Entropy production in pp and Pb-Pb collisions at energies available at the CERN Large Hadron Collider

We use experimentally measured identified particle spectra and Hanbury Brown-Twiss radii to determine the entropy per unit rapidity $dS/dy$ produced in $\sqrt{s} = 7$ TeV pp and $\sqrt{s_{\rm NN}} = 2.76$ TeV Pb-Pb collisions. We find that $dS/dy = 11335 \pm 1188$ in 0-10% Pb-Pb, $dS/dy = 135.7 \pm 17.9$ in high-multiplicity pp, and $dS/dy = 37.8 \pm 3.7$ in minimum bias pp collisions and compare the corresponding entropy per charged particle $(dS/dy)/(dN_{\rm ch}/dy)$ to predictions of statistical models. Finally, we use the QCD kinetic theory pre-equilibrium and viscous hydrodynamics to model entropy production in the collision and reconstruct the average temperature profile at $\tau_0 = 1$ fm/$c$ for high multiplicity pp and Pb-Pb collisions.


I. INTRODUCTION
Ultrarelativistic collisions of nuclei as studied at RHIC and the LHC are typically modeled assuming rapid thermalization within a timescale of 1-2 fm/c [1]. The subsequent longitudinal and transverse expansion of the created quark-gluon plasma (QGP) is then described by viscous relativistic hydrodynamics [2]. In this picture the bulk of the entropy is created during the thermalization process and the later stages of the evolution add relatively little [3]. By correctly accounting for the entropy production in different stages of the collisions, one can therefore relate the measurable final-state particle multiplicities to the properties of system, e.g. initial temperature, at the earlier stages of the collisions.
Two different methods are frequently used to estimate the total produced entropy in nuclear collisions. In the first method, pioneered by Pal and Pratt, one calculates the entropy based on transverse momentum spectra of different particle species and their source radii as determined from Hanbury Brown-Twiss correlations [4]. The original paper analyzed data from √ s NN = 130 GeV Au-Au collisions and is still the basis of many entropy estimations at other energies [3]. The second method uses the entropy per hadron as calculated in a hadron resonance gas model to translate the final-state multiplicity dN/dy per unit of rapidity to an entropy dS/dy [5,6]. Even though the estimate of the entropy from the measured multiplicity dN ch /dη is relatively straightforward one finds quite different values for the conversion factor between the measured charged-particle multiplicity dN ch /dη and the entropy dS/dy in the literature [6][7][8][9].
This paper provides an up-to-date calculation of entropy production in pp and Pb-Pb collisions at the LHC energies and uses state-of-the modeling of the QGP to reconstruct the initial conditions at the earliest moments in the collision. In Sec. II we recap the method of Ref. [4], which we use in Sec. III A and Sec. III B to calculate the total produced entropy per rapidity, and the entropy per final-state charged hadron S/N ch ≡ (dS/dy)/(dN ch /dy) from the identified particle spectra and femtoscopy data for √ s = 7 TeV pp and √ s NN = 2.76 TeV Pb-Pb collisions at LHC [10][11][12][13][14][15][16]. In Sec. IV the result for the entropy per particle is then compared to different estimates of the entropy per hadron calculated in hadron resonance gas models at the chemical freeze-out temperature of T ch ≈ 156 MeV [17]. Finally in Sec. V A we use different models of the QGP evolution to track entropy production in different stages of the collisions and to determine the initial temperature profile at τ = 1 fm/c.

II. ENTROPY FROM TRANSVERSE MOMENTUM SPECTRA AND HBT RADII
In this section we recap the entropy calculation from particle spectra and femtoscopy [4]. In this method the entropy S for a given hadron species at the time of kinetic freeze-out is calculated from the phase space density f ( p, r) according to where + is for bosons and the − for fermions. The factor 2J + 1 is the spin degeneracy. The total entropy in the collision is then given by the sum of the entropies of the produced hadrons species. The integral in Eq. 1 can be evaluated using the series expansion The density profile of the source is parametrized by a three dimensional Gaussian so that the phase space density can be written as The radii in Eq. 4 are functions of the momentum p. Writing Eq. 3 in terms of the radii R out , R side , R long measured through Hanbury Brown-Twiss two-particle correlations [18] and using the source volume R 3 inv = γR out R side R long in the two-particle rest frame with γ = m T /m ≡ m 2 + p 2 T /m one arrives at where m the particle mass and + is for bosons and − for fermions. Note that Eq. 5 includes the terms up to f 4 i /12 of the Taylor expansion in Eq. 2. Pions have the highest phase space density of the considered hadrons and the approximation made in Eq. 5 is better than 1% for pions in central Pb-Pb collisions at √ s NN = 2.76 TeV.
In pp collisions at √ s = 7 TeV the maximum pion phase space density F(p T ) exceeds unity at low p T rendering the series expansion in Eq. 2 unreliable. For pions in pp collisions we therefore approximate the (1+f ) ln(1+ f ) term of Eq. 1 by a polynomial of order 8. This gives an approximate expression with numerical coefficients a i which is also valid for values of F obtained for pions in high-multiplicity pp collisions: III. RESULTS A. Entropy in Pb-Pb collisions at √ sNN = 2.76 TeV We determine the entropy in Pb-Pb collisions at √ s NN = 2.76 TeV for the 10% most central collisions considering as final-state hadrons the particles given in Table I. The calculation uses transverse momentum spectra of π, K, p [10], Λ [11], and Ξ, Ω [12] from the ALICE collaboration as experimental input. We also use HBT radii measured by ALICE [19].
For the entropy determination the measured transverse momentum spectra need to be extrapolated to p T = 0. To this end we fit different functional forms to the p T spectra (Tsallis, Bose-Einstein, exponential in transverse mass m T = p 2 T + m 2 , Boltzmann, as defined in [10]).  [20,21] is fitted to the spectrum to extrapolate to pT = 0. The onedimensional scaled HBT radii divided by (( √ γ + 2)/3) 1/2 [22] where γ = mT/m as a function of the transverse mass mT are parametrized by a power law function αm β T and by an exponential function a exp(−mT/b) + c.
In the entropy calculation we only use the extrapolations in p T regions where data are not available, otherwise we used the measured spectra. Differences of the entropy estimate for different functional form are taken as a contribution to the systematic uncertainty. We have checked that the p T -integrated π, K, p multiplicities (dn/dy) y=0 agree with the values published in [10].
The one-dimensional invariant HBT radii R inv are only available for π, K, and p. When plotted as a function of transverse mass m T = m 2 + p 2 T the R inv values for these particle do not fall on a common curve. However, in [22] it was shown that the HBT radii R inv divided by (( √ γ + 2)/3) 1/2 where γ = m T /m are approximately a function of m T only. We use this m T scaling of the scaled HBT radii to obtain R inv (m T ) for all considered particles. The bottom panel of Fig. 1 shows parametrizations of the scaled HBT radii with a power law function and with an exponential function which provide different extrapolation towards the pion mass. We propagate the systematic uncertainties of the measured HBT radii as well as the uncertainty related to the two different parametrizations to the uncertainty of the extracted entropy.
For the entropy calculation the particle species in Table I are considered stable. The entropy carried by neutrons, neutral kaons, η, η , and Σ baryons is estimated based on measured species assuming that the entropy per particle is similar for particles with similar masses. The entropy carried by neutrons is assumed to be the same as the entropy carried by protons. The entropy associated with neutral kaons and η mesons is determined from charged kaons, the entropy of η from protons, and the entropy of Σ baryons from Λ.
The yields of particles in Table I contain contributions from strong decays. To take into account mass differences and to estimate the contributions from strong decays to the different particle species we simulate particle decays with the aid of Pythia 8.2 [23,24]. To this end we generate primary particles available in Pythia 8.2 with rates proportional to equilibrium particle densities in a noninteracting hadron gas [25,26]: Here g = 2J + 1 is the spin degeneracy factor and K 2 the modified Bessel function of the second kind. The + is for bosons and the − for fermions. For the chemical potential we use µ = 0. For the temperature we take T = 156 MeV as obtained from statistical model fits to particle yields measured at the LHC [17]. We then simulate strong and electromagnetic decays of the primary particles. Particle ratios after decays are used to estimate the entropy of unmeasured particles. In case of the η meson we find that after decays the η/K + ratio is 0.69 while the primary ratio is η prim /K + prim = 0.79. For the η we find η prim /p prim = 0.45 and η /p = 0.25 after decays. The primary Σ − prim /Λ prim ratio is about 0.66. The entropy carried by the Σ baryons is derived from the ratios Σ − /Λ ≈ 0.26 and Σ 0 /Λ ≈ 0.27 after decays.
The η, η mesons and Σ 0 baryons decay electromagnetically. Decay products from these decays (η, η → pions, and Σ 0 → Λγ) are not subtracted from the experimentally determined particle spectra. As η, η , and Σ are considered stable in the entropy calculation (see Table I) we correct for this feeddown contributions. In the particle decay simulation described above we determine the feeddown fraction and find R fd (η → π + ) = 3.6%, R fd (η → π + ) = 1.2%, R fd (η → η) = 5.9%, and R fd (Σ 0 → Λ) = 27.0%. The entropies for the particle species considered stable are summarized in Table I. These values represent the average of the entropies obtained for the power law and the exponential parametrization of the scaled invariant HBT radii. In both cases the Tsallis function was used to extrapolate the measured transverse momentum spectra to p T = 0. We considered the uncertainties of the measured transverse momentum spectra, the choice of the parametrization of the p T spectra, the uncertainties of the measured HBT radii, and the choice of the parametrization of the HBT radii as a function of m T as sources of systematic uncertainties. The estimated total entropy in 0-10% most central Pb-Pb collisions at √ s NN = 2.76 TeV is 11335 ± 1188. The uncertainty of the estimated entropy is the quadratic sum of the uncertainties related to the transverse momentum spectra (σ spectra = 629) and invariant HBT radii (σ Rinv = 1007). It is interesting to calculate the entropy per charged hadron in the final state from the total entropy. From [27] we obtain for 0-10% most central Pb-Pb collisions at √ s NN = 2.76 TeV a charged-particle multiplicity at midrapidity of dN ch /dη = 1448 ± 54. From our parametrizations of the pion, kaon, and proton spectra we find a Jacobian for the change of variables from pseudorapidity to rapidity of (dN ch /dy)/(dN ch /dη) = 1.162 ± 0.008. This yields an entropy per charged hadron in the final state of S/N ch = 6.7 ± 0.8.
In the paper by Pratt and Pal the entropy was determined for the 11% most central Au-Au collisions at a center-of-mass energy of √ s NN = 130 GeV. The total entropy per unit of rapidity around midrapidity was found to be dS/dy = 4451 with an estimated uncertainty of 10%. Using dN ch /dy = 536 ± 21 from [28] and (dN ch /dy)/(dN ch /dη) ≈ 1.15 we find an entropy per charged particle of S/N ch ≡ (dS/dy)/(dN ch /dy) = 7.2 ± 0.8. This value for Au-Au collisions at a center-ofmass energy of √ s NN = 130 GeV agrees with the value of S/N ch = 6.7 ± 0.8 we obtain for the LHC energies in this paper. Not only in high-energy nucleus-nucleus collisions but also in proton-proton and proton-nucleus collisions transverse momentum spectra and azimuthal distributions of produced particles can be modeled assuming a hydrodynamic evolution of the created matter [29][30][31]. This provides a motivation to determine the entropy dS/dy with the Pal-Pratt method also in pp collisions. Moreover, the experimentally determination of the entropy is of interest in the context of models which are based on entropy productions mechanisms not related to particle scatterings (see, e.g., [32,33]). Here we focus on minimum bias and high-multiplicity pp collisions at √ s = 7 TeV.
Transverse momentum spectra for both minimum bias collisions (π, K, p [13], Λ [11], and Ξ, Ω [14]) and highmultiplicity pp collisions (π, K, p [15], Λ, Ξ, Ω [16]) are taken from the ALICE experiment. The high-multiplicity sample (class I in [16] and [15]) roughly corresponds to the 0-1% percentile of the multiplicity distribution measured at forward and backward pseudorapidities. HBT radii are taken from [34]. In minimum bias pp collisions there is little dependence of R inv on transverse mass and a constant value R inv = 1.1 ± 0.1 fm is assumed. For the high-multiplicity sample m T scaling of R inv is assumed and the same power law and exponential functional forms as in the Pb-Pb analysis are fit to the data from [34] (N ch = 42-51 class in [34]). Taking into account the uncertainty of associating the multiplicity class in [15,16] with the one in [34] we assume an uncertainty of R inv for the high-multiplicity sample of about 10%.
With the same assumptions for the contribution of unobserved particles and feeddown as in Pb-Pb collisions we obtain dS/dy| MB = 37.8 ± 3.7 in minimum bias (MB) collisions and dS/dy| HM = 135.7 ± 17.9 for the high-multiplicity (HM) sample. The contribution of the different particles species to the total entropy are given in Tables II and III. With dN ch /dη = 6.0 ± 0.1 [35] and (dN ch /dy)/(dN ch /dη) = 1.21 ± 0.01 for minimum bias pp collisions we obtain S/N ch | MB = 5.2 ± 0.5 for the entropy per final-state charged particle. For the high-multiplicity sample with dN ch /dη = 21.3 ± 0.6 [15] and (dN ch /dy)/(dN ch /dη) = 1.19 ± 0.01 we find S/N ch | HM = 5.4 ± 0.7.

IV. COMPARISONS TO STATISTICAL MODELS
In order to compare the S/N ch value determined from the measured final state particle spectra to calculations in which particles originate from a hadron resonance gas one needs to know the ratio N/N ch of the total number of primary hadrons N (≡ N prim ) to the total number of measured charged hadrons in the final state N ch (≡ N final ch ). The latter contains feed-down contributions from strong and electromagnetic hadron decays. If only pions were produced one would get N/N ch = 3/2. With the afore- mentioned Pythia 8.2 simulation and the list of stable hadrons implemented in Pythia (again with hadron yields given by Eq. 8 for T = 156 MeV and µ b = 0) we obtain a value of (N/N ch ) Pythia = 1.14. In this calculation particles with a lifetime τ above 1 mm/c were considered stable. Using the implementation of the hadron resonance gas of ref. [36] we find (N/N ch ) TF = 1.09. In following we use N/N ch = 1.115 ± 0.03, i.e., we take the average of the two results as central value and the difference as a measure of the uncertainty.
In the simplest form of the description of a hadron resonance gas the system is treated as a non-interacting gas of point-like hadrons where hadronic resonances have zero width. The entropy density for a primary hadron with mass m at thermal equilibrium with temperature T and vanishing chemical potential µ = 0 is then given Entropy per primary hadron S/N for a noninteracting thermal hadron resonance gas at a temperature of T = 156 MeV as given by Eq. 11 as a function of the upper mass limit for particles listed in the particle data book [37]. The entropy per hadron saturates for high upper mass limits at a value of S/N = 6.9.
by [26] where + is for bosons and − for fermions. K 1 and K 2 are modified Bessel functions of the second kind. Using Eqs. 8 and 10 the entropy per primary hadron in the thermal hadron resonance gas can be calculated as where the index i denotes the different particles species. For illustration, the entropy per hadron is shown in Figure 2 as a function of the upper limit on the mass for all particles listed in the particle data book [37]. More sophisticated implementations of the hadron resonance gas take the volume of the hadrons and the finite width of hadronic resonances into account [17,36,[38][39][40][41][42][43][44][45]. Some of these models implement chemical nonequilibrium factors which we do not consider here. Models can also differ in the set of considered hadron states. In the following we concentrate on the models by Braun-Munzinger et al. [17] ("model 1") and Vovchenko/Stöcker [36] ("model 2"). The corresponding values for the entropy per primary hadron S/N and the entropy per final state charged hadron S/N ch are given in Tab. IV. The S/N ch values for these models are somewhat larger than the measured value of S/N ch = 6.7±0.8, but the deviations are not larger than 1-2σ. We note here that the two approaches calculate slightly different quantities. Our estimate is based on the non-equilibrium  (11) sums the entropy contributions of all primary hadrons in a thermal state before the decays. Although on general grounds we expect the total entropy to increase during the decays and re-scatterings in the hadronic phase, there are some decay products, e.g. photons, which are not included in our current entropy count. Accounting for such differences between the two approaches might bring the estimates closer together.

A. Pb-Pb collisions
The entropy in nuclear collisions, which we calculated in previous sections, is not created instantaneously, but rather the entropy production takes place in several stages in nuclear collisions [3]. In this section we will use different models to describe boost invariant expansion and, in particular, to determine the average initial conditions in 0-10% most central Pb-Pb collisions at √ s NN = 2.76 TeV at time τ 0 = 1 fm.
First, we can make a simple estimate of the initial entropy density s(τ 0 ) by assuming that the subsequent near ideal hydrodynamic evolution does not change the total entropy per rapidity dS/dy (which is also true for freestreaming expansion). The initial volume of the system created in a central ultra-relativistic nucleus-nucleus collision is determined by the geometry of the nuclei, i.e. the transverse area A. Then the initial entropy density is equal to Using for the transverse area A = πR 2 Pb with R Pb = 6.62 fm [47,48] gives an initial entropy density for the 0-10% most central Pb-Pb collisions at √ s NN = 2.76 TeV According to the lattice QCD equation of state [49,50], this corresponds to a temperature Here we would like to note that similar estimates done using the measured energy in the collisions instead of entropy underestimate the initial temperature [4] which gives an energy density e(τ 0 ) ≈ 13.9 GeV/fm 3 . This would correspond to much lower initial temperature T (τ 0 ) ≈ 305 MeV. This is because Eq. (15) is derived under the assumption of a constant energy per rapidity [53]. This holds for a free-streaming (or pressureless) expansion, but in hydrodynamics the system cools down faster due to work done against the longitudinal pressure. Taking τ f = R Pb as a rough estimate for the lifetime of the fireball, ideal hydrodynamics predicts an (τ f /τ 0 ) 1 3 ≈ 1.9 times larger initial energy density, which would revise the initial temperature estimate upwards to T (τ 0 ) ≈ 355 MeV and closer to the value we obtained from the entropy method.
Instead of assuming a constant entropy density in a collision, it is more realistic to use an entropy density profile s(τ, r), where r is a two-dimensional vector in the transverse plane (we still assume boost-invariance in the longitudinal direction). We will employ the two-component optical Glauber model to construct the transverse profile of initial entropy density [54]. In this model the initial entropy is proportional to the participant nucleon number and the number of binary collisions. For a collision at impact parameter b, the entropy profile is then where κ s (1 − α)/2 is entropy per rapidity per participant and κ s α is entropy per rapidity per binary collision. The number densities are calculated using the nucleonnucleon thickness functions (see Appendix A for details), and the value α = 0.128 reproduces centrality dependence of multiplicity [48]. We average over the impact parameter | b| ≤ 4.94 fm to produce entropy profile corresponding to 0-10% centrality bin of Pb-Pb collisions at √ s NN = 2.76 TeV [48]. The overall normalization factor κ s is adjusted to reproduce the final state entropy estimated in Sec. III A, which depends on the expansion model. To simulate the evolution and entropy production in nucleus-nucleus collisions we employ two recently developed models: kinetic pre-equilibrium propagator KøMPøST [55,56], and viscous relativistic hydrodynamics code FluiduM [57] 1 . For simplicity we employ a constant value of specific shear viscosity η/s and vanishing bulk viscosity ζ/s throughout the evolution.
KøMPøST uses linear response functions obtained from QCD kinetic theory 2 to propagate and equilibrate the highly anisotropic initial energy momentum tensor, which can be specified at an early starting time τ EKT = 0.1 fm. We specify the initial energy-momentum tensor profile to be 3 At the end of KøMPøST evolution all components of the energy momentum tensor, the energy density, transverse 1 We neglect the entropy production in the hadronic phase and match the entropy on the freeze-out surface. 2 Current implementation of KøMPøST uses results of pure glue simulations, but recent calculations with full QCD degrees of freedom indicate that the evolution of the total energymomentum tensor will not be significantly altered by chemical equilibration [58,59]. 3 As a purely practical tool we use lattice equation of state to convert entropy density profile obtained from the Glauber model Eq. (16) to the energy density needed to initialize KøMPøST, even though the system at τ EKT = 0.1 fm is not in thermodynamic equilibrium.
flow and the shear-stress components, are passed to the hydrodynamic model at fixed time τ hydro = 0.6 fm. The FluiduM package solves the linearized Israel-Stewart type hydrodynamic equations around an azimuthally symmetric background profile. In this work we propagate the radial background profile until the freezeout condition is met, which we define by a constant freeze-out temperature T fo = 156 MeV. Above this temperature the equation of state is that of lattice QCD [50]. Unless otherwise stated, we use a constant specific shear viscosity η/s = 0.08 and vanishing bulk viscosity ζ/s = 0.
We start by showing the temperature evolution in the hydrodynamic phase in Fig. 3. The solid and dotted white lines represent the freeze-out line at T fo = 156 MeV. The dashed horizontal line indicates the spatial slice of the fireball at some fixed time τ and above the freeze-out temperature. We now can define entropy as an integral of the entropy current su µ over a hypersurface Σ i where Σ i is one or more of the contours shown in Fig. 3. We define the total entropy in the QGP state at time τ as the integral over the contour 2: To include the entropy outflow from the QGP due to freeze-out we also define entropy on the contours Σ 1 +Σ 2 : Due to viscous dissipation S(τ )| QGP+freeze-out increases until the temperature in every hydro cell drops below the freeze-out temperature and the maximum value is simply the entropy current integral over the freeze-out surface Σ 1 + Σ 3 . In Fig. 4(a) we show the time dependence of entropy per rapidity in the QGP phase (yellow line) and including freeze-out outflow (green line) in hydrodynamically expanding plasma. The solid lines are for the simulation with with η/s = 0.08 and dashed lines correspond to η/s = 0.16. In both cases the initial entropy profile, Eq. (16), is adjusted so that after the preequilibrium (KøMPøST) and hydrodynamic (FluiduM) evolution the final entropy on the freeze-out surface is equal to dS/dy = 11335 estimated in Sec. III A. We see that at early times entropy is produced rapidly, but there is little entropy outflow through the freeze-out surface. At τ ≈ 2 fm the entropy in the hot QGP phase starts to drop because matter is crossing the freeze-out surface and at τ ≈ 10 fm there is no hot QGP phase left.
Here we note that the early time viscous entropy production in the hydrodynamic phase depends strongly on the initialization of the shear-stress tensor. In this work we use the pre-equilibrium propagator KøMPøST, which provides all components of energy-momentum tensor at hydrodynamic starting time and the shear-stress tensor approximately satisfies the Navier-Strokes constitutive equations [55,56]. We determine that for evolution with η/s = 0.08 the entropy per rapidity at time τ 0 = 1.0 fm is ≈ 95% of the final entropy on the freeze-out. For twice larger shear viscosity the entropy production doubles and to produce the same final entropy we need only ≈ 90% at τ 0 = 1.0 fm. Such entropy production is neglected in the naive estimate of Eq. (12).
Analogously to entropy, we use the same contours to define energy in the collision, that is, as integrals of the energy current eu µ . In Fig. 4(b) we show the energy per rapidity in different phases of the collision. We confirm that the energy per rapidity decreases rapidly in the hydrodynamic phase and at τ 0 = 1.0 fm is nearly twice larger than on the freeze-out surface and therefore invalidating the naive initial energy density estimates using Bjorken formula Eq. (15). However we do note that the magnitude of the final energy per rapidity in our event is below the measured value. In addition we show points for the energy per rapidity in the pre-hydro phase simulated by KøMPøST. Despite the large anisotropy in the initial energy-momentum tensor (T zz ≈ 0 initially), the energy per rapidity is rapidly decreasing in this phase. We note that at the same time there is a significant entropy production in the kinetic pre-equilibrium evolution [56].
Next in Fig. 5(a) we look at the transverse entropy density profile τ s(τ, r) at different times τ = 1.0, 3.0, 6.0 fm in the hydrodynamic evolution with η/s = 0.08. We see that the profile changes only little between 1 fm and 3 fm, which is due to an approximate one dimensional expansion and viscous entropy production. At later times the profile expands radially and drops in magnitude. The black-dotted line indicates the naive estimate of entropy density τ 0 s(τ 0 ) = 82.3 fm −2 for a disk-like profile with radius R Pb = 6.62 fm, see Eq. (13). Despite an overestimation of the net entropy at τ 0 = 1 fm, the actual density at the center of entropy profile is twice larger than the naive estimate. Correspondingly, the transverse temperature profile at τ 0 = 1 fm, shown in Fig. 5(b), is larger than the simple estimate and can reach 400 MeV in the center of the fireball.

B. Central pp collisions
In this section we present a similar analysis of entropy production in ultra-central pp collisions. Because of much smaller initial size, the QGP fireball (if created), has a much shorter life-time than the central Pb-Pb collisions. This should enhance the relative role of the preequilibrium physics of QGP formation.
To model the initial entropy density in pp collision, we use a Gaussian parametrization of the transverse entropy distribution s(τ 0 , r) = κ s τ 0 2πσ 2 e − r 2 2σ 2 (20) with a width σ = 0.6 fm, as used in other parametrizations [60]. We use a fixed value of η/s = 0.08 and, in view of the range of applicability of the linearized preequilibrium propagator, we use KøMPøST for a shorter time from τ EKT = 0.1 fm to τ hydro = 0.4 fm. First we show the temperature evolution in Fig. 6 and indicate the freeze-out contour (lines 1 and 3). We note that because of the compact initial size, the transverse expansion is so explosive that the center of the fireball actually freezes-out before the edges. Next in Fig. 7(a) we show the entropy evolution in the QGP phase and together with the outflow from through the freeze-out surface. In a smaller system, the radial flow builds up faster and the QGP and the combined QGP+freeze-out surface contributions starts to deviate early. This does not capture the entropy which already left T > T fo region in the KøMPøST phase, but for the early hydro starting time τ hydro , this fraction is small. We see that as a fireball of QGP ultra central pp collisions have a lifetime just above τ = 2 fm. Therefore the τ 0 = 1 fm reference time is no longer adequate time to discuss the "initial conditions" in such collisions. Next, in Fig. 7(b) we show the energy per rapidity in the hydrodynamic and preequilibrium stages. Here again we see that energy per rapidity decreases more rapidly in comparison of entropy production. tropy density is much smaller than in 0-10% centrality collisions and only at τ = 0.5 fm the temperature at the center reaches above T = 300 MeV.

VI. SUMMARY AND CONCLUSIONS
We provide independent determination of the finalstate entropy dS/dy in √ s = 7 TeV pp and √ s NN = 2.76 TeV Pb-Pb collisions from the final phase space density calculated from the experimental data of identified particle spectra and HBT radii. In addition, we have calculated the entropy per final-state charged hadron (dS/dy)/(dN ch /dy) in different collision systems. We find the following values for pp and Pb-Pb collisions: We compare our results for (dS/dy)/(dN ch /dy) ratio based on experimental data, to the values obtained from the statistical hadron resonance gas model at the chemical freeze-out temperature of T ch = 156 MeV. For the 0-10% most central Pb-Pb collisions statistical model values are systematically higher than our estimate, but in agreement at the 1-2σ level. However the measured (dS/dy)/(dN ch /dy) values in minimum bias and highmultiplicity pp collisions at √ s = 7 TeV are below the theory predictions of chemically equilibrated resonance gas, perhaps indicating that full chemical equilibrium is not reached in these collisions. Here we note that, interestingly, in pp collisions the estimated soft pion phasespace density exceeds unity. The precise knowledge of the total produced entropy in heavy ion collisions and the entropy per final-state charged hadron is important for constraining the bulk properties of the initial-state from the final state observables [55,59,61]. In order to determine the initial medium properties for high multiplicity pp and Pb-Pb collisions, we performed simulations of averaged initial conditions starting at τ 0 = 0.1 fm/c with kinetic preequilibrium model KøMPøST [55,56,62] and viscous relativistic hydrodynamics code FluiduM [57]. Importantly, these calculations take into account the produced entropy and work done in both the pre-equilibrium and hydrodynamic phases of the expansion [61]. We find that for simulations with the specific shear viscosity value η/s = 0.08 the initial pre-equilibrium energy per unity rapidity is about three times larger than at the final state in 0-10% most central Pb-Pb collisions at √ s NN = 2.76 TeV, and approximately twice larger in high multiplicity pp collisions at √ s = 7 TeV. At the time τ = 1 fm/c, the temperature in the center of the approximately equilibrated QGP fireball is about T ≈ 400 MeV for Pb-Pb and T ≈ 250 MeV for high-multiplicity pp collision systems. Finally, we note that in our simulations of Pb-Pb collisions with η/s = 0.08 only about 5% of the total final entropy is produced after τ = 1 fm/c, meaning that most of entropy production occurs in the pre-equilibrium phase.  7. (a) Entropy per rapidity in viscous hydrodynamic expansion with specific shear viscosity η/s = 0.08 for √ s = 7 TeV pp collisions (0-1% collisions with the highest multiplicity). The yellow line corresponds to entropy in the QGP phase (T (τ, r) > T fo ) (contour 2 in Fig. 6), whereas the green line shows the total cumulative entropy (contour 1 + 2 in Fig. 6). The initial conditions, i.e. parameter κs in Eq. (20), was tuned to reproduce the final freeze-out entropy dS/dy = 135.7 after the pre-equilibrium and hydrodynamic evolutions. (b) Analogous plot for energy per rapidity in hydrodynamic expansion with η/s = 0.08. Additional points show energy per rapidity in the pre-equilibrium stage. Then the collision probability of two nuclei with N A and N B nucleons is given by where the radius is implicitly assumed to be in the transverse plane and σ N N inel = 6.4 fm 2 is the inelastic nucleon-nucleon cross-section. The number of participant nucleons per transverse area is given by These probabilities are combined in the two-component Glauber model [48,54] where α is an adjustable parameter (sτ ) 0 = κ s 1 − α 2 dN part ( r, b) d 2 r + α dN coll ( r, b) d 2 r (A5)