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Microcanonical, canonical and grand canonical pains with the Hagedorn spectrum Luciano G. Moretto, L. Ferroni, J. B. Elliott, L. Phair UCB and LBNL Berkeley

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Can a “thermostat” have a temperature other than its own? Is T 0 just a “parameter”? According to this, a thermostat, can have any temperature lower than its own! T = T c = 273K or 0 ≤ T ≤ 273K ?

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Summary The true canonical temperature The true microcanonical temperature (are they the same?) The grand canonical mass spectrum The gas of bags Fluctuations and criticality. Is there a critical point? Do bags have surface energy?

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The (too) many ways of obtaining the Hagedorn spectrum ( given the experimental evidence!!) 1.Bootstrap 1.Mit Bag 1.Regge Trajectories 1.Fractal shapes ( if no surface energy)

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The partonic world (Q.G.P.) (a world without surface?) The M.I.T. bag model says the pressure of a Q.G.P. bag is constant: ; g: # degrees of freedom, constant p = B, constant. The enthalpy density is then which leads to an entropy of and a bag mass/energy spectrum (level density) of. This is a Hagedorn spectrum: Partonic vacuum Hadronic vacuum ? m m0m0 ln (m)

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Bag equilibrium with Fermionic obligatory constituents with For P 0 = B the bag is stabilized by the quantum pressure and T H = 0. n T

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Fractal Shapes Fisher Model g(A) lng(A,s)-C s s/T Lng(A,s) s If C s =0 S = KA ρ(M) = exp KA

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For 3d animals

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Regge Trajectories M2M2 I M 2 = kI M 2 = kn h In how many ways can one write Euler’s Partitio Numerorum

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Can a “thermostat” have a temperature other than its own? Is T 0 just a “parameter”? According to this, a thermostat, can have any temperature lower than its own! T = T c = 273K or 0 ≤ T ≤ 273K ?

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Thermal equilibrium System ASystem B No thermostat: any temperature One thermostat: one temperature Two thermostats: no temperature S E S E S E S E S E S E

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The partonic world (Q.G.P.) a constant temperature world The M.I.T. bag model : The Hagedorn spectrum: Implications?The Hagedorn Bag H(m)H(m)

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Equilibrium with Hagedorn bags: Example #1: the one dimensional harmonic oscillator BEWARE! A linear dependence of S on E spells danger to the unaware! Beware of the canonical and grand canonical ensembles! When in doubt, Go MICRO-canonical ! For a one dimensional harmonic oscillator with energy in contact with a Hagedorn bag of energy E: The probability P( ) is: E ln P The most probable value of : For E : T H

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The resistible troubles with microcanonicity Harmonic Oscillator ( 1 dim.) ctd The microcanonical partition : is this a Temperature ? E 1.0 T/T H

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Let us look at the spectra ln P(ε) ε E1 E2E3 Slope= 1/T H ln P(ε) ε Slope= 1/T H Identical results for an ideal gas Conclusion: Beware of microcanonical temperatures

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Microcanonical quirks for a gas of bags Microcanonical Partition m= M 0 - ε ε

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The total level density: Most probable energy partition: T H is the sole temperature characterizing the system: A Hagedorn-like system is a perfect thermostat. If particles are generated by the Hagedorn bag, their concentration is: Volume independent! Saturation! Just as for ordinary water, but with only one possible temperature, T H ! Equilibrium with Hagedorn bags: Example #2: an ideal vapor of N particles of mass m and energy

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The radiant Hagedorn bag (initial energy E 0, initial radius R 0 ) At coexistence: flux in in = flux out out = particles : Energy flux E : With no containing volume the Hagedorn bag radiates itself away. Upper limit for lifetime of Hagedorn bag: Similarities with compound nucleus: Same spectra and branching ratios Differences with compound nucleus: All quantities calculated at fixed T H

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Saturated vapor with free mass If we allow the mass of the particle to be free, its most probable value is:

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1.Anything in contact with a Hagedorn bag acquires the temperature T H of the Hagedorn bag. 2.If particles (e.g. s) can be created from a Hagedorn bag, they will form a saturated vapor at fixed temperature T H. 3.If different particles (i.e. particles of different mass m) are created they will be in chemical equilibrium. H(E)H(E) The story so far...

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Stability of the Hagedorn bag against fragmentation If no translational or positional entropy, then the Hagedorn bag is indifferent to fragmentation. H(m)H(m) H(mk)H(mk) H(m3)H(m3) H(m2)H(m2) H(m4)H(m4) H(m5)H(m5) H(m5)H(m5) H(m1)H(m1) H(m6)H(m6) indifferent

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Resonance gas - A gas without pressure No intrinsic energy and/or entropy penalty for aggregation How many particles? 1 ≤ N ≤ N max Ideal gas law:

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Resonance Gas Cont’d

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% diss. n n n -5/2 1.0 p almost constant ! n -3/2

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The evaporating bag The decoupling between the vapor concentration and m and T H occurs when the bag has evaporated completely:. The disappearance of the bag allows the vapor concentration to decrease as: As the volume increases further While the temperature remains constant:.

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The total level density: Most probable energy partition: T H is the sole temperature characterizing the system: A Hagedorn-like system is a perfect thermostat. If particles are generated by the Hagedorn bag, their concentration is: Volume independent! Saturation! Just as for ordinary water, but with only one possible temperature, T H ! Equilibrium with Hagedorn bags: Example #2: an ideal vapor of N particles of mass m and energy

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T < T H T = T H Non saturated gas of π etc. Gas of bags + saturated gas of π etc. One big bag

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T THTH

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The EoS for the Gas of Hagedorns For the mass spectrum Is the most probable concentration of H with mass m Energy conservation defines the maximal mass of Hagedorns m max and fixes the number of Hagedorns N tot and pressure P of the system Maximal number of Hagedorns

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The EoS for the Gas of Hagedorns Energy conservation defines the maximal mass of Hagedorns m max And gives EoS as P(E/V) : for M = 15.9 GeV to fit N tot of all resonances with masses < 1.85 GeV For plots E = 2 GeV Is speed of sound square

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A bag with a surface? Remember the leptodermous expansion: Notice that in most liquids a S ≈ -a V However, in the MIT bag there is only a volume term Should we introduce a surface term? Although we may not know the magnitude of as, we know the sign (+). The consequences of a surface term: V T TcTc V 0 C p=0 V VV

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Stability of a gas of bags Bags of different size are of different temperature. If the bags can fuse or fission, the lowest temperature solution at constant energy is a single bag. The isothermal solution of many equal bags is clearly unstable. A gas of bags is always thermodynamically unstable. A bag decays in vacuum by radiating (e.g. pions). As the bag gets smaller, it becomes HOTTER! Like a mini-black hole. The decay of a bag with surface

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Is the surface energy temperature dependent? From the bag stability condition: TcTc For V very small T T c TV T TcTc T indep T dep

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The perfect “Granulator” TcTc THTH

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Phase transitions from Hadronic to Partonic Worlds L. G. Moretto, K. A. Bugaev, J. B. Elliott and L. Phair Lawrence Berkeley National Laboratory Nuclear Science Division Phase transitions in the Hadronic world Pairing (superconductive) Transition finite size effects: correlation length Shape transition all finite size effects, shell effects Liquid-vapor (with reservations) van der Waals-like finite size effects due to surface T c ≈ 18.1 MeV c ≈ 0.53 0 p c ≈ 0.41 MeV/fm 3 Phase transitions in the partonic world Q. G. P.... Finite size effects?

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Fractal Shapes Fisher Model g(A) lng(A,s)-C s s/T Lng(A,s) s If C s =0 S = KA ρ(M) = exp KA

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For 3d animals

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Conclusion 1.Any system with a Hagedorn-like spectrum is a perfect thermostat. 2.If such a system evaporated non-Hagedorn particles they constitute a saturated vapor in physical and chemical equilibrium. 3.A vapor of Hagedorn bags is indifferent to aggregation/fragmentation P=0. 4.Surface energy further destabilizes the vapor. Smaller bags are at higher temperature.

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Conclusions Nuclear dropletsIsing lattices Surface is simplest correction for finite size effects (Rayleigh and Clapeyron) Complement accounts for finite size scaling of droplet For ground state droplets with A 0 <

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Origin of the bag pressure To make room for a bubble of volume V an energy E = BV is necessary. To stabilize the bubble, the internal vapor pressure p(T) must be equal to the external pressure B. Notice that the surface energy coefficient in this example is not obviously related to the volume energy coefficient. 10 m B = 1 atm

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Firm and fleeting features of phase transitions Usually very firm: Change in aggregation state: Solid - Liquid - Vapor Change in symmetry: Different crystalline structure Appearance of new features: Superconductivity - Superfluidity Evidence for hadronic to partonic (QGP) transition is (so far) not as firm as the above examples. Perhaps there are features so prominent that they have escaped the detailed inspection of experimental data. sulfur

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Origin of the bag pressure To make room for a bubble of volume V an energy E = BV is necessary. To stabilize the bubble, the internal vapor pressure p(T) must be equal to the external pressure B. Notice that the surface energy coefficient in this example is not obviously related to the volume energy coefficient. 10 m B = 1 atm

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Tantalizing telltales Chemical equilibrium at fixed temperature Physical equilibrium at fixed temperature This is observed in soft emission and in the center of mass of jets. Equilibria are a necessary condition… Is the constant temperature a signal? Usually not: phase coexistences are univariant (one one variable) Thus an entire range of temperatures Examples of univariant equilibirum: Liquid-vapor Solid-liquid Black body - its radiation However: fixing pressure leads to a unique temperature.

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