Observables of non-equilibrium phase transition
EEPJ manuscript No. (will be inserted by the editor)
Observables of non-equilibrium phase transition
Boris Tom´aˇsik , , Martin Schulc , Ivan Melo , , and Renata Kopeˇcn´a Univerzita Mateja Bela, FPV, Tajovsk´eho 40, 97401 Bansk´a Bystrica, Slovakia ˇCesk´e vysok´e uˇcen´ı technick´e v Praze, FJFI, Bˇrehov´a 7, 11519 Prague 1, Czech Republic ˇZilinsk´a univerzita, Elektrotechnick´a fakulta, Akademick´a 1, 01026 ˇZilina, Slovakia Abstract.
A rapidly expanding fireball which undergoes first-order phase transition will supercool andproceed via spinodal decomposition. Hadrons are produced from the individual fragments as well as theleft-over matter filling the space between them. Emission from fragments should be visible in rapiditycorrelations, particularly of protons. In addition to that, even within narrow centrality classes, rapiditydistributions will be fluctuating from one event to another in case of fragmentation. This can be identifiedwith the help of Kolmogorov-Smirnov test. Finally, we present a method which allows to sort events withvarying rapidity distributions in such a way, that events with similar rapidity histograms are groupedtogether.
PACS.
Experiments at NICA aim to explore the region of thephase diagram where highly compressed and excited mat-ter may undergo a first-order phase transition. It is arguedelsewhere in this volume that such a phase transition in arapidly expanding system may bring it out of equilibriumand end up in its spinodal decomposition. Such a processthen generates enhanced fluctuations in spatial distribu-tions of the baryon density and the energy density.In this paper we focus on observables which could helpto identify such processes.Before we explain various possible observables, we in-troduce DRAGON: the Monte Carlo tool suited for gen-eration of hadron distributions coming from a fragmentedfireball [1]. Then, we report on an idea proposed in [2,3]and further elaborated in [4]: clustering of baryons can bevisible in rapidity correlations of protons. Further, we turnour attention to the whole rapidity distributions of pro-duced hadrons and present an idea to search for nonstatis-tical differences between them with the help of Kolmogorov-Smirnov test [5]. Finally, we propose a novel treatmentnow being developed which also compares momentum dis-tributions from individual events and sorts events accord-ing to their similarity with each other [6].
In order to test the effects of fireball fragmentation intodroplets it is useful to have Monte Carlo tool for the gener-ation of artificial events with such features included. One possibility is to construct hydrodynamic models which in-clude such a behaviour in the evolution [7,8,9,10]. Theyallow to link the resulting effects in fireball evolution withthe underlying properties of the hot matter. On the otherhand, they offer less freedom for systematic investigationof how the fragmentation is indeed seen in data. Interest-ing questions of this kind are: what is the minimum sizeand abundance of fragments that can be seen? What ex-actly is their influence on spectra, correlations, anisotropies,and femtoscopy? How are these observables influenced bythe combination of droplet production and collective ex-pansion?Such questions can be conveniently explored with thehelp of Monte Carlo generator that uses a parametrization of the phase-space distribution of hadron production. Sucha tool has been developed in [1] under the title DRAGON(DRoplet and hAdron Generator fOr Nuclear collisions).All studies presented here have been performed on eventsgenerated with its help.The bedding of the generator is the blast-wave model.The probability to emit a hadron in phase-space is de-scribed by the emission function S ( x, p ) d x = g (2 π ) m t cosh( y − η ) exp (cid:18) − p µ u µ T (cid:19) × Θ ( R − r ) exp (cid:18) − ( η − η ) ∆η (cid:19) δ ( τ − τ ) × τ dτ dη r dr dφ . (1)It is formulated in Milne coordinates τ = √ t − z , η =(1 /
2) ln(( t + z ) / ( t − z )) and polar coordinates r , φ inthe transverse plane. Emission points are distributed uni-formly in transverse direction within the radius R and a r X i v : . [ nu c l - t h ] D ec Boris Tom´aˇsik, Martin Schulc, Ivan Melo, and Renata Kopeˇcn´a: Observables of non-equilibrium phase transition freeze-out occurs along the hypersurface given by con-stant τ = τ . Azimuthal anisotropy has not been used instudies presented here although the model includes such apossibility. There is collective longitudinal and transverseexpansion parametrized by the velocity field u µ = (cosh η cosh η t , cos φ sinh η t , sin φ sinh η t , sinh η cosh η t ) (2) η t = η t ( r ) = √ ρ rR . (3)The fireball is locally thermalized with the temperature T . A part of the hadrons, which can be specified in themodel, is emitted from the droplets. The droplets stemfrom the fragmentation of the same hypersurface as as-sumed in eq. (1). The actual picture is that when the fire-ball fragments, some free hadrons are born between theproduced droplets. The volume of droplets is distributedaccording to [11] P V ( V ) = Vb e − V/b . (4)The average volume of droplets is then 2 b . The minimalmass is practically set by the lightest hadron in simulation:usually the pion. The probability to emit hadron from adroplet drops exponentially in droplet proper time τ d P τ ( τ d ) = 1 R d e − τ d /R d , (5)where R d is the radius of the droplet. Momenta of hadronsfrom droplets are chosen from the Boltzmann distributionwith the same temperature as bulk production. Currently,neither momentum nor charge conservation is taken intoaccount in droplet decays, but an upgrade of the modelincluding these effects is envisaged.DRAGON also includes production of hadrons fromresonance decays. Baryons up to 2 GeV and mesons upto 1.5 GeV of mass are included. Chemical composition isspecified by chemical freeze-out temperature and chemicalpotentials for baryon number and strangeness. (Chemicalpotential for I should also be introduced but is practicallyvery small and thus neglected in the simulations.) Hadrons emitted from the same droplet will have similarvelocities. This should be seen in their correlations [2,3].Protons appear best suited for such a study. Their mass ishigher than that of most mesons, so their deflection fromthe velocity of the droplet due to thermal smearing willbe less severe. Pions would have better statistics thanks totheir high abundance, but their smearing due to thermalmotion and resonance decays is too big.Correlation function can be measured as a function ofrapidity difference ∆y = y − y or (better) of the relativerapidity y = ln (cid:20) γ + (cid:113) γ − (cid:21) (6) with γ = p · p /m m .The correlation function is conveniently sampled as C ( y ) = P ( y ) P , mixed ( y ) (7)where P ( y ) is the probability to observe a pair of pro-tons with relative rapidity y . The reference distribution P , mixed ( y ) in the denominator is obtained via the mixedevents technique.It is instructive to first consider a simple model wherethe rapidities of droplets follow Gaussian distribution ζ ( y d ) = 1 (cid:112) πξ exp (cid:18) − ( y d − y ) ξ (cid:19) . (8)Within the droplet i which has rapidity y i , rapidities ofprotons are also distributed according to Gaussian ρ ,i ( y ) = ν i √ πσ exp (cid:18) − ( y − y i ) σ (cid:19) . (9)This distribution is normalized to the number of protonsfrom droplet i , which is denoted as ν i .The resulting correlation function in this simple modelis [3,4] C ( ∆y ) − ξ (cid:104) N d (cid:105)(cid:104) ν ( ν − (cid:105) M (cid:104) N d ( N d − (cid:105)(cid:104) ν (cid:105) M (cid:115) σ ξ σ exp − ∆y σ (cid:16) σ ξ (cid:17) (10)where (cid:104) N d (cid:105) is the average number of droplets in one eventand (cid:104)· · · (cid:105) M denotes averaging over various droplets. Natu-rally, the width of the correlation function depends on σ ,as might have been expected. However, it also depends onthe width of the rapidity distribution of droplets: throughthe factor (1+ σ /ξ ), growing ξ leads to narrower protoncorrelation function.As an illustration relevant for NICA we generated setsof events with the help of DRAGON. On these sampleswe studied the influence of droplet size and the share ofparticles from droplets on the resulting correlation func-tions. It turns out that the relative rapidity y yields bet-ter results, so we have mainly used this observable in ouranalyses. A more detailed study, though not with specificNICA fireball settings, can be found in [4].DRAGON was set with Gaussian rapidity distributionwith the width of 1. Within the rapidity acceptance win-dow − < y < η f = 0 .
4. Chemi-cal composition was according to T ch = 140 MeV and µ B = 413 MeV. Recall that resonance decays are includedin the model. The same kinetic temperature and chemicalcomposition was assumed for the droplets. Total mass ofeach droplet is given by its size and the energy density oris Tom´aˇsik, Martin Schulc, Ivan Melo, and Renata Kopeˇcn´a: Observables of non-equilibrium phase transition 3 | Relative rapidity |y | )- D c o rr e l a t i on C ( | y no droplets
50% 20 fm
25% 50 fm
75% 20 fm
Fig. 1.
Proton correlation functions for four different settingsof hadron production from droplets. . Transverse size of the fireball was set to10 fm and the lifetime τ = 9 fm/ c , but these parametershave no influence on the presented results. Note that wehave imposed acceptance cut in rapidity − < y <
1, sothat we do not show results that would not be measurabledue to limited acceptance.In order to see the effect of droplet formation on thecorrelation function we simulated one data set with nodroplets and three sets which differ in droplet settings.We have sets with: b = 50 fm and the fraction of 25% ofhadrons from droplets, b = 20 fm and 50%, b = 20 fm and 75%. Recall that the mean droplet volume is 2 b .The resulting proton correlation functions in y areplotted in Fig. 1. As expected, without fragmentation thecorrelation function is flat. The widths of the correlationfunctions are given by the smearing of the momenta ofprotons within one droplet, mainly due to temperature.The level of correlation is expressed in the height ofthe peak at y = 0. Naturally, this is expected to growif a larger number of protons is correlated. This can beachieved in two ways: by increasing droplet sizes so thatmore protons come from each droplet, or by increasing thenumber of droplets by enhancing the share of particles pro-duced by droplets. By coincidence we thus obtained verysimilar results for the cases with droplet fractions 25% and75%, since the latter one assumes smaller droplets.Note the width scale of the correlation function whichis larger than the typical scale of strong interactions. Thusany modification due to final state interactions which havenot been included here is expected to be concentratedaround the peak of our correlation functions. The fragmentation of the fireball actually leads to event-by-event fluctuations of rapidity distributions. In each eventhadrons are produced from a different underlying rapiditydistribution. In [5] it was proposed to use a standard sta-
Fig. 2.
Definition of the distance between two events. Themeasured values of variable x are indicated on horizontal axis.Lines of different thickness represent two different events. tistical tool for the comparison of hadron rapidity distri-butions from individual events: the Kolmogorov-Smirnov(KS) test. The KS test has been designed to answer thequestion, to what extent two empirical distributions seemto correspond to the same underlying probability density.To apply the test on empirical distributions one firsthas to define a measure of how much they differ. For thesake of clarity and brevity we shall call empirical distri-butions events and the measure of difference will be theirscaled distance, to be defined later. A distance is definedin Fig. 2. Consider measuring the quantity x (this may bee.g. the rapidity) for all particles in two different events.We mark the values of x on the horizontal axis. Then, inthe same plot we draw for each event its empirical cumu-lative distribution function. It is actually a staircase: westart at 0 and in each position where there is measured x we make a step with the height 1 /n i , where n i is the mul-tiplicity of the event. The maximum vertical distance D between the two obtained staircases is taken as the mea-sure of difference between the two events. For further workone takes the scaled distance d = (cid:114) n n n + n D (11)where n , n are the multiplicities of the two events.Next one defines Q ( d ) = P ( d (cid:48) > d ) (12)i.e. the probability that the scaled distance d (cid:48) determinedfor a pair of random events generated from the same un-derlying distribution will be bigger than d . The formulasfor obtaining Q ( d ) for any d are given in the Appendixof [5]. Thus defined, for large d , the value of Q will besmall because there is little chance that two events will beso much different. If all events come from the same un-derlying distribution, then the Q ’s determined on a largesample of event pairs will be distributed uniformly . Boris Tom´aˇsik, Martin Schulc, Ivan Melo, and Renata Kopeˇcn´a: Observables of non-equilibrium phase transition h3Entries 99992Mean 0.3958RMS 0.3062 Q h3Entries 99992Mean 0.3958RMS 0.3062 (cid:47) h3Entries 99992Mean 0.4252RMS 0.3035h3Entries 99992Mean 0.3969RMS 0.3077h3Entries 99992Mean 0.5085RMS 0.2901 , R=135 h3Entries 99992Mean 0.2438RMS 0.2842 Q h3Entries 99992Mean 0.2438RMS 0.2842h3Entries 99992Mean 0.2849RMS 0.2934h3Entries 99992Mean 0.2411RMS 0.2825h3Entries 99992Mean 0.4868RMS 0.294 , R=381 charged hadrons Fig. 3. Q -histograms for samples of 10 simulated events. Ra-pidities of charged pions (top) and all charged hadrons (bot-tom) are taken into account. In a sample of events where the shape and dynamicalstate of the fireballs fluctuate, e.g. due to fragmentation,large scaled distance d will be more frequent. This is thentranslated into higher abundance of low Q values. Thusnon-statistical differences between events will show up asa peak at low Q in the histogram of Q values for largenumber of event pairs. In order to quantify the signifi-cance of the peak above the usual statistical fluctuationswe introduce R = N − N tot B σ = N − N tot BN tot B (13)where N is the number of event pairs in the first Q -bin, N tot is the number of all event pairs and B is the numberof Q -bins.To illustrate the application at NICA, we have usedevent samples with the same settings as in the previousSection and show in Fig. 3 the Q -histograms for pion ra-pidity distributions as well as rapidity distributions of allcharged hadrons. The signal is very strong and the onefor charged hadrons is generally more pronounced thanthe one for pions. The comparison of different data sets isconsistent with results for correlation functions from the previous section. Note that there is basically very weaksignal for the case without droplets, which shows thatclustering effect due to resonance decays cannot mask theinvestigated mechanism. In presence of fireball fragmentation, rapidity distribu-tions of different events show large variety. This motivatesthe quest to select among them groups of events which willbe similar. Such groups allow to appreciate the range offluctuations of the momentum distribution. They also maybe useful for the construction of mixed events histogramsused in correlation functions.A method for sorting events according to their simi-larity with each other has been proposed [12,6]. The ap-plication in [6] was on azimuthal angle distributions. Herewe use it for rapidity distributions. Details can be foundin [6]; here we only shortly explain the sorting algorithm.An event is characterized when all its bin entries n i are given; i numbers the bins in rapidity. Full bin recordwill be denoted { n i } .1. Events are initially sorted in a chosen way and dividedinto N quantiles of the distribution. We use deciles,numbered by Greek letters.2. For each event, characterized by record { n i } , calcu-late the probability that it belongs to the event bin µ , P ( µ |{ n i } ), using the Bayes’ theorem P ( µ |{ n i } ) = P ( { n i }| µ ) P ( µ ) P ( { n i } ) . (14)The probability P ( { n i }| µ ) that the event with binrecord { n i } belongs to the event bin µ can be expressedas P ( { n i }| µ ) = M ! (cid:89) i P ( i | µ ) n i n i ! (15)where M is the event multiplicity, the product goesover all (rapidity) bins, and P ( i | µ ) is the probabilitythat a particle falls into bin i in an event from eventbin µ P ( i | µ ) = n µ,i M µ . (16)( M µ is the total multiplicity of all events in event bin µ and n i,µ is the total number of particles in bin i .)Coming back to eq. (14): P ( µ ) = 1 /N is the prior and P ( { n i } ) = N (cid:88) µ =1 P ( { n i }| µ ) P ( µ ) . (17)3. For each event determine¯ µ = N (cid:88) µ =1 P ( µ |{ n i } ) µ (18)and re-sort all events according to ¯ µ . Then divide againinto quantiles. oris Tom´aˇsik, Martin Schulc, Ivan Melo, and Renata Kopeˇcn´a: Observables of non-equilibrium phase transition 5 -1 -0.5 0 0.5-1 -0.5 0 0.5-1 -0.5 0 0.5-1 -0.5 0 0.5024681012141618024681012141618 N u m b e r o f p a r t i c l e s N u m b e r o f p a r t i c l e s -1 -0.5 0 0.5 Fig. 4.
Average rapidity histograms of the 10 event bins afterthe sorting algorithm with 5000 events (with rapidity flip - seetext) converged. Droplet fraction 25% and b = 50 fm .
4. If the ordering of events changed, re-iterate from point2. In a less strict version of the algorithm, the orderingis re-iterated only if the assignment to quantiles haschanged.This iterative algorithm organizes events in such a way,that those which are similar to each other by the shapesof their histograms end up close to each other. It is notspecified a priori , however, whether there is any specificobservable according to which the sorting proceeds. Thealgorithm itself picks the best ordering automatically. Themethod actually provides a more sophisticated version ofthe Event Shape Engineering.We have tested the algorithm on a set of events gen-erated by DRAGON with the same parameters as in pre-vious two Sections. For illustration, we show in Fig. 4 theaverage histograms in different event bins after the sortingalgorithm. We have chosen the data set with droplet frac-tion 25% and b = 50 fm and the algorithm works withrapidity distributions of pions. As a result of the fluctu-ations in rapidity distributions, the differences betweenevent bins are large. On one end there are events with al-most symmetric distributions, whereas on the other endthere are events with strong emphasis on one side.It should be noted that the simulation setting assumessymmetric Gaussian rapidity distribution and correspon-ded to symmetric nuclear collisions. Consequently, there isno reason to favour one rapidity direction over the other.The resulting sorting in Fig. 4 is obtained when in themiddle of the iteration process one half of the events isflipped over the mid-rapidity.The difference between event bins is much bigger herethan in a sample of events where no droplets are present. We have sketched and explained two kinds of observablesthat can be used for identification of the fragmentationprocess: proton correlations in rapidity [3,4] and the Kol-mogorov-Smirnov test comparing the event-by-event ra-pidity distributions [5]. The motivation to look for thefragmentation comes from the fact that a first order phasetransition actually should proceed this way.It should be mentioned that in [13,14] it has been ar-gued that potentially there is a mechanism which may leadto fireball fragmentation even in absence of the first orderphase transition. A sharp peak of the bulk viscosity as afunction of temperature may suddenly cause resistance ofthe bulk matter against expansion. Driven by the inertia,the fireball could choose to fragment. This possibility putsthe uniqueness of the fragmentation process as the signa-ture for the first order phase transition under question.Nevertheless, it is still certainly worthwhile to investigatethe consequences of such a process.A process that could mask the signals of fragmenta-tion is rescattering of hadrons emitted from droplets. Itwould be interesting to combine the presented methodswith models including such a possibility.Finally, we presented a method which is still beingdeveloped and which allows to sort the measured eventsautomatically according to the most pronounced featuresin their histograms and build groups of similar events [6].This would allow to study such groups, where event-by-event fluctuations are suppressed, in more detail.
We acknowledge partial support by grants APVV-0050-11,VEGA 1/0469/15 (Slovakia). BT was also supported by grantsRVO68407700, LG15001 (Czech Republic). RK acknowledgessupport from SGS15/093/OHK4/1T/14.
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