Comparison of heavy-ion transport simulations: Collision integral in a box

Simulations by transport codes are indispensable to extract valuable physical information from heavy-ion collisions. In order to understand the origins of discrepancies among different widely used transport codes, we compare 15 such codes under controlled conditions of a system confined to a box with periodic boundary, initialized with Fermi-Dirac distributions at saturation density and temperatures of either 0 or 5 MeV. In such calculations, one is able to check separately the different ingredients of a transport code. In this second publication of the code evaluation project, we only consider the two-body collision term; i.e., we perform cascade calculations. When the Pauli blocking is artificially suppressed, the collision rates are found to be consistent for most codes (to within $1\%$ or better) with analytical results, or completely controlled results of a basic cascade code. In orderto reach that goal, it was necessary to eliminate correlations within the same pair of colliding particles that can be present depending on the adopted collision prescription. In calculations with active Pauli blocking, the blocking probability was found to deviate from the expected reference values. The reason is found in substantial phase-space fluctuations and smearing tied to numerical algorithms and model assumptions in the representation of phase space. This results in the reduction of the blocking probability in most transport codes, so that the simulated system gradually evolves away from the Fermi-Dirac toward a Boltzmann distribution. Since the numerical fluctuations are weaker in the Boltzmann-Uehling-Uhlenbeck codes, the Fermi-Dirac statistics is maintained there for a longer time than in the quantum molecular dynamics codes. As a result of this investigation, we are able to make judgements about the most effective strategies in transport simulations for determining the collision probabilities and the Pauli blocking. Investigation in a similar vein of other ingredients in transport calculations, like the mean-field propagation or the production of nucleon resonances and mesons, will be discussed in the future publications.

Figure 1

Momentum distributions at time $t=0$, 20, and $140\text{fm}/c$ for simulations without Pauli blocking initialized with Fermi-Dirac distributions of $T=0$ (upper panels) and $5\text{MeV}$ (lower panels) for the different BUU codes (left panels) and QMD codes (right panels). The thick dashed lines (red) represent ideal nonrelativistic Boltzmann distributions for $T_B=14.753$ and $15.833\text{MeV}$ corresponding to $T=0$ and $T=5\text{MeV}$, respectively.

Figure 2

Time evolution of the collision rate $d{N}_{\text{coll}}/dt$ without Pauli blocking for initializations with $T=0$ (upper panels) and $T=5\text{MeV}$ (lower panels) for BUU codes (left panels) and QMD codes (right panels). The codes are identified in the legend.

Figure 3

Collision rates, for systems initialized at $T=0$ (CT0, top, solid symbols) and $T=5\text{MeV}$ (CT5, bottom, open symbols), for BUU-type codes (left) and QMD-type codes (right). The lines represent the reference values given in Table 3: dashed (green) and dotted (purple) are the nonrelativistic and relativistic Boltzmann rates, respectively; solid (black) is the basic relativistic cascade code results. The symbols show the results of the codes averaged over time interval from 60 to $140\text{fm}/c$; cf. Fig. 2. The symbols (and their colors) indicate to which reference values the results of a code should be compared: circle (green) and triangle (purple) to the nonrelativistic and relativistic Boltzmann, respectively; square (black) to the relativistic cascade (see text).

Figure 4

Momentum distributions in cascade calculations with Pauli blocking at $t=0$, 20, 80, and 140 fm/$c$ from top to bottom for $T=5$ MeV for BUU codes (left panels) and QMD codes (right panels) as identified in the legend. The dashed lines are ideal Boltzmann distributions at $T_B=15.833$ MeV.

Figure 5

Top: Center-of-mass energy distribution of successful collision number averaged over time for $T=5$ MeV in CBOP1. Bottom: Averaged blocking factors as a function of center-of-mass energy for colliding pairs. BUU (QMD) codes are shown on the left (right), as identified in the legend. The thick dashed-purple lines are for the ideal Fermi-Dirac distribution at $T=5$ MeV in the relativistic case.

Figure 6

Scatter plots for momentum space occupation for the final states of all collisions in the first time step of the simulation (see text) for the SMF and ImQMD models initialized at $T=5$ MeV. From left to right, the panels represent in sequence: (a) results from SMF with (effectively) about 12 000 test particles per nucleon; (b) results from SMF with 100 test particles per nucleon and one run; (c) results from SMF with 10 test particles per nucleon and 10 runs; and (d) results from ImQMD for 100 runs. The red curves in the panels represent the Fermi-Dirac distribution at $T$ = 5 MeV; the green curve in the left panel is the Fermi-Dirac distribution at $T$ = 7 MeV.

Figure 7

Distribution of occupation probabilities (blue) in the first time step of the simulation for the $T$ = 5 MeV initialization with the mean and variance shown by the blue curve and the blue error bars. Left panels show results for BUU-type codes and right panels for QMD-type codes. The average blocking probabilities are shown as the black curve (see text). The Fermi-Dirac distribution with $T$ = 5 MeV used for initialization is represented with the solid line (red). The gray line and error bars for CoMD are explained in the text.

Figure 8

Successful collision rates from the different models in simulations with Pauli blocking for $T$ = 5 MeV initializations (CBOP1T5 mode). The square symbols show the results averaged over the time interval 60–140 fm/c, while stars are the successful collision rates for the first time step. The black line represents the reference value calculated with the basic cascade code for a fixed Fermi-Dirac blocker (CBOP2T5 mode).

Figure 9

Collision rates in calculations without Pauli blocking vs the average kinetic energy $\overline{\epsilon }$ for the different codes averaged over the time interval 60–140 fm/$c$. The thick lines (blue and green for relativistic and nonrelativistic codes, respectively) connect the values for the two initial conditions of $T=0$ and $T=5$ MeV. The values for the reference cases from Tables 2 and 3 are given as thin lines on which the vertical bars indicate the values of $\overline{\epsilon }$ for $T=0$ and $T=5$ MeV.

Figure 10

The same as Fig. 9 but for the attempted collision rates when the collisions are Pauli blocked with the probabilities evaluated with the analytic Fermi-Dirac distribution function (CBOP2T0 and CBOP2T5). Results are not available for GiBUU. The blue thick line for CoMD ($d{N}_{\text{att.}}/dt=136.6$ and 139.0 $c$/fm for CBOP2T0 and CBOP2T5, respectively) is outside of the plotted region.