PHYSICAL REVIEW C VOLUME 40, NUMBER 6 DECEMBER 1989 Collisional relaxation in simulations of heavy-ion collisions using Boltxmann-type equations G. Welke, R. MalAiet, * C. Gregoire, and M. Prakash Physics Department, State University of New York at Stony Brook, Stony Brook, New York I I 794 E. Suraud Grand Accelerateur National d'Ions I. ourds, Boite Postale No. 5027, F-14021, Caen, France (Received 14 April 1989) We compare three test-particle methods currently used in numerical simulations of Boltzmann- type equations for the analysis of intermediate-energy heavy-ion collisions with an exact solution of the Krook-Wu model. These methods are the full ensemble, parallel ensemble, and hybrid tech- niques. We find that collisional relaxation is sensitive to the method of simulation used. The full ensemble approach is found to agree with the exact results of the Krook-Wu model. The parallel ensemble procedure provides a reasonable approximation to the analytical relaxation rate for a wide range of systems, while the hybrid method overestimates the relaxation rate. We further compare transverse How data from the first two of these methods in a cascade simulation of heavy-ion col- lisions, and find reasonable agreement provided the two-body cross section is not enhanced by a large factor over its free space value. This has implications for quantitative comparisons of calcula- tions to experimental data. I. INTRODUCTION Heavy-ion experiments below E~, b=2 GeV/nucleon are currently being analyzed in terms of Boltzmann-type kinetic equations. One such equation for the time evolu- tion of the phase-space distribution function f(r, p, t ) of a nucleon that incorporates both the mean-field U and a collision term with Pauli blocking of final states is (see, for example, Ref. 1) +V UV f V„UV f- at 6 Jd &2d P2;d& 8'I[ffz(1 pfi )(1 pf2 ) — fi fp (1 pf )(1 pf2— )](2~)'6— '(p+p, — p, — p, . )) . Above, p is the density of nucleons, da. z&/dA is the differential nucleon-nucleon cross section, and g is the relative velocity. Equation (1.1) has been variously re- ferred to as the Boltzmann-Uehling-Uhlenbeck (BUU), Vlasov-Uehling-Uhlenbeck (VUU), Boltzmann-Nord- heim, or Landau-Vlasov ' equation. In general, the mean-field U depends on both the density p and the momentum p of the nucleon. The cascade model ignores mean-field e6'ects, the particles moving without interac- tion between collisions. The opposite extreme involves dropping the hard collisions, but retaining soft interac- tions, such as in the Vlasov Equation (see, for example, Ref. 1). Equation (1. 1) contains effects due to both hard collisions and soft interactions, albeit at a semiclassical level. In the several e6'orts to date, ' ' one of the main objec- tives has been to pin down the equation of state of dense nuclear matter. Simple parametrizations (see, for exam- ple, Refs. 9 and 10) that mimic results of more micro- scopic calculations" of the mean-field U are often used as inputs in simulating f (r, p, t ) using Eq. (1. 1). Another in- put is the energy-dependent differential nucleon-nucleon cross section which is usually taken' from experiments. Recently, the importance of medium modifications of the free space cross sections has also been emphasized. '' Where possible, comparisons of results from numerical simulations of Eq. (1. 1) with the observed patterns of matter, momentum, and energy Aow have been made and have led to some useful insights. " ' With increasing selectivity in the data and improved theoretical analyses the fulfillment of the stated objective appears promising. Our aim in this paper is to provide checks on the accu- racy of the numerical methods used in the simulation of Eq. (1. 1). Many of the current methods (see, for example, Refs. 1 and 5) rely on Monte Carlo simulations using the test-particle method. Reliable, i.e. , statistically sig- nificant estimates of observables usually require a large number of test particles. Equally important is the intrin- 40 2611 1989 The American Physical Society