Evolution of observables in a numerical kinetic model

The numerical solutions of the nonrelativistic and relativistic Boltzmann equations have been studied at various initial conditions. Particularly, the known analytical solution of the nonrelativistic Boltzmann equation at spherically symmetric Gaussian initial conditions has been recovered and the corresponding conservation of particle spectra and correlation functions have been confirmed. Similar conservation properties have also been found for a more general class of solutions describing the nonrelativistic and relativistic systems at anisotropic initial conditions.

Figure 1

Distributions of x components of particle three-momentum (a) and three-coordinate (b), particle momentum (c) and the corresponding CFs as functions of $q_{\mathrm{inv}}$ (d) obtained in the UKM-R simulation of the kinetic evolution of $N=400$ heavy spin-0 bosons of mass $m=0.938$ GeV/$c^2$ at the evolution time $t=0$, 50, and 100 fm/c. Elastic cross section ${\sigma }_{\mathrm{el}}=400$ mb, initial Gaussian radius $R_0=7$ fm, and initial temperature $T_0=0.130$ GeV. Gaussian fits of the momentum distributions and the CFs agree with the input initial values $\sigma =({\mathit{mT}}_0){}^{1/2}=0.3492$ GeV/c and $\lambda =1,R_0=7$ fm, respectively.

Figure 2

CFs obtained from four-coordinates of free streaming (${\sigma }_{\mathrm{el}}=0$) heavy spin-0 bosons of mass $m=0.938$ GeV/$c^2$ obtained in the UKM-R simulation for the same conditions as in Fig. 1 at two evolution times.

Figure 3

Initial and final distributions of x components of particle three-momentum (a), spatial x (b) and time (c) coordinates of the emission or last collision points, and CFs corresponding to these points as functions of $q_{\mathrm{inv}}$ (d) obtained in the UKM-R simulation of $N=400$ heavy spin-0 bosons of mass $m=0.938$ GeV/$c^2$ at the evolution time $t=$ 200 fm/c. Histograms 1, 2, and 3 in (b) and (c) correspond to ${\sigma }_{\mathrm{el}}=$ 0 (initial x distribution), 40, and 400 mb, respectively. The shown results of Gaussian fits of the momentum distributions and the CFs agree with the input initial values $\sigma =({\mathit{mT}}_0){}^{1/2}=0.3492$ GeV/c and $\lambda =1,R_0=7$ fm, respectively.

Figure 4

Final distributions of x and y coordinates of the emission or last collision points (dashed histograms) obtained in the UKM-R simulation on the same conditions as in Fig. 3 except for the initial distributions of spatial coordinates (solid histograms): $\left|x\right|\le 14$ fm; $\left|y\right|,\left|z\right|\le 7$ fm. Panels (a), (c), and (b), (d) correspond to ${\sigma }_{\mathrm{el}}=$ 400 and 40 mb, respectively.

Figure 5

Initial (thick) and final (thin) histograms of the components of particle three-momentum obtained in the UKM-R simulation for the same conditions as in Fig. 4.

Figure 6

Initial and final CFs corresponding to the UKM-R coordinate distributions in panels (a) and (c) of Fig. 4 (${\sigma }_{\mathrm{el}}=400$ mb). CFs are shown as functions of $q_x$ ($q_t\le 10$ MeV/c) (a), $q_t=({q_y}^2+{q_z}^2){}^{1/2}$ ($q_x\le 7$ MeV/c) (b), and $q_{\mathrm{inv}}$ (c). Results of Gaussian fits are shown. Fitted initial Gaussian radii in (a) and (b) are close to the r.m.s. values 8.08 and 4.04 fm of the corresponding initial x and y coordinate distributions in (a) and (c) of Fig. 4.

Figure 7

Final distributions of the x component (b) and the magnitude of particle three-momentum (d) obtained in the UKM-R simulation with ${\sigma }_{\mathrm{el}}=$ 400 mb for the same conditions as in Fig. 3 except for the initial three-momentum distribution (a) and (c) defined by $\left|p_x\right|,\left|p_y\right|,\left|p_z\right|\le (T_{0}m){}^{1/2},T_0$ = 0.130 GeV, $m=$0.938 GeV/$c^2$. Fitted Gaussian widths of final distributions, shown in (b) and (d), are close to the r.m.s. value $({\mathit{mT}}_0/3){}^{1/2}$ = 0.2016 GeV of the initial $p_x$ distribution.

Figure 8

Initial and final CFs corresponding to the UKM-R simulation specified in Fig. 7, as functions of $q_x$ ($q_t\le 10$ MeV/c) (a), $q_t=({q_y}^2+{q_z}^2){}^{1/2}$ ($q_x\le 7$ MeV/c) (b) and $q_{\mathrm{inv}}$ (c). Results of Gaussian fits are shown. Fitted correlation radii are close to the initial system Gaussian radius $R_0=7$ fm.

Figure 9

Distributions of particle energy (a) and (c), momentum (d), and x component of particle three-momentum (b) obtained in the UKM-R simulation of the kinetic evolution of $N=400$ light spin-0 bosons of mass $m=0.140$ GeV/$c^2$ and elastic cross section 400 mb. Initial three-coordinates and three-momenta have been, respectively, distributed according to Gaussian for radius $R_0=7$ fm and relativistic thermal distribution with temperature $T_0=0.130$ GeV. Initial distributions are solid histograms in (a), (b), and (d) Final ones are the solid histogram in (c) and dashed histograms in (b), (d). Fitted values of initial and final temperatures shown in (a) and (c) coincide with $T_0$.

Figure 10

Initial and final CFs corresponding to the UKM-R simulation as specified in Fig. 9. CFs are functions of $q_{\mathrm{inv}}$ (a) and $q=\left|q\right|$ (b).

Figure 11

Final distributions of x (a) and y (b) coordinates of the emission or last collision points (dashed histograms) obtained in the UKM-R simulation for the same conditions as in Fig. 9 except for the initial distributions of spatial coordinates (solid histograms): $\left|x\right|\le 14$ fm; $\left|y\right|,\left|z\right|\le 7$ fm.

Figure 12

Initial (solid) and final (dashed) histograms of components of particle three-momentum obtained in the UKM-R simulation for the same conditions as in Fig. 11.

Figure 13

Initial and final CFs corresponding to the UKM-R simulation as specified in Fig. 11. CFs are functions of $q_x$ ($q_t\le 10$ MeV/c), $q_t=({q_y}^2+{q_z}^2){}^{1/2}$ ($q_x\le 7$ MeV/c), and $q=\left|q\right|$.

Figure 14

Final distributions of components (b), and (d) and magnitude (f) of particle three-momentum obtained in the UKM-R simulation with ${\sigma }_{\mathrm{el}}=$ 400 mb for the same conditions as in Fig. 9, except for the initial three-momentum distribution (a), (c), and (e) defined by $\left|p_x\right|\le 0.7$ GeV/c, $\left|p_y\right|,\left|p_z\right|\le 0.35$ GeV/c. Fitted Gaussian widths of final distributions are in (b), (d), and (f).

Figure 15

Initial and final CFs corresponding to the UKM-R simulation as specified in Fig. 14. CFs are functions of $q_x$ ($q_t\le 10$ MeV/c), $q_t=({q_y}^2+{q_z}^2){}^{1/2}$ ($q_x\le 7$ MeV/c), and $q=\left|q\right|$.