ≥ T slope (Tsallis) is smaller than T(MB). The behaviour hasbeen studied in the Ref. [31]. The Tsallis distribu-tion describes a near-thermal equilibrium situationfor q value close to unity. For the same particle t (fm/c) / nd f χ p (TS) (TS) Λ p (MB) (MB) Λ p (TS) (TS) Λ p (MB) (MB) Λ = 30A GeV lab Au+Au, Eb = 2 fm, |y| < 1.0 (A GeV) lab E
10 15 20 25 30 35 40 T he r m a li z a t i on t i m e ( f m / c ) Petersen 2008This work
FIG. 5: (Color online) (Upper panel) The chi-square perdegrees of freedom ( χ /ndf ) at different times for theTsallis (filled symbols) and Maxwell-Boltzmann (opensymbols) distribution which are fitted to the energy spec-tra of Proton and Lambda in the central cell for Au +Aucollisions (b= 2 fm) at E lab = 30A GeV. (Lower panel)The thermalization time obtained in this work (bluesquare) is compared with the local thermalization time t start (red circle) used in the hybrid (UrQMD + Hydro-dynamics) model (Petersen 2008:[34]). The error barsare considered to be systematic. yield, Tsallis distribution leads to lower temperature(i.e. inverse slope parameter) than MB distributionfor q >
1. The Tsallis temperatue often interpretedas the superposition different MB temperatures andthe relaive width of fluctuation in T(MB) is relatedto non-extensivity parameter ( q −
1) [32]. We havechecked that ( q −
1) remains almost constant ∼ . D. A comparison with earlier work
We have compared our result with the local ther-malization time scale ( t start ) used by an earlier workof hybrid model of Boltzmann transport and hydro-dynamics by Petersen et al. [34]. The model hassuccessfully described the data of rapidity depen-dent yield, transverse mass spectra of hadrons atAGS and SPS experiments. The t start is consideredad hoc as the nuclear passage time in the center ofmass frame. The comparison can be found in Fig. 5.We have introduced a systematic uncertainty of ± . t = 1 fm.It has been found that our result decreases with in-creasing laboratory energy similar to t start but isabout 1.5 times larger in magnitude. The earlierwork has assumed that t start is the lowest possi-ble time needed for local thermalization however thecurrent study could provide a more realistic estimateof it. Nevertheless, the issue has been investigatedfurther in Ref. [34] and found that multiplicity andmean transverse momenta of particles do not changeappreciably when t start increases by factor of 2. IV. COMPARISON WITH STATISTICALTHERMAL MODEL
In the preceding sections we argued that thedense hadronic matter created in the collisions willachieve local thermal equilibrium on a certain timescale. Thus we can employ the statistical hadron gasmodel [35] to extract the intensive thermodynamicvariables like; temperature, chemical potential of thesystem during subsequent evolution. The statisticalmodel can not be applied prior to equilibrium, rathercan be applied beyond thermal freeze-out of the sys-tem. Traditionally thermal freeze-out is defined as:the average scattering rate between the constituentsbecomes smaller than the average expansion rate ofthe system. The system has become so dilute thathardly any collision between the constituents takesplace. Following this criterion, we have checked thetime evolution of the average number of collisions( h N coll ( t ) i ) suffered by different hadron species. TheFig. 6 shows that average number of collisions suf-fered by p, n, Λ, Σ baryons and K mesons almostsaturate for t &
17 fm/c at E lab = 10A GeV andt &
15 fm/c at E lab = 30A GeV. Considering theabove scenarios; we made the comparison of statis-tical model with UrQMD during the time interval 10fm/c ≤ t ≤
17 fm/c at 10A GeV and 8 fm/c ≤ t ≤
15 fm/c at 30A GeV laboratory energy.The expression for number density, energy densityfor the i’ th hadron species in the statistical hadron t T slope (Tsallis) T (MB)fm/c GeV GeV1 0.275 0.2672 0.252 0.2474 0.200 0.2236 0.140 0.1958 0.138 0.17810 0.121 0.16112 0.099 0.13814 0.083 0.116TABLE I: The inverse slope parameters for Tsallis and Maxwell-Boltzmann distributions at different times. Thedistributions are fitted to the energy spectra of Protons in the central cell for Au+Au collisions at E lab = 30A GeV. E lab = 10A GeV E lab = 30A GeVt T µ B µ s t T µ B µ s fm/c GeV GeV GeV fm/c GeV GeV GeV10 0.145 0.708 0.174 8 0.152 0.616 0.12311 0.136 0.697 0.148 9 0.145 0.601 0.10012 0.128 0.687 0.125 10 0.137 0.595 0.08113 0.120 0.680 0.102 11 0.129 0.593 0.06714 0.114 0.670 0.082 12 0.123 0.587 0.04715 0.108 0.664 0.070 13 0.115 0.586 0.03116 0.102 0.659 0.049 14 0.110 0.586 0.01917 0.097 0.656 0.041 15 0.105 0.585 0.011TABLE II: The time evolution of temperature (T), baryon chemical potential( µ B ), strange chemical potential ( µ s )in the central cell (2 × × ) for Au + Au collisions (b = 2 fm) at laboratory energies 10A and 30A GeV. Thethermodynamic parameters are obtained from energy density of baryons ( ε B ), number density of baryons ( n B ) andnumber density of strange hadrons ( n s ) using statistical hadron gas model. gas model are given by: n i = g i (2 π ~ ) Z πp f i ( T, µ i ) dp,ε i = g i (2 π ~ ) Z πp e i f i ( T, µ i ) dp, where e i is the energy, T is the temperature and µ i ischemical potential of the i’th hadron. The hadronsare considered relativistic, e i = ( p + m i ) . f i isthe distribution function of the i’th hadron (eitherFermi-Dirac or Bose-Einstein). However above dis-tributions are practically approximated to classicalMB distribution as; ( e i − µ i )/T >> µ i can be de-composed in terms of baryonic ( µ B ) and strange ( µ s )chemical potentials. The charge chemical potential( µ Q ), which is an order of magnitude smaller thanthe other two, has been neglected here. µ i = b i µ B + s i µ s ,b and s are the baryon and strangeness quantumnumber respectively. T , µ B , and µ s are extracted from the following equations; ε B = baryon X i ε i , n B = baryon X i b i n i , n s = baryon,meson X i s i n i (5)The quantities in the l.h.s. of the equation 5, namelyenergy density of baryons ( ε B ), number density ofbaryons ( n B ) and number density of strange hadrons( n s ) are obtained from the UrQMD. We have solvedthe above set of equations during the time inter-val stated earlier. The values are listed in Table II.We have plotted them in the QCD phase diagramin order to get an estimate about the chemical andthermal freeze-out time of the system (see Fig. 7).The chemical freeze-out line has been obtained em-pirically from the thermal model fit of particle ra-tios at different collision energies [36]. The ther-mal or kinetic freeze-out line has also been obtainedphenomenologically from the blast wave model fitsof the measured hadron spectra at different exper-iments [37]. It can be seen at low energies E lab = 10A GeV, the chemical and the kinetic freeze-out happens almost instantaneously at t ≈
17 fm/c. t (fm/c) 〉 c o ll N 〈 + K + K 〉 c o ll N 〈 Au+Au, b = 2 fm|y| < 1.0 Σ + Λ 〉 c o ll N 〈 Au+Au, b = 2 fm|y| < 1.0p + n = 10A GeV lab
E = 30A GeV lab E FIG. 6: (Color online) Time evolution of average num-ber of collisions ( h N coll ( t ) i ) suffered by (a) Protons andNeutrons, (b) Lambda and Sigma baryons, (c) Kaons forAu + Au collisions (b = 2 fm) at laboratory energies 10Aand 30A GeV. At higher energy E lab = 30A GeV, system under-goes first chemical freeze-out at t ≈
13 fm/c, thenkinetic freeze-out at t ≈
15 fm/c. The feature hasalready observed in low energy collision experimentsat RHIC [38]. We would also like to add that ourestimation of temperatures at the kinetic freeze-outtimes closely agree with the values given by blastwave model fit to the Λ baryon spectra from NA49Collaboration at the similar laboratory energies [15].We are interested in computing bulk properties ofa baryon rich hadronic medium, thus strange me-son contribution can be neglected as µ s ρ SM ≈ fewMeV. Using the values of temperature and chemicalpotential listed in Table II, we have calculated thepressure of baryons with the statistical hadron gasmodel: P = baryon X i g i (2 π ~ ) Z πp dp p p + m i ) f i ( T, µ i ) , (6)and the entropy density ( s ) for baryons using thethermodynamic relation: T s = ε B + P − µ i ( ρ NSB + ρ SB ) , (7)where µ i is the chemical potential, defined earlierin this section. We studied the time evolution of (GeV) B µ T ( G e V ) = 10A GeV lab E = 30A GeV lab
E Chemical freeze-outThermal freeze-outt = 10 fm/ct = 17 fm/ct = 8 fm/ct = 15 fm/c
Nuclear matter
CBM
FIG. 7: (Color online) The evolution of temperature ( T )and baryo-chemical potential ( µ B ) in the central cell forAu + Au collisions (b = 2 fm) at E lab = 10A GeVand 30A GeV.The solid (blue) line denotes the chem-ical freeze-out and the dashed (red) line denotes ther-mal freeze-out boundary in relativistic heavy ion colli-sions [36, 37]. The dotted circle denotes the expectedregion probed by the CBM experiment ( √ s NN = 4–10GeV) at FAIR. entropy density at E lab = 10A and 30A GeV tillthe thermal decoupling. Our aim is to get some in-sight about the fluidity of the dense baryonic mat-ter created in these collisions. In recent times sev-eral calculations [39–41] have been reported on thetransport properties of hadronic matter at finitebaryo-chemical potential, including the effect highmass resonances, etc. However the fluidity of densehadronic matter has possibly first discussed in [40]and subsequently in [42, 43]. The authors of Ref. [40]have argued that the fluid behaviour of a baryonrich ( µ B ∼
500 MeV) hadron gas is closer to theideal fluid limit than the corresponding gas withzero baryon number. Following their observation, wehave compared the entropy densities at E lab = 10Aand 30A GeV with the ideal fluid limit reached at thehighest RHIC energy ( √ s NN = 200 GeV). We haveparameterized temporal evolution of entropy densityof hadronic matter from an ideal hydrodynamic sim-ulation [44] for central Au +Au collisions at √ s NN = 200 GeV. The entropy density at r = 3 fm from thecenter has been found to scale with proper time ( τ )as ∼ τ − . for τ ≥
10 fm/c. The results are depictedin Fig. 8 along with the parameterization from idealhydrodynamics. It is heartening to see that the evo-lution of entropy density at E lab = 30A GeV closelyresembles with ideal hydrodynamic limit at zero netbaryon density. The entropy density at E lab = 10AGeV falls even little faster than the aforementionedlimit. It may imply that the hadronic matter pro-duced at 10A GeV beam energy is more ideal thanthe same at 30A GeV beam energy. The observation0 t (fm/c) ) - s ( f m = 10A GeV lab E = 30A GeV lab
E = 0) B µ Ideal hydro (
FIG. 8: (Color online) The time evolution of entropydensity of baryonic matter inside the central cell forAu +Au collisions (b = 2 fm) at E lab = 10A GeV and30A GeV. The dashed line denotes the parameterizationof ideal hydrodynamic evolution of entropy density inthe central region for Au +Au collisions (b = 0 fm) at √ s NN = 200 GeV [44]. can be understood using the fact that shear viscosityto entropy density ratio ( η/s ) of a hadronic systemdecreases with increasing fugacity ( µ B /T ) of the sys-tem [21]. V. SUMMARY AND DISCUSSION
In this article, we have investigated the time scalefor local thermal equilibration of dense baryonicmatter created in central Au + Au collisions atthe proposed CBM experiment energies of E lab =10A, 20A, 30A, 40A GeV. The microscopic trans-port model UrQMD has been used for this purposein the default cascade mode. The net baryon den-sity has been found maximum at 30-40 GeV and thenet strangeness of the created hadronic matter isdominated by baryons for all energies stated above.We have studied the time evolution of longitudinal- to-transverse microscopic pressure anisotropy andinverse slope parameter of the energy spectra ofbaryons and mesons inside a cell of 8 fm in thecentral region of Au + Au system. The pressureanisotropy ratio of baryons and mesons has achieveda constant value close to unity, on a certain time.The time has been found to decrease with the in-crease in laboratory energy. The time scale obtainedfrom the evolution of inverse slope parameter of en-ergy spectra of baryons nearly agrees with the pres-sure (or momentum) isotropization time. Howevera small time difference (∆ t ∼ lab = 10A and 30A GeV. They arefound to agree qualitatively with the empirical rela-tion between T and µ B at the chemical freeze-out.In addition we have calculated the entropy densityof the baryonic matter inside the cell and found theevolution is quasi-isentropic, close to the ideal hy-drodynamic limit at zero net baryon density. VI. ACKNOWLEDGEMENT
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