Abstract
Different thermalization scenarios for systems with large fields have been proposed in the literature based on classical-statistical lattice simulations approximating the underlying quantum dynamics. We investigate the range of validity of these simulations for condensate driven as well as fluctuation dominated initial conditions for the example of a single component scalar field theory. We show that they lead to the same phenomenon of turbulent thermalization for the whole range of (weak) couplings where the classical-statistical approach is valid. In the turbulent regime we establish the existence of a dual cascade characterized by universal scaling exponents and scaling functions. This complements previous investigations where only the direct energy cascade has been studied for the single component theory. A proposed alternative thermalization scenario for stronger couplings is shown to be beyond the range of validity of classical-statistical simulations.
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References
J. Berges, S. Borsányi and C. Wetterich, Prethermalization, Phys. Rev. Lett. 93 (2004) 142002 [hep-ph/0403234] [INSPIRE].
J. Berges, A. Rothkopf and J. Schmidt, Non-thermal fixed points: effective weak-coupling for strongly correlated systems far from equilibrium, Phys. Rev. Lett. 101 (2008) 041603 [arXiv:0803.0131] [INSPIRE].
J. Berges and G. Hoffmeister, Nonthermal fixed points and the functional renormalization group, Nucl. Phys. B 813 (2009) 383 [arXiv:0809.5208] [INSPIRE].
R. Micha and I.I. Tkachev, Relativistic turbulence: a long way from preheating to equilibrium, Phys. Rev. Lett. 90 (2003) 121301 [hep-ph/0210202] [INSPIRE].
R. Micha and I.I. Tkachev, Turbulent thermalization, Phys. Rev. D 70 (2004) 043538 [hep-ph/0403101] [INSPIRE].
B. Nowak, D. Sexty and T. Gasenzer, Superfluid turbulence: nonthermal fixed point in an ultracold Bose gas, Phys. Rev. B 84 (2011) 020506 [arXiv:1012.4437] [INSPIRE].
B. Nowak, J. Schole, D. Sexty and T. Gasenzer, Nonthermal fixed points, vortex statistics and superfluid turbulence in an ultracold Bose gas, Phys. Rev. A 85 (2012) 043627 [arXiv:1111.6127] [INSPIRE].
J. Berges and D. Sexty, Strong versus weak wave-turbulence in relativistic field theory, Phys. Rev. D 83 (2011) 085004 [arXiv:1012.5944] [INSPIRE].
J. Berges and D. Sexty, Bose condensation far from equilibrium, Phys. Rev. Lett. 108 (2012) 161601 [arXiv:1201.0687] [INSPIRE].
B. Nowak and T. Gasenzer, On a new twist in the dynamics of Bose-Einstein condensation, arXiv:1206.3181 [INSPIRE].
T. Gasenzer, L. McLerran, J.M. Pawlowski and D. Sexty, Gauge turbulence, topological defect dynamics and condensation in Higgs models, arXiv:1307.5301 [INSPIRE].
J. Berges, S. Scheffler and D. Sexty, Turbulence in nonabelian gauge theory, Phys. Lett. B 681 (2009) 362 [arXiv:0811.4293] [INSPIRE].
S. Schlichting, Turbulent thermalization of weakly coupled non-abelian plasmas, Phys. Rev. D 86 (2012) 065008 [arXiv:1207.1450] [INSPIRE].
A. Kurkela and G.D. Moore, UV cascade in classical Yang-Mills theory, Phys. Rev. D 86 (2012) 056008 [arXiv:1207.1663] [INSPIRE].
J. Berges, K. Boguslavski, S. Schlichting and R. Venugopalan, Turbulent thermalization process in heavy-ion collisions at ultrarelativistic energies, Phys. Rev. D 89 (2014) 074011 [arXiv:1303.5650] [INSPIRE].
J. Berges, K. Boguslavski, S. Schlichting and R. Venugopalan, Universal attractor in a highly occupied non-Abelian plasma, arXiv:1311.3005 [INSPIRE].
F. Gelis, E. Iancu, J. Jalilian-Marian and R. Venugopalan, The color glass condensate, Ann. Rev. Nucl. Part. Sci. 60 (2010) 463 [arXiv:1002.0333] [INSPIRE].
T. Lappi and L. McLerran, Some features of the glasma, Nucl. Phys. A 772 (2006) 200 [hep-ph/0602189] [INSPIRE].
S. Mrowczynski, Plasma instability at the initial stage of ultrarelativistic heavy ion collisions, Phys. Lett. B 314 (1993) 118 [INSPIRE].
P. Romatschke and M. Strickland, Collective modes of an anisotropic quark gluon plasma, Phys. Rev. D 68 (2003) 036004 [hep-ph/0304092] [INSPIRE].
P.B. Arnold, J. Lenaghan and G.D. Moore, QCD plasma instabilities and bottom up thermalization, JHEP 08 (2003) 002 [hep-ph/0307325] [INSPIRE].
M. Attems, A. Rebhan and M. Strickland, Instabilities of an anisotropically expanding non-Abelian plasma: 3D + 3V discretized hard-loop simulations, Phys. Rev. D 87 (2013) 025010 [arXiv:1207.5795] [INSPIRE].
P. Romatschke and R. Venugopalan, The unstable glasma, Phys. Rev. D 74 (2006) 045011 [hep-ph/0605045] [INSPIRE].
K. Fukushima and F. Gelis, The evolving glasma, Nucl. Phys. A 874 (2012) 108 [arXiv:1106.1396] [INSPIRE].
J. Berges and S. Schlichting, The non-linear glasma, Phys. Rev. D 87 (2013) 014026 [arXiv:1209.0817] [INSPIRE].
T. Epelbaum and F. Gelis, Pressure isotropization in high energy heavy ion collisions, Phys. Rev. Lett. 111 (2013) 232301 [arXiv:1307.2214] [INSPIRE].
T. Epelbaum and F. Gelis, Fluctuations of the initial color fields in high energy heavy ion collisions, Phys. Rev. D 88 (2013) 085015 [arXiv:1307.1765] [INSPIRE].
K. Dusling, T. Epelbaum, F. Gelis and R. Venugopalan, Instability induced pressure isotropization in a longitudinally expanding system, Phys. Rev. D 86 (2012) 085040 [arXiv:1206.3336] [INSPIRE].
T. Epelbaum and F. Gelis, Role of quantum fluctuations in a system with strong fields: spectral properties and thermalization, Nucl. Phys. A 872 (2011) 210 [arXiv:1107.0668] [INSPIRE].
L. Kofman, A.D. Linde and A.A. Starobinsky, Reheating after inflation, Phys. Rev. Lett. 73 (1994) 3195 [hep-th/9405187] [INSPIRE].
J. Berges and J. Serreau, Parametric resonance in quantum field theory, Phys. Rev. Lett. 91 (2003) 111601 [hep-ph/0208070] [INSPIRE].
A. Arrizabalaga, J. Smit and A. Tranberg, Tachyonic preheating using 2PI-1/N dynamics and the classical approximation, JHEP 10 (2004) 017 [hep-ph/0409177] [INSPIRE].
G. Aarts and J. Berges, Classical aspects of quantum fields far from equilibrium, Phys. Rev. Lett. 88 (2002) 041603 [hep-ph/0107129] [INSPIRE].
G.D. Moore, Problems with lattice methods for electroweak preheating, JHEP 11 (2001) 021 [hep-ph/0109206] [INSPIRE].
T. Gasenzer, B. Nowak and D. Sexty, Charge separation in reheating after cosmological inflation, Phys. Lett. B 710 (2012) 500 [arXiv:1108.0541] [INSPIRE].
A. Mueller and D. Son, On the equivalence between the Boltzmann equation and classical field theory at large occupation numbers, Phys. Lett. B 582 (2004) 279 [hep-ph/0212198] [INSPIRE].
S. Jeon, The Boltzmann equation in classical and quantum field theory, Phys. Rev. C 72 (2005) 014907 [hep-ph/0412121] [INSPIRE].
J. Berges, Introduction to nonequilibrium quantum field theory, AIP Conf. Proc. 739 (2005) 3 [hep-ph/0409233] [INSPIRE].
J. Berges and T. Gasenzer, Quantum versus classical statistical dynamics of an ultracold Bose gas, Phys. Rev. A 76 (2007) 033604 [cond-mat/0703163] [INSPIRE].
S. Jeon, Color glass condensate in Schwinger-Keldysh QCD, Annals Phys. 340 (2014) 119 [arXiv:1308.0263] [INSPIRE].
J. Berges, D. Gelfand and J. Pruschke, Quantum theory of fermion production after inflation, Phys. Rev. Lett. 107 (2011) 061301 [arXiv:1012.4632] [INSPIRE].
J. Berges, D. Gelfand and D. Sexty, Amplified fermion production from overpopulated Bose fields, Phys. Rev. D 89 (2014) 025001 [arXiv:1308.2180] [INSPIRE].
S.Y. Khlebnikov and I. Tkachev, Classical decay of inflaton, Phys. Rev. Lett. 77 (1996) 219 [hep-ph/9603378] [INSPIRE].
T. Prokopec and T.G. Roos, Lattice study of classical inflaton decay, Phys. Rev. D 55 (1997) 3768 [hep-ph/9610400] [INSPIRE].
J. Baacke, K. Heitmann and C. Patzold, Nonequilibrium dynamics of fermions in a spatially homogeneous scalar background field, Phys. Rev. D 58 (1998) 125013 [hep-ph/9806205] [INSPIRE].
P.B. Greene and L. Kofman, Preheating of fermions, Phys. Lett. B 448 (1999) 6 [hep-ph/9807339] [INSPIRE].
D. Boyanovsky, H. de Vega, R. Holman, D. Lee and A. Singh, Dissipation via particle production in scalar field theories, Phys. Rev. D 51 (1995) 4419 [hep-ph/9408214] [INSPIRE].
P.B. Greene, L. Kofman, A.D. Linde and A.A. Starobinsky, Structure of resonance in preheating after inflation, Phys. Rev. D 56 (1997) 6175 [hep-ph/9705347] [INSPIRE].
D.I. Podolsky, G.N. Felder, L. Kofman and M. Peloso, Equation of state and beginning of thermalization after preheating, Phys. Rev. D 73 (2006) 023501 [hep-ph/0507096] [INSPIRE].
K. Dusling, T. Epelbaum, F. Gelis and R. Venugopalan, Role of quantum fluctuations in a system with strong fields: Onset of hydrodynamical flow, Nucl. Phys. A 850 (2011) 69 [arXiv:1009.4363] [INSPIRE].
V.E. Zakharov, V.S. Lvov and G. Falkovich, Kolmogorov spectra of turbulence I: wave turbulence, Springer, Germany (1992).
J. Berges, Controlled nonperturbative dynamics of quantum fields out-of-equilibrium, Nucl. Phys. A 699 (2002) 847 [hep-ph/0105311] [INSPIRE].
J. Berges, S. Schlichting and D. Sexty, Dynamic critical phenomena from spectral functions on the lattice, Nucl. Phys. B 832 (2010) 228 [arXiv:0912.3135] [INSPIRE].
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Berges, J., Boguslavski, K., Schlichting, S. et al. Basin of attraction for turbulent thermalization and the range of validity of classical-statistical simulations. J. High Energ. Phys. 2014, 54 (2014). https://doi.org/10.1007/JHEP05(2014)054
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DOI: https://doi.org/10.1007/JHEP05(2014)054