Boltzmann-Langevin one-body dynamics for fermionic systems

A full implementation of the Boltzmann-Langevin equation for fermionic systems is introduced in a transport model for dissipative collisions among heavy nuclei. Fluctuations are injected in phase space and not, like in more conventional approaches, as a projection on suitable subspaces. The advantage of this model is to be specifically adapted to describe processes characterised by instabilities, like the formation of fragments from a hot nuclear system, and by dissipation, like the transparency in nucleus-nucleus collisions. 1 Fluctuations and bifurcations Dissipative nucleus-nucleus collisions are a unique probe for the in-medium nuclear interaction at densities away from saturation and at high nucleon momenta. In weakly-excited systems, Pauli-blocking factors of final orbitals largely suppress two-body direct interactions. In a semiclassical framework, the dynamics of such situation can be appropriately described within a Vlasov formalism, where the temporal evolution of the one-body distribution function f( r, p, t) (function of time t, space coordinates r and momentum p) in the self-consistent one body field is governed by the effective Hamiltonian H[f ]. However, for more violent collisions, direct two-body interactions become significant and require to be treated by an additional Boltzmann collision integral I[f ]. In the Uehling-Uhlenbeck form, this latter is introduced as a continuous-source term Ī[f ]. Within standard forms of BUU/VUU/BNV formalisms, the additional use of an ensemble-averaged mean field makes the EPJ Web of Conferences DOI: 10.1051/ , epjconf 2012 / 00027 (2012) 31 3100027 This is an Open Access article distributed under the terms of the Creative Commons Attribution License 2.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. C © Owned by the authors, published by EDP Sciences , 2012 SIF Article available at http://www.epj-conferences.org or http://dx.doi.org/10.1051/epjconf/20123100027 transport model particularly suited for one-body observables (for example the collective flow, or proton spectra), but suppresses bifurcations. Bifurcations are however crucial in processes characterised by volume instabilities, where fluctuations are the seeds of the formation of several intermediate-mass fragments. A possible approach to this scenario is the employment of a two-body Hamiltonian, as in Molecular Dynamics [1]. As an extension of the one-body dynamics, another approach is the introduction of one more term δI[f ] which produces Langevin fluctuations. The resulting temporal evolution of the one-body distribution function in a BoltzmannLangevin approach is given by ∂t f = {H[f ], f} + Ī[f ] + δI[f ] . (1) This latter is the method we follow. Many approaches to the Boltzmann-Langevin equation in nuclear transport models are projections on one suitable subspace, like the coordinate space. In this framework, a fluctuating term can be prepared by associating a Brownian force to a stochastic one-body potential [2] or it can correspond to kinetic equilibrium fluctuations of a Fermi gas [3]. The disadvantage of projection methods is that the amplitude of the fluctuation may have to be adjusted and fluctuations may appear from a specific time on (when equilibrium is attained or when entering spinodal conditions), as an amplification of the most unstable mode. The approach we follow consists in constructing a stochastic collision term which includes Boltzmann-Langevin fluctuations in phase space, supplemented by a strict treatment of Pauli-blocking factors, in the framework of semiclassical test-particle-based transport models. 2 Collision statistics and fragment formation We construct the Bolzmann-Langevin one body (BLOB) model by using the same semiclassical isospin-dependent mean-field employed in the Stochastic Mean Field (SMF) model [2,4]; this is characterised by an incompressibility of kinf = 200 MeV (soft) and the potential component of the symmetry energy is given either by a linear term as a function of the density (asystiff), or by a quadratic term, with negative curvature (asy-soft). In SMF, the collision term is introduced in the standard UehlingUhlenbeck form and fluctuations are injected in a reduced subspace, agitating the spatial density profile. On the other hand, in BLOB fluctuations are implemented in full phase-space, replacing the collision integral by EPJ Web of Conferences


Fluctuations and bifurcations
Dissipative nucleus-nucleus collisions are a unique probe for the in-medium nuclear interaction at densities away from saturation and at high nucleon momenta. In weakly-excited systems, Pauli-blocking factors of final orbitals largely suppress two-body direct interactions. In a semiclassical framework, the dynamics of such situation can be appropriately described within a Vlasov formalism, where the temporal evolution of the one-body distribution function f ( r, p, t) (function of time t, space coordinates r and momentum p) in the self-consistent one body field is governed by the effective Hamiltonian H[f ]. However, for more violent collisions, direct two-body interactions become significant and require to be treated by an additional Boltzmann collision integral I[f ]. In the Uehling-Uhlenbeck form, this latter is introduced as a continuous-source termĪ [f ]. Within standard forms of BUU/VUU/BNV formalisms, the additional use of an ensemble-averaged mean field makes the transport model particularly suited for one-body observables (for example the collective flow, or proton spectra), but suppresses bifurcations.
Bifurcations are however crucial in processes characterised by volume instabilities, where fluctuations are the seeds of the formation of several intermediate-mass fragments. A possible approach to this scenario is the employment of a two-body Hamiltonian, as in Molecular Dynamics [1]. As an extension of the one-body dynamics, another approach is the introduction of one more term δI[f ] which produces Langevin fluctuations. The resulting temporal evolution of the one-body distribution function in a Boltzmann-Langevin approach is given by This latter is the method we follow. Many approaches to the Boltzmann-Langevin equation in nuclear transport models are projections on one suitable subspace, like the coordinate space. In this framework, a fluctuating term can be prepared by associating a Brownian force to a stochastic one-body potential [2] or it can correspond to kinetic equilibrium fluctuations of a Fermi gas [3]. The disadvantage of projection methods is that the amplitude of the fluctuation may have to be adjusted and fluctuations may appear from a specific time on (when equilibrium is attained or when entering spinodal conditions), as an amplification of the most unstable mode.
The approach we follow consists in constructing a stochastic collision term which includes Boltzmann-Langevin fluctuations in phase space, supplemented by a strict treatment of Pauli-blocking factors, in the framework of semiclassical test-particle-based transport models.

Collision statistics and fragment formation
We construct the Bolzmann-Langevin one body (BLOB) model by using the same semiclassical isospin-dependent mean-field employed in the Stochastic Mean Field (SMF) model [2,4]; this is characterised by an incompressibility of k inf = 200 MeV (soft) and the potential component of the symmetry energy is given either by a linear term as a function of the density (asystiff), or by a quadratic term, with negative curvature (asy-soft).
In SMF, the collision term is introduced in the standard Uehling-Uhlenbeck form and fluctuations are injected in a reduced subspace, agitating the spatial density profile. On the other hand, in BLOB fluctuations are implemented in full phase-space, replacing the collision integral by the Boltzmann-Langevin approach described in the following. Like in the method introduced in ref. [5], the collision term of BLOB involves entire nucleons. The test-particle method is used and a number of N test test-particles per nucleon is employed. A 'nucleon' is represented by an agglomerate of N test test-particles of identical isospin which share the same volume in coordinate and in momentum space, as shown in fig. 1.
The number of attempted collisions is weighted on the probability of satisfying the mean-free-path condition for the two agglomerates of test particles in a given interval of time. Such probability is proportional to the mean density in momentum space, the average relative momentum between the two clouds and the nucleon-nucleon cross section. This latter is divided by a factor N test in order to keep a correspondence with the Uehling-Uhlenbeck collision term. In fig. 2, SMF and BLOB are used to simulate the same system 124 Sn+ 124 Sn at 50 AMeV and b = 0. We use here a simplified ver-    ing the beginning of the process of fragment formation, characterised by an initial increase of the number of collisions and a successive stabilisation and levelling off due to the separation of the fragments. This figure shows that, as expected, the number of attempted collisions per test particle averaged over a statistics of dynamical trajectories should not differ in the present formalism with respect to a conventional BUU approach when the initial rising side of the spectrum is considered. On the other hand, the amplitude of the fluctuation marks the difference between the two models. In correspondence with the phase of dilution (around 80-120 fm/c), fluctuations characterise not only the statistics of collisions but also the kinematics. The simulation was repeated for N t = 40 and N t = 100. The incomplete treatment of fluctuations used in SMF results in establishing a dependence of the collision statistics on the numerical parameter N t . When N t is reduced in SMF the fragment formation is anticipated, the dynamics becomes more explosive, and the fragment multiplicity increases. All these effects are even further enhanced when moving from SMF with N t = 40 to BLOB, but in the latter case the dependence on N t disappears, as expected, because in BLOB a fluctuation-source term, corresponding to the Boltzmann-Langevin collision integral, has been implemented. Like in the approach followed in ref. [6], the effective-collision probability is weighted on the blocking factors of the destination sites, taking into account the full extension of the initial distributions. In BLOB, distributions are adapted to the occupation profile by the use of a numerical optimisation procedure. This latter converges first to the appropriate extension of the scattered regions (by expanding initial and final sites in momentum space) and then converges to the appropriate test-particle final distribution by reorganising the scattered test particles according to packets which can be modulated in shape; the principle of this method is shown in fig. 1. Due to the inclusion of blocking factors in the collision probability, the number of effective collisions drops to smaller values with respect to the number of attempted collisions. The evolution of the effective-collision number is not expected to be identical for SMF and BLOB calculations; the number is reduced with respect to the attempted-collision spectrum, keeping the shape mostly unchanged, but the effective collisions result more penalised by the Pauli blocking treatment in BLOB then in SMF. On the one hand, this difference comes from the fact that the Pauli blocking is applied to the scattering of test-particle couples in SMF, while in BLOB it is applied to extended distributions of test particles. On the other hand, it is important to remark that in BLOB the number of effective collisions is reduced with respect to SMF but in this latter case collisions are more efficient in pro-ducing phase-space fluctuations. Moreover, in the BLOB calculation the fluctuation is not gradually amplified (the SMF approach results in amplifying the most unstable mode) but it presents large and correct amplitudes since the initial instants: this favours the early appearing of bifurcations and explains the earlier fragment formation and the larger kinetic energy of the fragments. As a further outcome, a larger variety of exit-channel configurations are expected.
In fig. 3, like in fig. 2, SMF and BLOB are again used to simulate the system 124 Sn+ 124 Sn at 50 AMeV and b = 0. In full agreement with the analysis of the collision statistics and the properties of fragments, we find larger fragment kinetic energies for the BLOB simulation. The fragment configuration is more isotropic in the BLOB simulation, this could be due to a larger transparency. A similar comparison was done between SMF and  AMD in ref. [7]. Another calculation is shown in fig. 3 for the less excited system 124 Sn+ 124 Sn at 30 AMeV, close to the threshold between fusion and fragmentation. This analysis has the purpose of studying the effect of bifurcation. In the present example we find that in the SMF calculation one trajectory prevails (leading to fusion), while in the BLOB calculation a more complete phase-space sampling allows to access a variety of exit-channel configurations (from fusion to fragmentation).