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Table of contents
Résumé
Abstract
Remerciements
1 Introduction
1.1 Understanding the effectiveness of hydrodynamics in heavy ion collisions
1.2 How to read this thesis?
I Theoretical background
2 The problem of thermalization in heavy-ion collisions
2.1 The strong interaction
2.2 Looking inside a proton, first part
2.3 Heavy Ion collisions
2.4 Looking at a proton, second part
2.5 The quark-gluon-plasma: experimental evidences
2.6 Has the QGP ever existed in the history of the universe?
2.7 Quark-gluon-plasma: the puzzle
3 Theoretical tools to deal with the Quark-Gluon-Plasma
3.1 Kinetic Theory
3.2 Hydrodynamics
3.3 Strongly coupled techniques: fast « hydrodynamisation »
3.4 Quantum Chromodynamics
3.5 Specificities of heavy-ion collisions
3.6 The Color Glass Condensate (CGC) effective theory
3.7 JIMWLK equation
3.8 LO CGC results: Impossible matching with hydrodynamics
3.9 NLO CGC results: Weibel instabilities and secular divergences
3.10 Summary
Appendices
3.A Bjorken’s law for an ideal fluid
4 Beyond standard perturbation theory
4.1 Schwinger-Keldysh formalism
4.2 Resummation formula
4.3 The Classical-statistical approximation: a path integral approach
4.A Relation between Schwinger-Keldysh and Feynman generating functionals
II Study of a scalar field theory
5 Scalar field theory in a fixed volume
5.1 Setup of the problem, specificities of the scalar model
5.2 The physics of instabilities
5.3 Macroscopic observables: the formation of an EOS
5.4 Microscopic properties of fixed volume scalar field theory
5.5 Summary
4 TABLE DES MATIÈRES
5.A Instabilities in the fixed-volume case
5.B Appendix: Effective Hamiltonian
6 Expanding system
6.1 Expanding scalar theory
6.2 Numerical implementation
6.3 Independence with respect to the initial time
6.4 Resonance band
6.5 Occupation Number
6.6 Energy-momentum tensor
6.7 Hydrodynamical behavior
6.8 Summary
6.A Numerical considerations
7 Non Renormalizability of the Classical Statistical Approximation
7.1 Renormalization of Green’s functions
7.2 Renormalization of composite operators
7.3 The retarded-advanced basis
7.4 Eliminating the source term
7.5 Ultraviolet power counting in the full theory
7.6 Ultraviolet power counting in the CSA
7.7 Ultraviolet divergences in the CSA
7.8 Impact of the non-renormalizability of the CSA on Tmn
7.9 Cumulative effects of the non-renormalizability
7.10 Possible partial cure
7.11 Could the cure be implemented numerically?
7.12 Summary
7.A Calculation of G1112 and G1222
7.B Calculation of G1122
III Yang-Mills theory
8 Spectrum of fluctuations above the light cone
8.1 Spectrum of fluctuations: a new derivation
8.2 Known results for the background field
8.3 The axial gauge
8.4 Going to Fock-Schwinger gauge
8.5 Small fluctuations in the forward light cone
8.6 Summary
8.A Useful formulas to derive (8.35)
8.B Several checks on the step 3
9 Numerical results
9.1 Numerical implementation of the Yang-Mills Equations
9.2 Matrix multiplication on the lattice
9.3 Leap-frog algorithm
9.4 Initial conditions for the background field
9.5 Discretized form of the energy-momentum tensor
9.6 Numerical checks
9.7 Initial conditions for the small fluctuations
9.8 Monte-Carlo: speed versus storage
9.9 Enforcing the non-linear Gauss’s law
9.10 Renormalization
9.11 Numerical results: isotropization, anomalous viscosity
9.12 Summary
10 Conclusion
