Equation of State and Phase Transitions in the Nuclear
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National Academy of Sciences of Ukraine Bogolyubov Institute for Theoretical Physics Has the rights of a manuscript Bugaev Kyrill Alekseevich UDC: 532.51; 533.77; 539.125/126; 544.586.6 Equation of State and Phase Transitions in the Nuclear and Hadronic Systems Speciality 01.04.02 - theoretical physics DISSERTATION to receive a scientiﬁc degree of the Doctor of Science in physics and mathematics arXiv:1012.3400v1 [nucl-th] 15 Dec 2010 Kiev - 2009 2 Abstract An investigation of strongly interacting matter equation of state remains one of the major tasks of modern high energy nuclear physics for almost a quarter of century. The present work is my doctor of science thesis which contains my contribution (42 works) to this ﬁeld made between 1993 and 2008. Inhere I mainly discuss the common physical and mathematical features of several exactly solvable statistical models which describe the nuclear liquid-gas phase transition and the deconﬁnement phase transition. Luckily, in some cases it was possible to rigorously extend the solutions found in thermodynamic limit to ﬁnite volumes and to formulate the ﬁnite volume analogs of phases directly from the grand canonical partition. It turns out that ﬁnite volume (surface) of a system generates also the temporal constraints, i.e. the ﬁnite formation/decay time of possible states in this ﬁnite system. Among other results I would like to mention the calculation of upper and lower bounds for the surface entropy of physical clusters within the Hills and Dales model; evaluation of the second virial coeﬃcient which accounts for the Lorentz contraction of the hard core repulsing potential between hadrons; inclusion of large width of heavy quark-gluon bags into statistical description. I believe that the suggested mathematical solution of the freeze-out problem in relativistic hydrodynamic model and in hydro-cascade model has not only an academic interest, but also has some practical value. In addition I hope that the experience gained in working out some partly successful signals of deconﬁnement transition can be useful for other researchers to go further in this direction. 3 CONTENTS Abbreviations 7 Introduction 9 1 Nuclear Matter in Thermodynamic Limit 30 1.1. A Self-consistent Mean-Field EOS for Nuclear Matter . ............... 33 1.2. Statistical Multifragmentation Model in ThermodynamicLimit............ 39 1.3. Singularities of Isobaric Partition and the Mechanism of Phase Transitions . 43 1.4. The Critical Indices and Scaling Relations of the SMM . .............. 52 1.5. MajorConclusions ............................... ..... 57 2 Exactly Solvable Statistical Models for Finite Nuclear Systems 60 2.1. The Complement Method for Finite Drop . ........ 63 2.2. Constrained SMM in Finite Volumes . ........ 71 2.2.1. Isobaric Partition Singularities at Finite Volumes................ 73 2.2.2. NoPhaseTransitionCase. ..... 75 2.2.3. Finite Volume Analogs of Phases. ....... 78 2.3. GasofBagsinFiniteVolumes . ....... 81 2.4. Hills and Dales Model and Source of Surface Entropy . ............. 84 2.4.1. GrandCanonicalSurfacePartition. ......... 85 2.4.2. Special Ensembles for Surface Partition. ........... 88 2.4.3. The Lower Bounds for the Surface Entropy Coeﬃcients. ........... 92 2.5. Conclusions .................................... .... 96 3 Virial Expansion for the Lorentz Contracted Rigid Spheres and Relativistic VdW EOS for Hadronic Binary Mixtures. 99 4 3.1. The Van der Waals EOS for the Lorentz Contracted Hard Spheres. .. .102 3.2. Hadronic Binary Mixtures in Canonical Ensemble . .............109 3.2.1. The Van der Waals Excluded Volume Model. .......111 3.2.2. Generalization to the Two-component Case. ..........113 3.2.3. Comparison of Two-component VdW Approximations. ..........116 3.3. Grand Canonical Treatment of Binary Mixtures . ............118 3.3.1. The Two VdW Approximations. .119 3.3.2. Relativistic Excluded Volumes. ........122 3.3.3. Hard-core Radii from Particle Yield Ratios. ...........126 3.3.4. Intermediate Conclusions. .......131 3.4. RelativizationoftheVdWEOS . .......133 3.4.1. Proof of Causality in the High Pressure Limit. ...........138 3.4.2. A Few Remarks on Possible Observables. ........143 3.5. Conclusions .................................... .144 4 Exactly Solvable Phenomenological EOS of the Deconﬁnement PT 147 4.1. Microcanonical Analysis of the Hagedorn Model . .............149 4.1.1. Harmonic Oscillator Coupled to . .......................154 H 4.1.2. An Ideal Vapor Coupled to ...........................154 H 4.1.3. asaRadiantBag. ...............................156 H 4.1.4. Fragmentation of . ...............................158 H 4.2. HagedornThermostatModel. .......159 4.2.1. A Single Hagedorn Thermostat Case. .......160 4.2.2. TheRoleoftheMassCut-oﬀ. .162 4.2.3. Concluding Remarks on the HTM. .165 4.3. Mott-HagedornModelforQGP . .. .167 4.3.1. Resonance Width Model: Mott Transition. .........169 4.3.2. Applications to Lattice Quantum Chromodynamics and Heavy-Ion Collisions.172 4.3.3. Anomalous J/ψ Suppression............................174 5 4.4. QuarkGluonBagswithSurfaceTension . .........178 4.4.1. TheRoleofSurfaceTension.. .. .181 4.4.2. Generalization to Non-Zero Baryonic Densities. .............187 4.4.3. Surface Tension Induced Phase Transition. ...........190 4.5. Conclusions and Perspectives . .........194 5 Freeze-out Problem of Relativistic Hydrodynamics and Hydrokinetics 200 5.1. Freeze-out Problem in Relativistic Hydrodynamics . ................206 5.1.1. Decay of Perfect Fluid into the Gas of Free Particles. .............206 5.1.2. Conservation Laws at the Freeze-out Hypersurface. .............212 5.1.3. ConsistencyTheorem. .217 5.2. Equations for the Freeze-out Hypersurface . .............220 5.2.1. Freeze-out Calculus: General Scheme. ..........220 5.2.2. Freeze-out in 1+1 Dimensional Hydrodynamics. 224 5.2.3. Entropy Production in the Freeze-out Shock. ...........228 5.2.4. Freeze-out of the Simple Wave. .. .231 5.3. Relativistic Kinetic Equations for Finite Domains . ................241 5.3.1. Drift Term for Semi-Inﬁnite Domain. ........244 5.3.2. Collision Term for Semi-Inﬁnite Domain. ..........247 5.4. Derivation of Conservation Laws for Hydrokinetics . ...............250 5.4.1. Boundary Conditions at Σ∗ foraSingleDegreeofFreedom. 254 5.4.2. Boundary Conditions at Σ∗ forManyDegreesofFreedom. 258 5.5. ConcludingRemarks .............................. .262 6 Experimental Signals of the Deconﬁnement Transition 267 6.1. Statistical Production of Particles. ..............269 6.1.1. Particle Spectra and Apparent Temperature. ...........271 6.1.2. Freeze-out of J/ψ Meson. ............................273 6.1.3. Ω, J/ψ and ψ′ Early Hadronization at SPS Energies. 275 6 6.2. Hadron Spectra and QGP Hadronization in Au+Au CollisionsatRHIC . .281 6.3. Transverse Caloric Curves of Kaons as the Signal of Deconﬁnement Transition in A+ACollisions ...................................... 288 6.4. ConcludingRemarks .............................. .293 Conclusions 295 Appendices 301 AppendixA.......................................... 301 The Canonical Treatment of the Relativistic VdW EOS. .........301 Stability of the Non-linear Approximation. ..........304 AppendixB.......................................... 305 Derivation of the Lorentz Contracted Excluded Volumes . ...........305 The List of Used Sources 311 7 Abbreviations 1. PT phase transition 2. A+A nuclear heavy-ion (collision; reaction) 3. QGP quark-gluon plasma 4. EOS equation of state 5. HS hypersurface 6. FO freeze-out 7. SMM Statistical Multifragmentation Model 8. FDM Fisher Droplet Model 9.GBM GasofBagsModel 10. QGBST Quark-Gluon Bags with Surface Tension 11. GCE grand canonical ensemble 12. HDM Hills and Dales Model 13. VdW Van der Waals 14. GCSP grand canonical surface partition 15. CCSP canonically constrained surface partition 16. SGCSP semi-grand canonical surface partition 17. CSMM constrained Statistical Multifragmentation Model 18. GSMM generalized Statistical Multifragmentation Model 8 19. SBM Statistical Bootstrap Model 20. HTM Hagedorn Thermostat Model 21. BD Bass-Dumitru 22. TLS Teaney-Lauret-Shuryak 23. t.l. time-like 24. s.l. space-like 25. RFG referenceframeofgas 26. RFF rest frame of ﬂuid 27. AT apparent temperature 9 Introduction In this dissertation, the author investigates the equation of state of strongly interacting matter. The main attention is devoted to theoretical research of two major phase transitions (PT), which are accessible for the experimental studies in the nuclear heavy-ion (A+A) collisions. They are the liquid-gas PT in nuclear media observed in A+A reactions at intermediate energies known as nuclear multifragmentation, and the deconﬁnement PT from usual hadrons to quark-gluon plasma (QGP). The major diﬃculties of experimental and theoretical studies of these PTs are determined by the spatial and temporal ﬁniteness of strongly interacting systems created during the A+A collision process. Thus, the characteristic number of particles at the beginning of a collision is about a few hundred nucleons which collide in practically vanishing volume due to the Lorentz contraction of original nuclei. After a complicated evolution during 10 to 20 fm/c, which includes thermalization of highly excited hadronic/nuclear matter, its expansion accompanied by a possible PT and cooling down to free streaming particles, the system typically stops its nontrivial evolution in the volume of about a few thousand fm3 which is ﬁlled up with a few thousand secondary hadrons. Therefore, theoretical