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Fast local thermalization of gluons and quarks characterizes the initial stages of relativistic heavy-ion collisions. For a theoretical description, effective weakly-coupled kinetic theories that rely on the quantum Boltzmann equation have been proposed and solved numerically. In the present work, I aim to account for the time evolution during the rapid equilibration of partons in a simplified model through a nonlinear diffusion equation for the occupation-number distributions in the full momentum range. It is shown that in case of constant transport coefficients, the equation can be solved analytically in closed form through a nonlinear transformation. The occupation-number distribution is then obtained via the logarithmic derivative of a generalized (time-dependent) partition function.

Although the nonlinear boson diffusion equation (NBDE) for the thermalization of gluons had been proposed in [1], the analytical solution had initially been derived only for the free case. In order to obtain the Bose-Einstein distribution in the stationary limit, however, one has to consider the boundary condition [2] at the singularity p=\mu with the chemical potential \mu<0 for number-conserving elastic gluon scatterings, and \mu=0 for inelastic scatterings, which are essential for the thermalization. It is shown that analytical solutions of the NBDE can still be obtained [3].

The model is applied to the equilibration of gluons in heavy-ion collisions at LHC energies where initial central temperatures of 500-600 MeV are reached during local thermalization. Equilibrium is attained through the nonlinear evolution of the distribution functions at short times t<<1 fm/c in the infrared, whereas it takes more time in the large-momentum region to attain the Maxwell-Boltzmann tail of the distribution function. Thermalization in the IR occurs much faster through inelastic as compared to elastic gluon scatterings, thus preventing the formation of a gluon condensate through number-conserving elastic collisions. These results are consistent with QCD-based numerical findings in [4].

[1] Wolschin, G.: Equilibration in ?finite Bose systems. Physica A 499, 1 (2018).

[2] Wolschin, G.: Local thermalization of gluons in a nonlinear model. Nonlin. Phenom. Complex Syst. 23, 72 (2020).

[3] Wolschin, G.: Nonlinear diffusion of gluons, in preparation.

[4] Blaizot, J.P., Liao, J., Mehtar-Tani, Y.: The thermalization of soft modes in non-expanding isotropic quark gluon plasmas. Nucl. Phys. A 961, 37 (2017).

Topic | Heavy Ion Physics |
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