Kinetic description of Bose-Einstein condensation with test particle simulations

We present a kinetic description of Bose-Einstein condensation for particle systems being out of thermal equilibrium, which may happen for gluons produced in the early stage of ultrarelativistic heavy-ion collisions. The dynamics of bosons towards equilibrium is described by a Boltzmann equation including Bose factors. To solve the Boltzmann equation with the presence of a Bose-Einstein condensate we make further developments of the kinetic transport model BAMPS (Boltzmann approach of multiparton scatterings). In this work we demonstrate the correct numerical implementations by comparing the final numerical results to the expected solutions at thermal equilibrium for systems with and without the presence of Bose-Einstein condensate. In addition, the onset of the condensation in an overpopulated gluon system is studied in more details. We find that both expected power-law scalings denoted by the particle and energy cascade are observed in the calculated gluon distribution function at infrared and intermediate momentum regions, respectively. Also, the time evolution of the hard scale exhibits a power-law scaling in a time window, which indicates that the distribution function is approximately self-similar during that time.

Figure 1

Collision rates per particle as a function of temperature. The solid squares show the numerically calculated collision rates with $N_{\text{test}}=1600$, while the solid curve depicts the analytical results.

Figure 2

Dependence of the chemical potential $\mu$ and the temperature $T$ on $f_0$ and $Q_s$.

Figure 3

The time evolution of the momentum distribution function for $f_0=0.05$.

Figure 4

Comparisons between the calculated distributions at equilibrium times and the Bose-Einstein distributions with the expected temperatures and chemical potentials.

Figure 5

Same as Fig. 3, but for $f_0=0.154$.

Figure 6

Same as Fig. 3, but for $f_0=1$.

Figure 7

The particle distribution function multiplied by momentum squared.

Figure 8

The particle distribution function multiplied by momentum to a power of $7/3$.

Figure 9

The particle distribution function multiplied by momentum to a power of $7/4$.

Figure 10

Same as Fig. 6.

Figure 11

Comparison of the gluon momentum distribution at $6.692\mathrm{fm}/\mathrm{c}$ (black open circles and solid curve) with the Bose-Einstein distribution at thermal equilibrium (red curve).

Figure 12

Time evolution of the density of the gluon condensate.

Figure 13

Time evolution of the hard scale $\mathrm{\Lambda }$ and the Debye screening mass $m_D$.