Outlook of the seminar • Introduction. • Hadronic matter phase diagram (lQCD) • Golden probes. • Heavy Ion Collisions. • Experimental Results. • More on Heavy Flavour and Quarkonium. • Future ALICE program for run#3. • MFT project. • Conclusions.

Key Questions

ü What are the fundamental properties of strongly interacting matter as a function of temperature and density? ü How do hadrons acquire their mass and how is the mass modified by the medium they move in? ü What are the properties of the quark gluon plasma? ü To which extent did these properties govern the ü evolution of the universe? ü Are there colour superconductors and highly dense gluonic objects in nature?

Hadronic matter at high T ü Before the discovering of the QCD asymptotic freedom, Hagedorn tried to imagine the properties of matter at high T (in the 60's).

ü Hadronic resonances are created increasing the internal degrees of freedom of the system; ü Hagedorn realized that the number of hadron species increases exponentially with the hadron mass.

Hagedorn limit temperature

• When matter try to reach the T H temperature, energy is used to increase the number of hadronic resonances. • In consequence, infinite energy will be needed to reach a temperature T H (~170 MeV) of matter. T H becomes the limit temperature of the matter. Caveat for small QGP drops ü Body thermal radiation is a bad approximation for a QGP of 7 fm diameter since the mean path of a photon or lepton in QGP will be larger that its diameter. ü Therefore photon emitted by the QGP are not in equilibrium with the QGP. ü Yields scale by a factor ~αQED /α QCD ~0.02 (at GeV scale).

Quarkonium

• Bound state of Q and its anti-Q:

-Charmonium for c-cbar: J/ψ, ψ', χc; -Bottomonium for b-bar: upsilon family 1S, 2S and 3S.

• Properly described by QM: he interaction. This explains why (2S) resonance is easily melt hy ⌥(1S) would melt at higher temperatures than that of J/ . Yo ulation of an upper bound of the dissociation temperature of quark of A. Mocsy and P. Petreczky in the reference [Abreu 08]. Their are quoted in table II.

Parton-QGP interaction

• Depending on colour density in the QGP;

• Casimir factor (C R ): 4/3 for q , 3 for g • Dead cone effect: important for beauty 5 GeV/c 2 q,g E E-ΔE 25

ΔE

How is the energy dissipated in the medium?

Heavy Ion Collisions

Experimental Method

• Heating matter with Ultra Relativistic Heavy Ion Collisions. • The first problems to be solved:

-More complex than an incoherent superposition of individual parton-parton (nucleon-nucleon) collisions. -Full Thermal and Chemical Equilibration of the system.

-Nuclear Dynamics under control

• Looking for Probes of the equilibrated and highly excited matter. and the total energy in the volume considered will be :

E = dE dy y=0 ⇥ 2 d ⌧ ,
where dE/dy is the total energy created by the strong interaction For other rapidity domains, the previous expression can be easil total energy by the transverse energy E T . Finally, we can calcu the volume 12 :

✏(y) = dE T dy ⇥ 1 ⇡R 2 ⌧ ,
which links the energy density with the transverse energy produ 1. Formation.

The initial energy density can then be estimated assuming th production of particles, as ⌧ form ⇠ ⌧ strong ⇠ 1 fm/c 13 .

Bjorken estimated the energy density for heavy ion collision SppS collider at CERN, that were p s NN ⇠ 500 GeV per nucleo

Particle Decays I

• Strange Hadrons

• K 0 S →π + π -(494 MeV/c 2 , 2.7cm, 68.6%), • Λ→pπ -(1.
12 GeV/c 2 , 7.9cm, 63.9%),

• Ξ -→Λπ -(1.32 GeV/c 2 , 4.9cm, 99.9%),

• Ω -→ΛΚ -(1.67 GeV/c 2 , 2.5cm, 67.8%),

• Neutral Mesons

• π 0 →γγ (135 MeV/c 2 , 251Å, 98.8%),

• η→γγ (547 MeV/c 2 , 11Å, 39.2%),

• Meson vectors

• ρ→e + e -(770 MeV/c 2 , 8.3fm, 4.5 •10 -5 ),

• ω→e + e -(782 MeV/c 2 , 148fm, 7.1 •10 -5 ), -Charmonium

• φ→K + K -(1.02 GeV/c 2 ,
• J/Ψ→e+e-(3.10 GeV/c 2 , 0.14Å, 6.02%), µ + µ (6.01%),

• Ψ→e+e-(3.69 GeV/c 2 , 0.04Å, 8.5%), µ + µ (7.7%).

-Bottomium

• ϒ→e+e-(9.46 GeV/c 2 , 0.24Å, 2.52%), µ + µ (2.48(0.07)%),

• ϒ'→e+e-(10.0 GeV/c 2 , 0.28Å, 1.18%), µ + µ (1.31(0.20)%),

τ + τ -(1.7(1.6)%), • ϒ''→e+e-(10.4 GeV/c 2 , 0.47Å, seen), µ + µ (1.81(0.17)%). π 0 peak Pb+Pb @ 5.5A TeV Invariant Mass Analysis ρ→e + e - π 0 →γγ J/Ψ→e + e -, µ + µ - Combinatorial Background m ij (γ i γ j ) Physics Background m +-(µ + µ -)
Particles Decays III

• Open Charm :

• D +-→e +-X (1.87 GeV/c 2 , 317µm, 17.2(1.9)%),

» D + → K 0 e + ν e , D + → K -π + e + ν e ,
• D 0 →e + X (1.86 GeV/c 2 , 124µm, 6.8(0.3)%), µ + X(6.6(0.8)%),

K -π + (3.85%) » D 0 → K -e + ν e , D 0 → K -π 0 e + ν e ,
• Open Bottom :

• B +-→l +-ν X (5.28 GeV/c 2 , 495µm, 10.3(0.9)%), J/Ψ X (~0.5%)

» B + → D 0 l + ν l , B + → D * l + ν l ,
• B 0 →l + ν X (5.28 GeV/c 2 , 468µm, 10.5(0.8)%), J/Ψ X(~0.5%),

» B 0 → D -l + ν e , B 0 → D * l + ν e , cτ = 317 µm D + e + Momentum measurement B ~ constant X + B v c e dt p d ! ! ! × ⋅ = p = 0.3* B T [ ] * ρ m [ ] GeV/c
Measurement :

Uncertainty: R P 1 • e -and γ deposit their total energy in the Calorimeter

P 2 P 3 s L 8 2 ≈ ρ s s L L δ δ ρ δρ 2 ⊕ ≈ s s L L p p δ δ δ δ ⊕ ⊕ ≈ 2 B B L s 2 3 . 0 8 2 B B L B s p L L p p * * ⊕ ⊕ ≈ δ δ δ δ p b p p × ⊕ ≈ a δ L=3m, s=10cm ρ=11
• Hadronic calorimeter (may be in the future at mid-rapidity)

• Zero Degree Calorimeters are largely used

• High Multiplicity :

• Small R M ~ 2-5 cm • Distance 4-5 m from IP • Spectrometer

• Sampling Calorimeters:

• cheap (acceptance)

• Lead+Scintillator ] for the formation of hot QCD matter. After a formation time ⌧ form a volume with a high energy density is created. After equilibration at ⌧ ther , the evolution of the hot QCD matter follows the laws of the relativistic hydrodynamics. First, there is a longitudinal expansion until the system reaches a longitudinal size close to its transverse size, then a tridimensional expansion starts until the density is so low that no more inelastic (elastic) collision

Heavy Ion Collisions at RHIC and LHC

Experimental study of the hadronic-matter phase diagram by means of the heavy ion collisions at ultra relativistic energies. source and is less affected by experimental uncertainties, an increase is observed beyond systematic errors (Fig. 3-b). At lower energies a rather flat behavior with a shallow minimum between AGS and SPS energies was observed and interpreted as due to the transition from baryon to meson dominance at freeze-out [44]. An increase of R side at high energy is consistent with that interpretation. Available model predictions are compared to the experimental data in Figs. 2-d and3. Calculations from three models incorporating a hydrodynamic approach, AZHYDRO [45], KRAKOW [46,47], and HKM [48,49], and from the hadronic-kinematics-based model HRM [50,51] are shown. An in-depth discussion is beyond the scope of this Letter but we notice that, while the increase of the radii between RHIC and the LHC is roughly reproduced by all four calculations, only two of them (KRAKOW and HKM) are able to de- [36][37][38]30,43]. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this Letter.) locity gradient of the expanding system. The longitudinal velocity gradient in a high energy nuclear collision decreases with time as 1/τ [52]. Therefore, the magnitude of R long is proportional to the total duration of the longitudinal expansion, i.e. to the decoupling time of the system [31]. Quantitatively, the decoupling time τ f can be obtained by fitting R long with

R 2 long (k T ) = τ 2 f T m T K 2 (m T /T ) K 1 (m T /T ) , m T = m 2 π + k 2 T , (2)
where m π is the pion mass, T the kinetic freeze-out temperature taken to be 0.12 GeV, and K 1 andK Elliptic Flow (v 2 ) at RHIC

• Elliptic flow is important in HIC;

• Well understood at RHIC with Hydrodynamical models;

-sQGP concept has been suggested; The opacity of QGP Colour Screening at RHIC all centralities. Thus, the observed centrality dependence of the v 2 for inclusive J/ψ with 1.5 ≤ p T < 10 GeV/c does not result from any bias in the sampled p T distributions. For J/ψ with p T < 1.5 GeV/c (not shown), the v 2 is compatible with zero within one standard deviation for the four centrality classes. The ⟨p T ⟩ uncor ranges from about 0.75 to 0.9 GeV/c. Recombination models predict larger R AA at low p T . At high pT suppression is very strong. sverse momentum for inclusive J/y production was computed by fitting the p T distribution of inclusive yields with the function ), all three numbers in the p T range (0-0.3) GeV/c, and J/y coherent photoproduction cross section in Pb-Pb collisons at p s NN = 2.76 TeV, with their statistical and uncorrelated systematic uncertainties.

f (p T ) = C ⇥ p T (1 + (p T /p 0 ) 2 ) n , (7
A correlated systematic uncertainty also applies to the cross section. In the most central classes, an upper limit (95% CL) on the J/y yield excess and on the cross section is given.

function of p T in a given centrality class was used:

dN h J/y AA dp T = N ⇥ ds h J/y pp dp T ⇥ R h J/y AA ⇥ (A ⇥ e) h J/y
AA .

(2)

The factor N is fixed by normalizing the integral of Eq. ( 2) in the p T range 1-8 GeV/c to the number of J/y measured in the same range, where the hadroproduction component is dominant. The second term is given by the fit of the J/y p T -differential cross section measured in pp collisions [25] using Eq. ( 1). Table 1: Raw number of J/y (N J/y AA ), expected raw number of hadronic J/y (N h J/y AA ) and measure number of J/y ( N excess J/y AA ), all three numbers in the p T range (0-0.3) GeV/c, and J/y coherent p cross section in Pb-Pb collisons at p s NN = 2.76 TeV, with their statistical and uncorrelated systemat A correlated systematic uncertainty also applies to the cross section. In the most central classes, (95% CL) on the J/y yield excess and on the cross section is given.

dN h J/y AA dp T = N ⇥ ds h J/y pp dp T ⇥ R h J/y AA ⇥ (A ⇥ e) h J/y AA .
The factor N is fixed by normalizing the integral of Eq. ( 2) in the p T range 1-8 GeV/c to ), all three numbers in the p T range (0-0.3) GeV/c, and J/y coherent photoproduction cross section in Pb-Pb collisons at p s NN = 2.76 TeV, with their statistical and uncorrelated systematic uncertainties.

A correlated systematic uncertainty also applies to the cross section. In the most central classes, an upper limit (95% CL) on the J/y yield excess and on the cross section is given.

dN h J/y AA dp T = N ⇥ ds h J/y pp dp T ⇥ R h J/y AA ⇥ (A ⇥ e) h J/y
AA .

(2)

The factor N is fixed by normalizing the integral of Eq. ( 2) in the p T range 1-8 GeV/c to the number of

Correlated systematic 14%

-Luminosity, MC input, trigger and tracking efficiencies, and matching.

Comparison with theoretical expectations

No model is available for a complete description of coherent photoproduction in Pb-Pb nuclear collisions.

-Part of the nuclei interacts.

- full nucleus spectator region

101 E γ = ħw γ N γ photon flux 82 Pb γ source 208 Pb nuclear target J/ψ <p T >~1/R Pb γ dp T dy dY J/ ,i AA dp T dy = N J/ ,i AA (p T bin) BR ⇥ (A ⇥ ✏ i )(p T bin) ⇥ p T ⇥ y ⇥ N i MB AA!J/ AA = X i= A,A i (w ) ⌦ Z R dN (w , b) dS dS
The aim of ultra-relativistic heavy-ion collisions is the study of nuclear matter at high temperature and pressure, where Quantum Chromodynamics (QCD) predicts the existence of a deconfined state of partonic matter, the Quark-Gluon Plasma (QGP). Heavy quarks are ex-5 pected to be produced in the primary partonic scatterings and to interact with this partonic matter, making them ideal probes of the QGP. According to the color screening model [1], quarkonium states are suppressed in the QGP, with di↵erent dissociation probabilities for the various 102 grated and differential cross section for the production of J/ψ in P b+ P b collisions TeV calculated with the help of the second approximation of the photon flux .4)) for the realistic form factor. The differential cross section is compared with imental data [16]. ∆y = 1.5 means y ∈ (2.5,4.0) for brevity. lower limit exceeds somewhat the ALICE data may be due to the fact that the coherent γA → J/ψA cross section was used in our EPA calculation whereas only spectators may be active in the production of J/Ψ. the ALICE data may be due to the fact that the coherent used in our EPA calculation whereas only spectators may be The measurement of the ψ(2S).

Improvement of the precision on ψ(2S).

S/B ratio increases by a factor 3-5 .

MFT half-disk design

• Two detection planes:

-hermetic detector, -coverage around the BP, -redundancy (50%).

• Two PCBs, containing the regulators, data, clock and slow control lines. Material budget per half-disk

• FPC is the main contributor to the material budget (38%), followed by the stiffener (27%) and the disk spacer (17%). • Silicon pixel sensors contributes to 17%.

• Water cooling represents 0.08% of X 0 (15%), with "spikes" of 0.3% of X 0 .

• Material budget per disk below 0.6% of X 0. MFT ladder assembly

• Preparation of the ladder elements:

-common ITS-MFT sensors, FPC, carbon plastic stiffeners, -soldering of SMD components and connector on the FPC.

• HIC Soldering (FPC and sensors):

-common semi-automatic assembly system for ITS inner barrel and MFT at CERN, -visual inspection and electrical tests.

• HIC and stiffener gluing.

• Qualification test.

• Production of ladders:

-an MFT represents 280 ladders: 16, 36,120, 92, 16 ladders of 1 to 5 sensors respectively, -5 half-disk spares and 20% of ladder spares: total of 506 ladders, -duration of ladders production is estimated to 10 months. -ladders, half-disk support, half-disk spacer, 2 PCBs.

• Positioning of the ladders on the half-disk:

-positioning (1/10 mm precision) of ladders on the front and back planes, -gluing on the half-disk spacer, -Electrical test.

• Qualification tests.

• Survey of the sensor positions wrt the half-disk support.

• Production of half-disks:

-MFT represents 10 half-disks + 5 spare half-disks, -duration of half-disk production is estimated to 3 months Full MFT data throughput 57 Gb/s. Frontal absorber cut p>4 GeV/c

Impact parameter resolution

• Resolution below 100 µm for p T >1 GeV/c.

• 10% worsening of the offset by increasing the material budget per disk from 0.6 to 0.8% of X 0 .

132 ALICE -LHCC referees meeting, november 18th 2014 • Central Pb-Pb collisions.

BlackFIG. 9 .

 9 FIG. 7.Critical behaviour for massless quarks and µ B = 0 of the order parameters of the deconfinement (left plot) and of the chiral (right plot) transitions as predicted by lattice QCD calculations. The order parameters are the Polyakov susceptibility ( L ) and the chiral susceptibility ( m ) [Karsch 02a]. Both transitions would indeed be the same one or would take place at the same critical temperature.

•

  Colour charges in the QGP could screen the potential between two heavy quarks: • Only if λ D (Debye Length)<r 0 (size of the quarkonum) screening occurs; • λ D depends on QGP temperature. Dissociation Temperature • ψ' and χ c melt at QGP transition. • J/ψ and ϒ(2S) melt at similar temperatures. • ϒ (1S) needs very hot QGP to be melted. < ⇠ T c ⇠1.2T c ⇠1.2T c ⇠1.3T c ⇠2.0T c per bound of dissociation temperatures T d of quarkonium states in uni perature T c obtained by A. Mocsy and P. Petreczky in [Abreu 08].

?

  of the rapidity distribution, dn/dy, is invariant Dynamical reason: • Plateau of the particle density per unit of rapidity in hadron hadron collisions

  FIG. 10. Bjorken scenario [Bjorken 83] for the formation of hot QCD matter. After a formation time ⌧ form a volume with a high energy density is created. After equilibration at ⌧ ther , the evolution of the hot QCD matter follows the laws of the relativistic hydrodynamics. First, there is a longiBjorken PRD27 140 (1983)

  280fm, 49.1%), →e + e -(3.0 •10 -4 ),

•

  and Drift Chambers are largerly used: 1. Can provide a submilimeter spatial resolution 2. Active volumes of several m 3 . • Position Sensitive Silicon Detectors 1. Close to the interaction vertex 2. An order of magnitud better position Optimized for dN ch /dy = 8000: -R min = 85cm; • Good resolution p T & resolution dE/dx:-R max = 2.5m; • Large acceptance |η|<0.9 -

•

  Heavy quarkonia states, heavy quark production and electroweak bosons in the muon channel; • Charmonia from p T =0 GeV/c; • Separation of Upsilon states; • Acceptance -2.5<η<-4.0 in full ϕ; • 1M read-out channels, fast trigger decision (RPC), 90 m2 of CSC, Dipole of 0.7 T (3Tm);

  FIG. 10. Bjorken scenario [Bjorken 83] for the formation of hot QCD matter. After a formation time ⌧ form a volume with a high energy density is created. After equilibration at ⌧ ther , the evolution of the hot QCD matter follows the laws of the relativistic hydrodynamics. First, there is a longitudinal expansion until the system reaches a longitudinal size close to its transverse size, then a tridimensional expansion starts until the density is so low that no more inelastic (elastic) collision

  FIG. 7.Critical behaviour for massless quarks and µ B = 0 of the order parameters of the deconfinement (left plot) and of the chiral (right plot) transitions as predicted by lattice QCD calculations. The order parameters are the Polyakov susceptibility ( L ) and the chiral susceptibility ( m ) [Karsch 02a]. Both transitions would indeed be the same one or would take place at the same critical temperature.

Fig. 3 .

 3 Fig. 3. Pion HBT radii at k T = 0.3 GeV/c for the 5% most central Pb-Pb at √ s NN = 2.76 TeV (red filled dot) and the radii obtained for central gold and lead collisions at lower energies at the AGS [35], SPS [36-38], and RHIC [39-42,30,43]. Model predictions are shown as lines. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this Letter.)

Fig. 4 .Fig. 5 .

 45 Fig. 4. Product of the three pion HBT radii at k T = 0.3 GeV/c. The ALICE result (red filled dot) is compared to those obtained for central gold and lead collisions at lower energies at the AGS [35], SPS [36-38], and RHIC[39][40][41][42]30,43]. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this Letter.)

Fig. 4 .

 4 Fig. 4. Product of the three pion HBT radii at k T = 0.3 GeV/c. The ALICE result (red filled dot) is compared to those obtained for central gold and lead collisions at lower energies at the AGS [35], SPS [36-38], and RHIC [39-42,30,43]. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this Letter.) ALICE,PLB 396, 328 (2011)

FIG. 14 .

 14 FIG. 14. Invariant cross section (pp) and invariant yield (Au-Au) of direct photons as a function of p T . The three curves on the pp data represent NLO pQCD calculations, and the dashed curves show

  FIG. 18. Left: Nuclear modification factor R AA (p T ) for ⇡ 0 in central (closed circles) and peripheral (open circles) Au-Au at p s NN = 200 GeV. Figure 3 from reference [PHENIX 03b]. Right: R AA of charged particles measured by ALICE in the most central Pb-Pb collisions (0-5%) in comparison to results from CMS and model calculations. Figure 4 from reference [ALICE 12a]

Figure 1 :

 1 Figure 1: Dimuon invariant-mass distributions from the pp (a) and PbPb (b) data at p s NN = 2.76 TeV. The same reconstruction algorithm and analysis criteria are applied to both data sets, including a transverse momentum requirement on single muons of p µ T > 4 GeV/c. The solid lines show the result of the fit described in the text.

Fig. 4 :

 4 Fig. 4: (color online) Inclusive J/ψ v 2 (p T ) for non-central (20%-60%) Pb-Pb collisions at √ s NN = 2.76 TeV (see text for details on uncertainties). The used p T ranges are: 0-2, 2-4, 4-6 and 6-10 GeV/c. Calculations from two transport models [26] and [14] are also shown (see text for details).

Fig

  Fig. 10: (Color online) Inclusive J/y R AA as a function of the J/y rapidity measured in Pb-Pb collisi 2.76 TeV [27], compared to theoretical calculations of CNM effects due to shadowing and/or coherent en 64].

Fig. 6 :ü

 6 Fig. 6: (Color online) Mean transverse momentum hp T i measured by ALICE [37] and PHENIX [21, 52, 56] as a function of the number of participant nucleons (left). r AA measured by NA50 [57], PHENIX and ALICE and compared to model calculations [13, 58], as a function of the number of participant nucleons (right).

Fig. 1 :

 1 Fig.1: (Color online) Raw OS dimuon pT distribution for the invariant mass range 2.8 < m µ + µ < 3.4 GeV/c 2 and centrality class 70-90%. The red line represents the pT distribution of coherently photo-produced J/y as predicted by the STARLIGHT MC generator[21] in Pb-Pb ultra-peripheral collisions and convoluted with the response function of the muon spectrometer. The normalization of the red line is given by the measured number of J/y in excess reported in Table1after correction for the y(2S) feed-down and incoherent contributions (see text).

-

  Photoproduction in Pb-Pb UPC at 2.76 TeV

  99

  first time of an excess in the opposite sign dimuon in peripheral Pb-Pb collisions Phys. Lett. B 718 (2013) 1273 -1283 J/ψ photoproduction (occuring at b < 2R) is proposed as the underlying physics at the origin of the J/ψ low p T yield excess observed in hadronic Pb-Pb collisions 5 ALICE Collaboration, PLB718 (2013) 1273 arXiv:1209.3715 and EPJ73 (2013) 11 arXiv:1305.1467Charmonium and e + e -/dy (µb 0-10 339 ± 85 ± 78 406 ± 14 ± 55 < 251 < 318 10-30 373 ± 87 ± 75 397 ± 10 ± 61 < 237 < 290 30-50 187 ± 37 ± 15 126 ± 4 ± 15 62 ± 37 ± 21 73 ± 44 +26 27

117J-

  /y measured in the same range, where the hadroproduction component is dominant. The second term Measured excess, -Incoherent J/ψ and coherent ψ(2S) contributions, -Centrality selection, occupancy, trigger, tracking.

  Photon flux of a charge distribution of the nucleus for b<2R Pb . -An extrapolation of measured J/ψ in UPC (b 1 =2R Pb ,b 2 =∞) to 70-90% (N part ~11) centrality class (b 1 70-90% ,b 2 70-90% ), provides a cross section of ~40 µb. Good qualitative agreement with the measured value 59±16 µb. -Two extreme cases for the photon flux:

  10 states depending on the temperature of the medium. On the other hand, regeneration models predict charmonium production via the (re)combination of charm quarks during [2-4] or at the end [5, 6] of the deconfined phase. AL-ICE measurements of the J/ nuclear modification factor 15 (R AA ) [7-10] and elliptic flow [11] in Pb-Pb collisions at tions from hadroproduction w 25 ation scenarios. A plausible e cess is caused by coherent ph this process, quasi real photo electromagnetic field of one coherently with all the gluon 30 to produce a J/ . The cohe maximum transverse moment of the order of one over the n duction occurs at very low p T tion is well known in Ultra-P 35 and has been studied also at measurements give insight int the incoming Pb nuclei over x values, providing informat study of J/ hadroproduction Model M.Klusek-Gawenda & A. Szczurek -"Calculation of coherent photoproduction for b<2R A is not clear." -Different ways to treat the overlap region. -Reasonable good agreement with our preliminary results. -Experimental data favour the scenario where only the spectator region contributes to the coherent photoproduction.

  FIG. 9: dσ/dy cross sections for different centrality bins. Theoretical results for different models of the photon flux are compared with the preliminary ALICE data[16]. The shaded area represents the experimental uncertainties.

  . Theoretical results for different models of ith the preliminary ALICE data[16]. The shaded area represents

( 0 )Figure 2 . 32 :(Figure 2

 02322 FIG. 17.Average R AA of D mesons in the 0-20% centrality class compared to modification factors of charged particles and non-prompt J/ from B decays measured

Figure 2

 2 Figure 2.40: Low -p T excess observed in the Pb-Pb at p s NN = 2.76 TeV in the centrality intervals 70-90% (left) and 50-70% (right). The solid line is based on scaling from pp collisions.

Figure

  Figure 2.19, MFT LoI

  Figure 3.10: Example of radiation length fraction distribution for a half-disk.

Figure 3 . 10 :

 310 Figure 3.10: Example of radiation length fraction distribution for a half-disk.

Figure 3 . 11 :

 311 Figure 3.11: Mean contributions to the fraction of X 0 of the di↵erent elements of disk1 along X and Y directions.

  127 ALICE -LHCC referees meeting, november 18th 2014 sensors connector MFT half-disk assembly • Preparation of the half-disk elements:

  128 ALICE -LHCC referees meeting, november 18th 2014 Read-out architecture • Between 128-264 high speed data signals (1.2 Gb/s) per disk. • Between 96-136 clock and slow control signals per disk. • Total of 1456 twinax cables for read-out. • Concentrator board ~ 4 m away, where TID about 10 krad. 129 ALICE -LHCC referees meeting, november 18th 2014 Identical to ITS inner barrel read-out. One single line per sensor. /s lines comply with MFT requirements.

Figure 6

 6 Figure 6.13: Algorithm track finding e ciency vers momentum (right), AliRoot simulations without noise, maton (CA) and Linear Track Finding (LTF). Only disks are considered here, see Figure 6.2 right.

Figure 6 Figure 6 Figure 6 . 15 :

 66615 Figure 6.13: Algorithm track finding e ciency versus transverse momentum (left) and total momentum (right), AliRoot simulations without noise, all multiplicities, with the Cellular Automaton (CA) and Linear Track Finding (LTF). Only tracks having hits in at least four MFT disks are considered here, see Figure 6.2 right.

Figure 6

 6 Figure 6.17: Probability of correct matching between MFT and MUON tracks versus the muon transverse momentum for the track-matching and the cluster-matching methods in 5 % more central Pb-Pb collisions at p s = 5.5 TeV for the entire rapidity range (left) and -for the

ITS Acceptance Radius (cm) Area (m 2 ) Channels

		Calorimeters
	• Secondary vertex resolution < 100µm • Electromagnetic Calorimeters
	(Heavy flavours);
	Silicon Pixel • Reconstruction of particle decays; SPD
	• SPD is important for 2009 run: triggering and measurement of the charged hadron	Pixels
	multiplicity
			Silicon Drift
			SDD
	Silicon Pixel	|η|<1.4	3.9	0.07	3.3M
	(SPD)		7.6	0.14	6.6M
	Silicon Drift (SDD)	|η|<0.9	15.0 23.9 Silicon Strip SSD	0.42 0.89	43K 90K
	Silicon Strip	|η|<0.97	37.8/38.4	2.09	1.1M
	(SSD)		42.8/43.4	2.68	1.5M

Physics goal in 2009: Charged hadron multiplicity distribution at 900 GeV and 10 TeV.

Table 1 :

 1 Raw number of J/y (N

	-5.4 (3.4)σ
	significance in
	70-90% (50-70%)
	centrality bins.
	-1.4σ in 30-50%.

Table 1 after correction for the y(2S) feed-down and incoherent contributions (see text). 2 Evaluation of the J/ψ excess at very low p T Excess in the yield of J/y at very low p T ALICE Collaboration Cent. (%) N 187 ± 37 ± 15 126 ± 4 ± 15 62 ± 37 ± 21 73 ± 44 +26

Table 1 :

 1 Raw number of J/y (N

		J/y AA	N AA h J/y	N AA excess J/y	ds coh J/y /dy (µb)
	0-10 10-30 30-50	339 ± 85 ± 78 406 ± 14 ± 55 373 ± 87 ± 75 397 ± 10 ± 61	< 251 < 237	< 318 < 290

117 J/y measured in the same range, where the hadroproduction component is dominant. Th 118 Excess in the yield of J/y at very low p T ALICE Collaboration Cent. (%) N 187 ± 37 ± 15 126 ± 4 ± 15 62 ± 37 ± 21 73 ± 44 +26 27 ± 10 50-70 89 ± 13 ± 2 39 ± 2 ± 5 50 ± 14 ± 5 58 ± 16 +8 10 ± 8 70-90 59 ± 9 ± 3 8 ± 1 ± 1 51 ± 9 ± 3 59 ± 11 +7 10 ± 8 J/y AA ), expected raw number of hadronic J/y (N h J/y AA ) and measured excess in the number of J/y ( N excess J/y AA

ALICE -LHCC referees meeting, november 18th 2014

ALICE -LHCC referees meeting, november 18th 2014

MFT project

New ITS,high resolution, low-materialthickness Inner Tracking System

Letter of Intent:

An internal Forward Tracker for the ALICE Muon Spectrometer.

The glasses of the ALICE Muon Spectrometer

MFT detector in a few words.

• Precision of the tracking: tens of µm.

• 2000 charged particles per unit of rapidity.

• Challenging integration and matching between muon and the MFT tracks.

MFT Layout

MFT design goals

Study QGP physics at forward rapidity in ALICE

• Vertexing for the ALICE Muon Spectrometer (MS) at forward rapidity:

-5 detection disks of silico n pixel sensors O(25 µm x 25 µm), -0.6% of X 0 per disk, --3.6 < η < -2.45, -Disk#0 at z=-460 mm, R in =25 mm (limited by the beam-pipe radius).

• Good matching efficiency between MFT and MS:

-disk#4 at z=-768 mm (limited by FIT and the frontal absorber).

• Fast electronics read-out:

-Pb-Pb interaction rate ~50 kHz, and pp interactions at 200 kHz.

Reminder of physics cases

Reminder of the MFT LoI

• Open Heavy Flavour:

-charm in the single muon channel down to p T =1 GeV/c, -beauty in the J/ψ channel, down to p T =0.

• Charmonium -separation between prompt and decay J/ψ, -measurement of the ψ(2S).

• Low mass dimuon:

-improvement of the invariant mass resolution, -higher sensitivity to the continuum. Improved performances (J/ψ from B decays)

Good discrimination between prompt and B-decay J/ψ down to p T =0 thanks to the Lorentz boost along the z-axis. t z is weakly dependent on p T for p T <M J/ψ .

119 ALICE -LHCC referees meeting, november 18th 2014 • Rejection of secondary interactions in the absorber.

• Rejection of muons from π and K weak decays.

• Separation of the beauty and charm.

• Charm measurement down to p T =1 GeV of the HFM.

Pixel results

Example of measurement at PS test beam with pALIPIDEfs

Measurements at PS: 5 -6 GeV π-, read-out rates 10-40 kHz Results refer to 50 µm thick chips: non irradiated and irradiated with neutrons (0.25 x 10 13 and 10 13 1MeV n eq / cm2)

123 ALICE -LHCC referees meeting, november 18th 2014

Joint MFT-ITS Silicon Pixel Sensor Strategy

• Requirements of ITS inner-barrel and MFT are almost identical.

• ITS-MFT common sensor benefits:

-minimize sensor cost and manpower resources, -similar flex printed circuit, -same bonding technique (laser soldering), -same read-out architecture.

• ITS sensor(s) will be selected by the ITS project beginning 2015, following an internal review involving also the MFT project.

• ALPIDE sensor fits the requirements of the MFT detector:

-pile-up is reduced due to the shorter integration time. Pile-up deteriorates MS-MFT matching efficiency (see MFT LoI), -Power consumption is below 50 mW/cm 2 , easier integration of the MFT cooling system.

ALICE -LHCC referees meeting, november 18th 2014 This section points out some design choices and their possible impact on the global project cost.

Cost Assessment 81

1646

The costs for R&D are not included in the assessment, neither the personnel cost, nor basic 1647 infrastructures of the participating institutes. A summary of the major tasks responsibilities

Very difficult to conclude when the history is being written.

Many intriguing results from the HIC at the LHC. Digesting run1 (2010-2013) and preparing run2 (2015)(2016)(2017)(2018).

We are willing to continuing this fascinating adventure of ALICE at the LHC, with the ALICE upgrade projects for run3/run4 (2018-2025).


Antoine Moinet

Michele Starnini

Romualdo Pastor-Satorras

Random walks in non-Poissoinan activity driven temporal networks

The interest in non-Markovian dynamics within the complex systems community has recently blossomed, due to a new wealth of time-resolved data pointing out the bursty dynamics of many natural and human interactions, manifested in an inter-event time between consecutive interactions showing a heavy-tailed distribution. In particular, empirical data has shown that the bursty dynamics of temporal networks can have deep consequences on the behavior of the dynamical processes running on top of them. Here, we study the case of random walks, as a paradigm of diffusive processes, unfolding on temporal networks generated by a non-Poissonian activity driven dynamics. We derive analytic expressions for the steady state occupation probability and first passage time distribution in the infinite network size and strong aging limits, showing that the random walk dynamics on non-Markovian networks are fundamentally different from what is observed in Markovian networks. We found a particularly surprising behavior in the limit of diverging average inter-event time, in which the random walker feels the network as homogeneous, even though the activation probability of nodes is heterogeneously distributed. Our results are supported by extensive numerical simulations. We anticipate that our findings may be of interest among the researchers studying non-Markovian dynamics of time-evolving complex topologies.

I. INTRODUCTION

Temporal networks [1,[START_REF] Lambiotte | A Guide to Temporal Networks[END_REF] constitute a recent new description of complex systems, that, moving apart for the classical static paradigm of network science [START_REF] Newman | Networks: An introduction[END_REF], in which nodes and edges do not change in time, consider dynamic connections that can be created, destroyed or rewired at different time scales. Within this framework, a first round of studies proposed temporal network models ruled by homogeneous Markovian dynamics [START_REF] Gauvin | [END_REF]. A prominent example is represented by the activity-driven model [START_REF] Perra | [END_REF], in which nodes are characterized by a different degree of activity, i.e. the constant rate at which an agent sends links to other peers, following a Poissonian process. The memoryless property implied by the Markovian dynamics greatly simplifies the mathematical treatment of these models, regarding both the topological properties of the time-integrated network representation [6], and the description of the dynamical processes unfolding on activity-driven networks [7][8][9][10][11][12].

However, the Markovian assumption in temporal network modeling has been challenged by the increasing availability of time-resolved data on different kinds of interactions, ranging from phone communications [13] and face-to-face interactions [14], to natural phenomena [15,16] and biological processes [17]. These empirical observations have uncovered a rich variety of dynamical properties, in particular that the inter-event times t between two successive interactions (either the creation of the same edge or two consecutive creations of an edge by the same node), ψ(t), follows heavytailed distributions [14,18,19]. This bursty dynamics [18] is a clear signature that the homogeneous temporal process description is inadequate and that non-Markovian dynamics lie at the core of such interactions. As a consequence, the interest in non-Markovian dynamics within the complex systems community has recently blossomed, from the point of view of both mathematical modeling [20][21][22][23][24][25][26] and dynamical processes, especially regarding epidemic spreading [27][28][29][30].

Within the framework of non-Markovian networks modeling, the Non-Poissoinan activity driven (NoPAD) model [20,31] offers a simple, mathematically tractable framework aimed at reproducing empirically observed inter-event time distributions, overcoming the limitations of the classical activitydriven paradigm.

The bursty nature of temporal networks can have a deep impact on dynamical processes running on top of them, ranging form epidemic spreading, percolation, social dynamics or synchronization; see Ref. [START_REF] Porter | Dynamical Systems on Networks: A Tutorial[END_REF] for a bibliographical summary. Among the many dynamical processes studied on temporal networks, the random walk stands as one of the most considered, due to its simplicity and wide range of practical applications [START_REF] Weiss | Aspects and Applications of the Random Walk[END_REF][START_REF] Masuda | [END_REF]. Traditional approaches are based on the concept of continuous time random walks [START_REF] Klafter | First Steps in Random Walks: From Tools to Applications[END_REF], where the random walk is represented as a renewal process [START_REF] Cox | Renewal Theory[END_REF], in which the probability per unit time that the walker exits a given node through an edge is constant. This Poissonian approximation [START_REF] Klafter | First Steps in Random Walks: From Tools to Applications[END_REF], which translates in a waiting time of the walker inside each node with an exponential distribution, permits an analytic approach based on a generalized master equation [START_REF] Weiss | Aspects and Applications of the Random Walk[END_REF]. The Poissonian case has been considered in particular for activity-driven networks [7,10,[START_REF] Sousa Da Mata | [END_REF].

However, if the inter-event time distribution ψ(t) is not exponential, as empirically observed, the waiting time of the random walker shows aging effects, meaning that the time at which the walker will leave one node depends on the exact time at which it arrived at the considered node. Such memory effects are particularly important when the inter-event time distribution lacks a first moment [38]. A way to neglect these aging effects is by considering active random walks, in which the inter-event time of a node is reinitialized when a walker lands on it, in such a way that intervent and waiting time distributions coincide. In opposition, in passive random walks the presence of the walker does not reinitialize the inter-event times of nodes or edges, and thus the waiting time depends on the last activation time [39]. The non-Poissoinan scenario has been considered in the general context of a fixed network in which edges are established according to a given interevent time distribution ψ ij (t) for active walkers [39][40][41] and for passive walkers [39], usually with the assumption of a finite average inter-event time distribution, with the exception of Ref. [41].

In this paper, we contribute to this endeavor with the study of passive random walks on temporal networks characterized by non-Markovian dynamics, by considering the case of networks generated by the NoPAD model. In the NoPAD model, nodes establish connections to randomly chosen neighbors following a heavy-tailed inter-event time distribution ψ c (t) ∼ t -1-α , with 0 < α < 2, depending on an activity parameter c assigned to each node [20,31]. We show that the dynamics of passive random walks on NoPAD networks fundamentally departs from the one observed on classical Poissonian activity-driven networks. For the case α > 1, when the average inter-event time is finite, we observe that the passive random walk behaves in the infinite network limit as an active one with inter-event time distribution ψ c (t) ∼ t -α . For the more interesting case α < 1, we argue that a passive random walk behaves, in the large time limit, as a walker in a homogeneous network. Our results are checked against extensive numerical simulations.

The paper is organized as it follows: In Section II we present the definition of passive random walks on NoPAD networks. Sect. III presents a general formalism for the walker occupation probability and first passage time distribution, that can be further elaborated in Laplace space in the case α > 1, corresponding to an inter-event time distribution with finite first moment. We present the application of this formalism for the standard Poissonian activity driven model in Sec. IV, recovering previously known results. In Sec. V we consider NoPAD networks with finite average inter-event times. The case of infinite average inter-event times is discussed in Sec. VI. Our conclusions are finally presented in Sec. VII.

II. PASSIVE RANDOM WALKS ON NOPAD NETWORKS

In the NoPAD model [20,31], nodes establish instantaneous connections with randomly chosen peers by following a renewal process. Each node is activated independently from the others, with the same functional form of the interevent time distribution ψ c (t) ∼ t -α-1 , with α > 0, between consecutive activation events, which depends on an activity parameter c, heterogeneously distributed among the population with a probability distribution η(c). The dynamics of a random walk on NoPAD networks is defined as follows: A walker arriving at a node i at time t remains on it until an edge is created joining i and another randomly chosen node j at a subsequent time t > t, after a waiting time tt has elapsed. The walker then jumps instantaneously to node j and waits there until an edge departing from j is created at a subsequent time t > t . To simplify calculations, here we will focus on directed random walks: a walker can leave node i only when i becomes active and creates an edge pointing to another node [7,[START_REF] Sousa Da Mata | [END_REF]. We consider the case of a passive ran-dom walk: the internal clock of the host node i is not affected by the walker's arrival, and it must wait there until i creates a new connection. With this definition, a directed random walk on a NoPAD network can be mapped to a continuous time random walk on a fully connected network in which each node has a different distribution of waiting times. We assume that all nodes are synchronized at a time -t a < 0 (i.e., the internal clock of all nodes is set to zero at time -t a or, in other words, we assume the all nodes become active at time -t a ) and that the random walk starts at time t = 0 from a node with activity c, chosen for generality with probability distribution H(c).

For a general inter-event time distribution ψ c (t), it is important to recall that the relevant quantity to describe the motion of a random walker is the waiting time distribution of residence inside each node. If ψ c (t) takes an exponential form, the activation rate is constant, implying that the time to the next activation is independent of the time of the last one. In this case, the waiting time distribution coincides with ψ c (t) and memory effects are absent [START_REF] Cox | Renewal Theory[END_REF]. When ψ c (t) has a non-exponential form, the waiting time distribution is different from ψ c (t) and indeed it takes a non-local form: A walker arriving at a node with activity c at time t, it will jump out of it at the next activation event of this node. Assuming the previous activation event took place at time t p < t, the next one will take place at time t n , where the inter-event time ∆t = t nt p is randomly distributed according to ψ c (∆t). The waiting time of the walker in node with activity c is thus given by τ = t nt and depends explicitly on the immediately previous activation time t p . An exact description of the passive walker will thus require knowledge of the complete trajectory of the walker in the network, and of the whole sequence of activation times of all nodes [39]. This requirement can be however relaxed in the case of NoPAD networks. In the class of activity driven networks, after an activation event, the walker jumps to a randomly chosen node. Thus, in the limit of an infinite network, each node traversed in the path of the walker is essentially visited for the first time. Therefore, the random walker waiting time distribution depends not on the whole walker path, but only on the temporal distance to the synchronization point. In these terms, we consider as the waiting time distribution the forward interevent time distribution [START_REF] Klafter | First Steps in Random Walks: From Tools to Applications[END_REF], h c (t a + t , t), defined as the probability that a walker arriving on a node with activity c at time t (hence at a time t a + t measured from the synchronization point of all nodes in the network) will escape from it, due to the activation of the node, at a time t + t, or, in other words, that it will wait at the node for a time interval t. When the inter-event time distribution is exponential, corresponding to a memory-less Poisson process, one has h c (t a +t , t) ≡ ψ c (t), independent of both the aging time t a and the arrival time t of the walker [START_REF] Cox | Renewal Theory[END_REF]. For general forms of the inter-event time distribution ψ c (t), aging effects take place and the function h c depends explicitly on the arrival time t [38].

III. GENERAL FORMALISM

In this Section we develop a general formalism to compute the steady state occupation probability and the first passage time properties of the passive random walk in infinite NoPAD networks. In the case of an inter-event time distribution with finite first moment, and in the limit of an infinitely aged network (t a → ∞) we can pass to Laplace space to provide closed-form expressions.

A. Occupation probability

We consider here the occupation probability P (c, t|c 0 ), defined as the probability that a walker is at a node with activity c at time t, provided it started at time t = 0 on a node with activity c 0 . To compute it, let us define the probability Φ n (c, t|c 0 ) that a walker starting at c 0 has performed n hops at time t, landing at the last hop, made at time t , with 0 < t < t, at a node with activity c. These two probabilities are trivially related by the expression

P (c, t|c 0 ) = ∞ n=0 Φ n (c, t|c 0 ). (1)
For n = 0 (the node c 0 does not become activated during the whole time t), we have

Φ 0 (c, t|c 0 ) = hc0 (t a , t)δ c,c0 , (2)
where δ c,c is the Kronecker symbol and we have defined

hc (t a + t , t) = ∞ t h c (t a + t , τ ) dτ (3)
as the probability that a walker arriving at time t on a node c has not left it up to time t + t .

To calculate Φ n (c, t|c 0 ) for n ≥ 1 we make use of a selfconsistent condition. Defining Ψ n (t|c 0 ) as the probability that the n-th jump of a walker starting at c 0 takes place exactly at time t, we can write

Φ n (c, t|c 0 ) = t 0 Ψ n (t |c 0 )η(c) hc (t a + t , t -t ) dt . (4)
This equation expresses the sum of the probabilities of the events in which the walker performs its n-th jump at any time t < t, arrives in this jump at a node c, given by the probability η(c), and rests at that node c for a time larger than tt . To compute Ψ n (t|c 0 ) we apply another self-consistent condition, namely

Ψ n (t|c 0 ) = c t 0 Ψ n-1 (t |c 0 )η(c )h c (t a + t , t -t ) dt .
(5) This equation implies the (n -1)-th jump taking place at time t , and landing on a node c , with probability η(c ), and the last jump taking place, from c , at time tt . The expression is averaged over all possible values of the activity c of the intermediate step. The iterative Eq. ( 5), complemented with the initial condition Ψ 1 (t|c 0 ) = hc0 (t a , t), provides a complete solution for the steady state probability, via Eqs. ( 4) and (1).

B. First passage time distribution

We now consider the first passage time probability F (t, c|c 0 ), defined as the probability that a walker starting at a node of activity c 0 arrives for the first time at another node of activity c exactly at time t. To compute it, we define Ψn (t|c, c 0 ) as the probability that the walker performs his nth hop at time t, irrespective of where it lands, in a trajectory that has never visited before a node of activity c. We can thus write 1

F (t, c|c 0 ) = h c0 (t a , t)η(c) + t 0 dt ∞ n=1 Ψn (t |c, c 0 ) c =c η(c )h c (t a + t , t -t )η(c). (6)
In this equation, the first term accounts for the walker arriving at c in a single hop, while for the second term we consider that the walker has performed an arbitrary number of hops n ≥ 1 at a time t , that the last of these hops lands on a node with activity c = c, and from this node the walker performs a final hop, after a waiting time tt that lands it on a node with activity c. The probability Ψn (t|c, c 0 ) can be recovered from 1 We neglect here the case c 0 = c. Its consideration will imply an additional term δ(t)δc 0 ,c in Eq. ( 6), where δ(t) is the Dirac delta function.

the recurrent relation

Ψn (t|c, c 0 ) = t 0 dt Ψn-1 (t |c, c 0 ) c =c η(c )h c (t a +t , t-t ), (7)
which considers the (n -1)-th hop taking place at time t , landing at a node of activity c = c, and performing a last hop after a time tt .

C. Inter-event time distributions with finite average

While the previous formalism is exact for NoPAD networks of infinite size, it cannot be developed further in absence of detailed information about the functional form of the forward inter-event time distribution h c (t , t), which is in general very hard to obtain [START_REF] Klafter | First Steps in Random Walks: From Tools to Applications[END_REF]. Progress is possible, however, when the first moment of the inter-event time distribution ψ c (t), defined as

τc = ∞ 0 u ψ c (u) du, (8)
is finite. When this condition applies, and in the limit of very large aging time t a → ∞, the forward inter-event time distribution does no longer depend on its first argument and it is given by [START_REF] Klafter | First Steps in Random Walks: From Tools to Applications[END_REF][START_REF] Cox | Renewal Theory[END_REF]42]

h c (t) = 1 τc ∞ t ψ c (u)du. (9)
In this double limit of infinite network size and aging time, the passive random walker behaves effectively as an active random walker in which the waiting time distribution is given the forward inter-event time distribution h c (t).

Under assumption of a finite average inter-event time, defining the Laplace transforms

Φ n (c, s|c 0 ) = ∞ 0 Φ n (c, t|c 0 )e -st dt, (10)
Ψ n (s|c 0 ) = ∞ 0 Ψ n (t|c 0 )e -st dt, (11)
h c (s) = ∞ 0 h c (t)e -st , (12)
hc (s) = ∞ 0 hc (t)e -st dt, (13)
we can write Eq. ( 4) in Laplace space as

Φ n (c, s|c 0 ) = η(c)Ψ n (s|c 0 ) hc (s), (14)
while Eq. ( 5) takes the form

Ψ n (s|c 0 ) = c η(c )Ψ n-1 (s|c 0 )h c (s). (15)
Eq. ( 15) can be easily solved, yielding

Ψ n (s|c 0 ) = c η(c )h c (s) n-1 Ψ 1 (s|c 0 ). (16)
Considering that Ψ 1 (t|c 0 ) = h c0 (t), we can combine Eqs. ( 16) and ( 14) to obtain

P (c, s|c 0 ) = ∞ n=0 Φ n (c, s|c 0 ) = hc0 (s)δ c,c0 + η(c) hc (s)h c0 (s) ∞ n=1 c η(c )h c (s) n-1 = hc0 (s)δ c,c0 + η(c) hc (s)h c0 (s) 1 -c η(c ) hc (s) . ( 17
)
From Eq. ( 17) we can obtain the probability P (c, t) of observing the walker at a node c at time t, irrespective of the position c 0 of origin, as

P (c, t) = c0 H(c 0 )P (c, t|c 0 ), (18)
which, from Eq. ( 17), can be written in Laplace space as

P (c, s) = H(c) hc (s) + η(c) hc (s) c0 H(c 0 )h c0 (s) 1 -c η(c )h c (s) . (19)
For the first passage time distribution, Laplace transforming Eq. ( 7), we obtain

Ψn (s|c, c 0 ) = Ψn-1 (s|c, c 0 ) c =c η(c )h c (s). (20)
Since Ψ1 (s|c, c 0 ) = h c0 (s), we have, solving the recursion relation Eq. (20),

Ψn (s|c, c 0 ) = h c0 (s)   c =c η(c )h c (s)   n-1 . (21)
Introducing Eq. ( 21) into the Laplace space counterpart of Eq. ( 6), we finally have

F (s, c|c 0 ) = h c0 (s)η(c) + h c0 (s)η(c) c =c η(c )h c (s) 1 -c =c η(c )h c (s) = h c0 (s)η(c) 1 -c =c η(c )h c (s) (22)
The mean first passage time (MFPT), defined as

T (c|c 0 ) = ∞ 0 t F (t, c|c 0 )dt, (23)
can be obtained, from the Laplace transform in Eq. ( 22), as [START_REF] Klafter | First Steps in Random Walks: From Tools to Applications[END_REF] T

(c|c 0 ) = - dF (s, c|c 0 ) ds s=0 . (24)

IV. POISSONIAN ACTIVITY-DRIVEN NETWORKS

To check the expressions obtained in the previous Section, we start by considering Poissonian AD networks, for which the random walk problem has been already studied [7,[START_REF] Sousa Da Mata | [END_REF]. In this case, the inter-event time distributions for each node takes an exponential form, ψ c (t) = ce -ct , where the value of c is extracted from the distribution η(c). This fact renders the results in Section III exact. The system completely lacks memory, and it holds h c (t , t) ≡ ψ c (t) = ce -ct , while hc (t |t) = e -ct . With the corresponding Laplace transforms h c (s) = c/(c + s) and hc (s) = 1/(c + s), the probability P (c, t) of observing the walker in a node c at time t can be written, from Eq. ( 19), as

P (c, s) = H(c) c + s + η(c) c + s c0 H(c 0 ) c0 c0+s 1 -c η(c ) c c +s . ( 25
)
The steady state occupation probability P ∞ (c) = lim t→∞ P (c, t) can be obtained in Laplace space as the alternative limit P ∞ (c) = lim s→0 sP (c, s), leading to

P ∞ (c) = 1 c -1 η(c) c , (26)
independent of the initial distribution H(c 0 ), where c -1 = c η(c )/c is the average of the inverse activity in the network. Thus, as time increases, the average occupation probability crosses over from the initial distribution at time t = 0 of random walkers, P 0 (c) = H(c), to the steady state occupation probability, P ∞ (c) ∼ η(c)/c, at large times [START_REF] Sousa Da Mata | [END_REF].

The first passage time probability in Laplace space, Eq. ( 22), in the Poissonian case, reads

F (s, c|c 0 ) = c 0 η(c) (c 0 + s) 1 -c =c η(c ) c c +s . (27)
The first derivative of Eq. ( 27) evaluated at s = 0 leads to the MFPT on a node with activity c, when the walker starts from a node with activity c 0 ,

T (c|c 0 ) = 1 η(c) c -1 - η(c) c + 1 c 0 . ( 28
)
The MFPT of nodes with activity c, irrespective of the activity c 0 of the starting node, can be obtained by averaging over the initial position of the walker, T (c) = c0 H(c 0 )T (c|c 0 ). If such position is chosen uniformly at random in the network, H(c 0 ) = η(c 0 ), Eq. ( 28) becomes

T (c) = c -1 η(c) - 1 c + c -1 . (29)
This expression provides a correction to the result in Ref. [7], derived by a pure mean-field calculation. The first term of Eq. ( 29) can be obtained by following the mean field argument in Ref. [7], and it indicates that the MFPT of nodes with activity c will be inversely proportional to the density of nodes in that activity class c, given by η(c). The second term takes into account the probability of not arriving earlier on nodes with activity c, while the third term, which is constant in c, accounts for the escape time from the starting node of the walker. We note that, while the second term is always negligible with respect to the others, the third constant term can be relevant for nodes of small activity c, if the activity distribution η(c) is power law distributed, η(c) ∼ c -γ [START_REF] Perra | [END_REF].

A. Numerical Application

To provide an example application, we consider the simplest case of an AD network with two different activities 1 and < 1, with an activity distribution

η(c) = p δ c, + (1 -p) δ c,1 . (30)
If we assume H(c) = η(c), Eq. ( 25) reduces to

P (c, s) = η(c) s(c + s) c η(c ) c + s -1 . ( 31
)
From here, we can obtain

c η(c ) c + s = (1 -p) + p + s ( + s)(1 + s) , (32)
which leads to

P ( , s) = p(1 + s) s[(1 -p) + p + s] ≡ pt c 1 s -t c p(1 -p)(1 -) 1 t -1 c + s , (33)
with t c = p+(1-p) . This expression in Laplace space can be trivially anti-transformed, yielding the occupation probability

P ( , t) = P ∞ ( ) -t c p(1 -p)(1 -)e -t/tc , (34)
with

P ∞ ( ) = η( )/[ c -1 ] = p/[(1 -p) + p].
The occupation probability relaxes exponentially to the steady state with a time scale t c that can become very small when both p and tend to zero [START_REF] Sousa Da Mata | [END_REF].

For the first passage time distribution, application of Eq. ( 27) leads directly to

F (s, |1) = p p + s , (35)
indicating an exponential distribution in real time F (t, |1) = pe -pt . The MFPT is obtained as

T ( |1) = 1 p , (36)
independent of . On the other hand,

F (s, 1| ) = (1 -p) (1 -p) + s , (37)
leading to F (t, 1| ) = (1p)e -(1-p)t , from which one can obtain the MFPT

T (1| ) = 1 (1 -p) , (38)
diverging in the limits → 0 or p → 1.

V. NON-POISSONIAN ACTIVITY-DRIVEN NETWORKS WITH FINITE AVERAGE INTER-EVENT TIME

We now consider the more interesting case of non-Poissonian Activity-Driven (NoPAD) networks, in which the inter-event time distribution is different from exponential. To fix notation, we will focus in particular in the power law form

ψ c (t) = αc(ct + 1) -(1+α) , (39)
with α > 0 to allow for normalization. Here we will consider the case α > 1 corresponding to finite average inter-event time of value

τc = ∞ 0 u ψ c (u) du = (α -1)c(ct + 1) -α . (40)
In this case, for an infinitely aged network, t a → +∞, the forward recurrence time no longer depends on the aging time and one has, from Eq. ( 9),

h c (t) = 1 τc ∞ t ψ c (u)du = (α -1)c (ct + 1) -α , (41) hc (t) = ∞ t h c (u)du = (1 + ct) 1-α . ( 42
)
In the limit of large t 1, which correspond to s 1 in the Laplace space, and by virtue of the Tauberian theorems [START_REF] Klafter | First Steps in Random Walks: From Tools to Applications[END_REF], we can write, for 1 < α < 2,

ψ c (s) 1 -τc s + Γ 2-α α -1 s c α + o(s α ), (43)
h c (s) 1 -Γ 2-α s c α-1 + o(s α-1 ), (44)
hc (s) Γ 2-α c s c α-2 + o(s α-2 ), (45)
where o(x) denotes a function f (x), such that lim x→0 f (x)

x = 0, and Γ z ≡ Γ(z) is the Gamma function [START_REF] Abramowitz | Handbook of mathematical functions[END_REF]. These expressions, combined with Eq. ( 19), yield

P (c, s) η(c)c -(α-1) s c -(α-1) + o s -1 . ( 46
)
By taking the limit t → ∞, the steady state occupation probability finally reads In order to confirm this result, we have performed numerical simulations of the passive directed random walk on an NoPAD network with an inter-event time distribution given by Eq. ( 39) and power-law distributed activity η(c) ∝ c -γ , where the activity takes values in the interval [ , 1]. In Fig. 1 we show the theoretical prediction Eq. (47) (dashed lines), obtained in the limit of infinite network size, compared with direct simulation in finite networks of size N = 10 5 for different values of the inter-event time exponent α > 1 (hollow symbols). Simulations are performed in the limit of infinite aging time, t a → ∞. The first activation of every node takes thus place at a time, measured from the beginning of the random walk t = 0, given by the distribution in Eq. (41). We keep track of the whole history of the network, each successive activation of each node taking place at inter-event times given by Eq. (39). Walks are stopped at time time t = 10 6 , where the occupation probability is computed. As we can see, even for the moderate network size considered here, the infinite network approximation provides an excellent approximation for the steady state distribution. Interestingly, when taking the limit α → 2 in Eq. (47), we recover the result established for Poissonian AD networks, i.e. P ∞ (c) = η(c)c -1 / c -1 . This result is general for any α ≥ 2, as can be seen from the corresponding leading order expansions in Laplace space for this range of α values, namely,

P ∞ (c) = η(c)c -(α-1) c -(α-1) . ( 47
)
ψ c (s) 1 -τc s + 1 2 τ 2 c s 2 , (48)
h c (s) 1 - 1 c(α -2) s, (49)
hc (s) 1 c(α -2) , (50)
which, substituted on Eq. ( 19), lead again to Eq. ( 26) in the steady state.

For the first passage time distribution, Eq. ( 22) may be expanded in the limit s 1. Inserting the expansion of the with inter-event time distribution given by Eq. ( 39), for different values of the exponent α. We consider a bi-valued activity distribution with c = (with η( ) = p) or c = 1 (with η(1) = 1 -p). We plot the first passage time distribution to nodes with c = , with the walker starting from nodes with c0 = 1 (hollow symbols), along with the expected behavior F (t, c|c0) ∼ t -α (dashed lines). Number of walkers W = 10 6 , p = 0.5, = 0.1. Network size N = 10 5 .

forward inter-event time distribution Eq. ( 43), we obtain

F (s, c|c 0 ) 1-   c 1-α 0 + 1 η(c) c =c η(c )c 1-α   Γ 2-α s α-1 .
(51) In the time domain, this translates into a power-law behavior at large times, F (t, c|c 0 ) ∼ t -α . This distribution lacks a first moment in the regime 1 < α < 2, implying that the MFPT is infinite. In Fig. 2 we perform numerical simulations to evaluate the first passage time probability when the interevent time distribution is power-law with 1 < α < 2 and the activity is bi-valued, with η(c) of the form Eq. (30). For both values of α considered, one can observe a power-law decay in the actual random walks (performed as described for Fig. 1), corresponding to the expected behavior in the infinite network limit F (t, c|c 0 ) ∼ t -α . At the mean-field level, the average time to reach a node with a given activity c is equal to the average number of independent trials required to land on a node with activity c (equal to 1/η(c)), times the average waiting time spent on a node. Therefore, this time trivially diverges when the average waiting time is infinite, as indicated here by a first passage time distribution lacking the first moment.

VI. NON-POISSONIAN ACTIVITY-DRIVEN NETWORKS WITH INFINITE AVERAGE INTER-EVENT TIME

We consider now an inter-event time distribution of the form Eq. (39) with 0 < α < 1, which implies that the average time between two consecutive activation of an agent with activity c is infinite. For such values of α, the dependency of the forward waiting time distribution on the aging time cannot be eliminated even in the limit of strongly aged networks, so that the use of the Laplace transform does not yield any substantial simplification. Nevertheless, some insight may be obtained concerning the dynamics of the random walk starting on a strongly aged network.

Let us recall the expression of the double Laplace transform of the forward waiting time distribution, namely [START_REF] Klafter | First Steps in Random Walks: From Tools to Applications[END_REF]38,[START_REF] Barkai | [END_REF]45],

h c (u, s) = ∞ 0 dt ∞ 0 dt h c (t , t)e -ut e -st = 1 1 -ψ c (u) ψ c (u) -ψ c (s) s -u . ( 52
)
Let us first consider the limit of strongly aged network and very large t, with ct ct a 1, corresponding to s u 1. In this case, one can expand

h c (u, s) u α -s α u 1+α , (53)
which, upon inverse transformation, leads to

h c (t a , t) t α a sin(πα) π t -α-1 . ( 54
)
On the other hand, in the limit of strong aging, ct a 1, but small t t a , one can expand ψ c (u) ∼ 1 + αΓ -α u α c -α in Eq. ( 52) to obtain

h c (u, s) - 1 s - 1 -ψ c (s) sαΓ -α u α c -α , (55)
which, disregarding a constant term, leads to

h c (t a , t) c (c t a ) α-1 sin(πα) π ψc (t), (56)
where ψc (t) = ∞ t ψ c (t ) dt = (ct + 1) -α . The behavior of ψc (t) can be approximated to ψc (t) 1 if ct 1, while for ct 1, it holds ψc (t) (ct) -α . The behavior of h c (t a , t) can thus be summarized in the following three regimes:

h c (t a , t)      sin(πα) π c (c t a ) α-1 for c t 1 sin(πα) π t α-1 a t -α for 1 c t c t a sin(πα) π t α a t -α-1 for t t a .
(57) Interestingly, at large times, i.e. ct 1, the forward waiting time distribution is independent of c. Besides, the tail of the distribution is proportional to t α a t -α-1 , so that the probability that the forward waiting time is greater than t a is constant and does not depend on t a . This means that the interval [t a , +∞[ carries a constant probability weight with respect to the other two terms, although its size decreases when t a grows. This, along with the fact that the total weight is constant and equal to 1 because h c is normalized, implies that the weight carried in a time window [0, t 0 ] tends to zero when t a tends to infinity. In fact, one could argue that the weights calculated from Eq. ( 57) are not exact because they neglect higher order corrections (in particular the distribution in Eq. ( 57) is not normalized). The reasoning is thus true under the implicit assumption that the weights calculated from Eq. ( 57) and carried in the intervals [0, t 0 ] and [t 0 , +∞[ are proportional to their corresponding real weights, which is not guaranteed.

In order to check these assumptions, on Fig. 3 we compare the ratio of the real weights R(t a , t 0 ) ≡ hc (t a , 0) -hc (t a , t 0 ) / hc (t a , t 0 ), evaluated from a numerical simulation of a renewal process with an inter-event time distribution of the form Eq. ( 39), and the ratio evaluated from Eq. (57), whose dominant order, with the conditions 1 ct 0 ct a , is equal to α(t 0 /t a ) 1-α . We observe a good agreement between the simulations and the analytical estimation, which allows us to make the following reasoning: let us consider a NoPAD network with inter-event time distribution given by Eq. ( 39) with α < 1, and an arbitrary activity distribution η(c) excluding zero-valued activities. Then there exists a node with a minimum activity c min > 0, and also a time t 0 , such that c min t 0 1. Then if the nodes are synchronized at t = -t a with t a t 0 and we start an activated random walk dynamics at time t = 0, the probability that the time t 1 at which the walkers escape from their first hosts is greater than t 0 is almost equal to 1. This holds a fortiori for all the following waiting times of the walker occurring at times t = t 2 , t 3 , ..., t k because t k is extracted from the distribution h c (t a + t k-1 , τ ). Besides, the conditional probability that t 1 = τ given that t 1 ≥ t 0 is independent of c as we see from Eq. (57), which means that all the hops for all the walkers are performed with waiting times that practically do not depend on the activity of the hosts.

As a result, after its first jump, the probability that a walker is at a node of activity c is constant and equal to η(c). In other words, if the initial distribution of the walkers is H(c), the probability P (c, t) that the walker is at a node with activity c at time t is equal to η(c) if the walker has escaped from its first host and H(c) otherwise, i.e.

P (c, t) H(c) hc (t a , t) + η(c)(1 -hc (t a , t)). (58)
In the limit of infinite t, hc (t a , t) vanishes, and the steady state of the walker is given by P ∞ (c) = η(c). That is, in the large time regime, the walker behaves as in a completely homogeneous network, in which jumps were performed independently of the node activity. This result generalizes the observation made in Ref. [39]. In order to check the validity of the time dependence expressed in Eq. ( 58 , along with their expected value hc (t a , t). This last curve is evaluated from an independent numerical simulation of a renewal process, and is found to be independent of the activity c. We observe that the result stated in Eq. (58) perfectly match the numerical simulations in networks of finite size.

VII. CONCLUSIONS

In this paper we have explored the behavior of a passive node-centric random walk unfolding on non-Markovian temporal networks generated by the NoPAD model, which considers a power-law form ψ c (t) ∼ (ct + 1) -1-α of the interevent time distribution between consecutive activation events of nodes with activity c. We have focused in particular on the behavior of the occupation probability and first passage time distribution, in the case of a very large aging time t a , that is, when the time elapsed between the initial synchronization of all nodes in the network and the start of the random walker is very large. The nature of the NoPAD model allows to simplify calculations in the limit of infinite network size, in which every node in the path of the walker is visited for the first time. In this approximation, we develop a general theory for the walker dynamics, that can be analytically solved in Laplace space if the inter-event time distribution of the nodes has a finite first moment. In this case, in the limit t a → ∞, the waiting time of the walker inside a node becomes independent of its arrival time, and a passive random walk with inter-event time distribution φ c (t) ∼ t -1-α , with α > 1, behaves essentially as a active random walk with φ c (t) ∼ t -α , in which the internal clock of each node is reset after the lading of the walker. Numerical simulations show that the actual passive random walk process is very well described by our theory for a sufficiently large network size.

If the inter-event time distribution lacks a first moment, which happens in the case α < 1, our theory is not valid, since the waiting time inside a node cannot be decoupled from the landing time. In the limit of very large t a , however, we develop arguments hinting that the random walker will "feel" a network with homogeneous activity distribution, which im-plies that the probability that the walker is at a node of activity c is equal to η(c) in the large time limit. This result is straightforwardly extended to arbitrary aging times t a (including nonaged networks t a = 0) because after a transient regime of duration t , the forward waiting time distribution h c (t a + t , τ ) will meet the conditions expressed in Eq. (57), and the system will be in the same situation as before, i.e. evolving as if the network was homogeneous. This observation generalizes the results in Ref. [39] referred to networks with identical inter-event time for all nodes. Interestingly, this result is also recovered taking the limit α → 1 in the equation describing the occupation probability in the case of an inter-event time distribution with finite first moment, a fact that provides additional evidence for its relevance.

Figure 1 .

 1 Figure 1. Steady state occupation probability of a directed passive random walk on an infinitely aged NoPAD network, P∞(c), with inter-event time distribution given by Eq. (39), for different values of the exponent α. We consider an activity distribution η(c) ∼ c -γ with = 10 -2 and exponent (a) γ = 1.80 and (b) γ = 2.50. Dashed lines correspond to the analytical result of Eq. (47). The steady state distribution is computed from W = 10 6 different walks, stopped at time t = 10 6 . Network size N = 10 5.

Figure 2 .

 2 Figure2. Distribution of the first passage time for a directed passive random walk on an infinitely aged NoPAD network, F (t, c|c0), with inter-event time distribution given by Eq. (39), for different values of the exponent α. We consider a bi-valued activity distribution with c = (with η( ) = p) or c = 1 (with η(1) = 1 -p). We plot the first passage time distribution to nodes with c = , with the walker starting from nodes with c0 = 1 (hollow symbols), along with the expected behavior F (t, c|c0) ∼ t -α (dashed lines). Number of walkers W = 10 6 , p = 0.5, = 0.1. Network size N = 10 5 .

Figure 3 .

 3 Figure 3. Ratios R(ta, t0) = [ hc(ta, 0) -hc(ta, t0)]/ hc(ta, t0) as a function of ta, obtained from numerical simulations of a renewal process with an inter-event time distribution of the form Eq. (39) with different values of α. The asymptotes α(t0/ta) 1-α calculated from Eq. (57) are plotted as dashed lines. Reference time t0 = 10 3 and c = 1.

Figure 4 .

 4 Figure 4. Random walk dynamics on an aged NoPAD network with three-valued activity and infinite average waiting time. Activity values are c = 0.1, c = 1 and c = 10, with η(c) = 1/3. Initial distribution H(c) = δc,10. We plot for the three values of c the reduced occupation probability Pr(c, t) = (P (c, t)-η(c))/(H(c)η(c)) as a function of time. (a): α = 0.2 and ta = 10 3 . (b): α = 0.5 and ta = 10 5 . The behavior predicted by Eq. (58) is plotted in dashed lines. Network size N = 10 5 and number of walkers W = 10 6 .

1 / 3 .

 13 ), we have performed numerical simulations of the activated random walk on a NoPAD network of size N = 10 5 where the activity takes three values, c = 0.1, 1 or 10, each with probability η(c) = Walkers are initially hosted by nodes with activity equal to c = 10 , i.e. H(c) = δ c,10 . Fig. 4 shows the reduced occupation probability P r (c, t) = [P (c, t)η(c)]/[H(c)η(c)] as a function of the time t for α = 0.2 and t a = 10 3 , Fig. 4(a), and for α = 0.5 and t a = 10 5 , Fig. 4(b)

ACKNOWLEDGMENTS

This work was supported by the Spanish Government's MINECO, under project FIS2016-76830-C2-1-P. R.P.-S. acknowledges additional funding by ICREA Academia, funded by the Generalitat de Catalunya regional authorities.