1 Preface

Violent Nuclear Reactions: Large-Amplitude Nuclear Dynamic Phenomena in Fermionic Systems

La verità è una giovinetta tanto bella quanto pudica e perciò va sempre avvolta nel suo mantello.

(The truth is a young maiden as modest as she is beautiful, and therefore she is always seen cloaked.)


– UMBERTO ECO, “L‘isola del giorno prima”



Le docteur examina donc l’empreinte deux fois, trois fois, et il fut bien obligé de reconnaître son origine extraordinaire.

(Thereupon he looked twice, three times, at the print, and he was obliged to acknowledge its extraordinary origin.)


– Jules Verne, “Les Aventures du capitaine Hatteras”



Contents:

Chapter 1 Preface

This review was written in order to be conferred the French accreditation to supervise research (“Habilitation à diriger des recherches”).

I wish to start by thanking all the members of the jury, Maria Colonna, Virginia De La Mota, Eric Suraud, Elias Khan and Oliver Lopez for accepting to read and review this document, on a subject they pioneered. Thanks to them and to many researchers who prepared the terrain, I could also add some contribution to an already well paved avenue. In other words, having them in my jury makes me as happy as a three-year-old kid meeting his favourite team of astronauts!

As a doctoral student, oscillating between Orsay and Darmstadt, I could finally dedicate fanatically to my favourite amusement, nuclear reactions. For a few years, I was among the fission and fusion veterans, experimentalists and visiting theoreticians. I will always be grateful to my first research group in GSI (Darmstadt) for teaching me how to distinguish a light- from a heavy-ion beam from the ramping noise of the transformers of the synchrotron; students were betting German superheavy chocolate snacks on the mass number of the ion beam. I learnt that when the logbook falls on a keyboard in the control room of a particle accelerator, a relativistic beam of heavy ions can end up on a block of copper, traverse it anyway, and produce a grandiose stream of heavy particles and a blast of neutrons which activate all radiation-safety systems and determines the interruption of the whole machine. In a millisecond scale, this was my electrifying introduction to nuclear physics. In better controlled conditions, we discovered several new isotopes, observed new nuclear processes and discoveries were coming daily. Too many discoveries (as it used to be in Darmstadt still some years after 2000) inspired new huge projects, huge projects imposed regulations and restrictions, restrictions spoiled the daily-discovery dream. Never mind, the theoretical understanding of what had been observed, and how it relates to other fields, should be the natural completion of a successful expedition.

During my stay in Caen, starting in 2004, I joined a new very exciting research: nuclear pasta phases in compact stars. At that time, it was believed that the low-density nuclear matter constituting the outer shells of proto-neutron stars could develop a series of topologies of nucleon arrangements which recall spaghetti, lasagne, maccaroni (and maybe also bucatini, fettuccine, farfalline, gnocchetti, penne, pennette, vermicelli, zitoni, orecchiette…). All these configurations were supposed to manifest phase transitions and the overall pasta-phase region was supposed to end-up in a critical point which, eventually, could even be related to a supernova explosion of type II, due to the consequence of a diverging susceptibility on neutrino trapping. After two years of statistical modelling, I had the privilege to destroy the field, or at least part of it, because I discovered that stellar matter can not present any second-order phase transition, nor first order. Pasta phases survived as a set of crossover structures, but they looked somehow less interesting, at first from a thermodynamic point of view.

To find back first-order phase transitions in nuclear matter, with all related instabilities, I returned from the stars to my motherland: heavy-ion collisions. I have always been thrilled by the fact that in this field, galvanising discussions on the reaction mechanisms, were ending up in a theoretical béhourd among myriads of different interpretations. These last years, on the basis of some experience on both statistical and dynamical modelling, and also thanks to the surrounding experimental environment, I could absorb many ideas, recommendations, points of view, indirect experiences from the pioneers of the field in Caen, Catania, Nantes, Orsay and other partner laboratories, and I could also participate to data analyses and discuss new experimental observations. Finally, I could crystallise this information in a new comprehensive description of violent nuclear reactions starting from a microscopic approach.

In this habilitation report, I decided to select only one activity, the dynamical modelling of violent nuclear processes from a one-body point of view, and to present it synthetically in relation to new progress and to the theoretical framework which I contributed to develop, sometimes connecting back to experimental projects where I was, in more than one case, involved as the spokesperson. The aim is to produce a useful and readable document which rather briefly draws conclusions on topics that, though being still source of developments, can be very efficiently handled at a high degree of accuracy. In particular, neither pure theory, nor experimental analysis is put forward in this report, and a phenomenological line is followed. The text is a re-edition of published and to-be-submitted material from my bibliographic list with updates, additional content and connections, all reorganised into a monograph. Most of the works I’m reviewing in this report are the results of a long and fruitful collaboration with Maria Colonna, to whom I would like to express my deep gratitude.

Finally, I’m indebted to all my colleagues, my family and friends for the encouragement on my research. When my two very small kids will be able to read, they will learn that I thank them too, for so numerous and inspiring macroscopic analogies of chaotic dynamics involving catastrophic irreversible processes which correspond, in a nuclear context, to the core subject of this report.

Chapter 2 Introduction: context and some history

2.1 Scope and plan

The splitting of a nuclear complex into fragments constitutes a rich phenomenology in many-body physics which led to a century of discoveries, ranging from nuclear fission to a myriad of different nuclear-reaction channels. Correspondingly, a broad spectrum of interpretations have been stimulated. The process, when regarding nuclear reactions, can be studied in a laboratory and controlled up to a certain extent, yielding insights on the nuclear interaction. Less directly, extrapolations to dense matter in extended astrophysical objects can be searched, connecting microphysics to macrophysics. Beside this suggestive bridge between small and large scales, similar behaviours are displayed in other classical and quantum systems in condensed-matter physics. The interconnection between these distant fields invited to establish common theoretical frameworks, from energy density functionals to statistical approaches for the equation of state.

Given the breadth of the subject, rather than undertaking an interdisciplinary overview, we focus on a more specific topic: the fragmentation phenomenology in a nuclear system from nuclear matter to atomic nuclei. This topic is nowadays attracting increasing attention due to the recent capability of handling new degrees of freedom, from bosonic states to strangeness but, at least in the nuclear-physics sector, such step forward is still in a developing phase. As a second restriction, we focus therefore on aspects of fermionic dynamics in nucleonic systems (and we neglect for instance topics like alpha clustering at very low density, as well as very-large-density effects which characterise relativistic regimes. As a third restriction, we focus on “violent” processes, and hence an energy range situated around the Fermi regime. Due to analogies that will be elucidated in the text, we also include highly excited heavy remnants resulting from ion-ion or nucleon-ion collisions at relativistic energies: these systems are the useful intermediate step for moving from nuclear matter to heavy-ion collisions.

Such energy selection corresponds to situations where the formation of fragments in nucleonic matter is induced through a violent external action (mechanical or thermal). Without such external action, only a few radioactive isotopes of heavy elements can divide spontaneously into lighter nuclides when subjected to alpha decay or, more seldom, to spontaneous binary fission. Even more rarely, some actinides and few other elements may undergo binary and even ternary splits where the lightest products are clusters ranging from He up to the region of Mg. The tendency towards cluster decay reflects nuclear-structure properties and it can be enhanced when more states are involved in the decay scheme. To access more excited states and channels beyond spontaneous decay, an external action is needed to warm up the system.

In practice, this situation is achieved in various types of particle-nucleus and nucleus-nucleus collisions, and it is explored in nucleonic matter in very hot astrophysical objects like supernovae and their precursors. If the system is excited up to the Fermi regime, the widths of the energy levels fuse into a continuum and footprints of nuclear structure are lost. In this case, the system can be completely rearranged through a thermodynamic process. In particular, the phase-space of the system dilutes, so that nucleonic collisional correlations are no more removed by the Pauli exclusion principle, and they become a continuous seed of dynamical fluctuations and bifurcations. Such systems are characterised by a prominent tendency to break into fragments, displaying a huge variety of possible configurations. For this reason, we focus on the Fermi regime. Indicatively, heavy-ion collisions at incident energies of around 10 to 50 MeV per nucleon correspond to such regime. For larger bombarding energies, collective modes are gradually overwhelmed by nucleonic collisional correlations. Those latter finally prevail and completely determine the dynamics when reaching relativistic energies. However, even at these high energies, collective modes can be restored in the decay sequence, when one large and warm nuclear remnant emerges from the ashes of the reaction.

The purpose of this report is to analyse the above phenomenology within a dynamical optics which keeps the equivalence to the statistical description at equilibrium: to do this, we start from nuclear matter and from the dispersion relation, which is the foundation of a dynamical approach for large-amplitude perturbations. Such strategy solves many long-standing questions related to the incomplete picture on the nuclear reaction mechanisms which emerges from statistical or equilibrium approaches alone.


This report starts with a discussion on how to handle fluctuations in the dynamics of nuclear matter; a conceptually simple approach based on the Boltzmann-Langevin equation is proposed and taken as a reference throughout the review. An analysis of fluctuations in nucleonic Fermi fluids will follow. We will then move from nuclear matter to open nuclear systems and study the fluctuation-bifurcation phenomenology in heavy-ion collisions: we focus first of all on spallation mechanisms at relativistic energies. We progress to the study of nucleus-nucleus collisions at Fermi energies. Anytime it is possible, we investigate connections between the microscopic (transport) description and the macroscopic (statistical) description. The conclusions are perspectives.

2.2 Fragmenting nuclei: a brief history

The process of fragment formation in an induced nuclear reaction was already inspiring pioneering research efforts in the middle of the 1930s. Bohr might have brought about the challenge in the conclusions of a renowned publication suggesting that [Boh] when an atomic nucleus is bombarded “with particles of energies of about a thousand million volts, we must even be prepared for the collision to lead to an explosion of the whole nucleus”. This sentence was so prophetic that it become one of the bestselling citations in prefaces of books and monographs in nuclear physics!

Indeed, few years later, such scenario started to be outlined in many experiments where light particles (neutrons, protons, deuterons) where directed on a heavy nucleus. Already in his doctoral dissertation (1937) and later, in a 1947 publication, Seaborg [Sea] coined the term spallation as a nuclear process where the entrance channel is a light high-energy projectile hitting a heavy ion, and the exit channel consists of several particles, where heavy residues or fission fragments combine with light ejectiles, including a large neutron fraction. The newly introduced reaction was soon associated to technological applications like neutron sources, secondary beams, hadron therapy and accelerator-driven systems for energy production and transmutation; in astrophysics, spallation was associated to the cosmic-ray isotopic spectrum.

The same year, Serber [Ser] proposed a schematic description where the process essentially translates in a fast stage of excitation followed by a second stage of statistical decay from a fully thermalised compound nucleus [Tho], thus completing the scenario initiated by Weisskopf in his famous 1937 publication [Wei]. This very general description resulted very successful but its shortcoming was soon determined by further new findings. The picture become more involved when, firstly Nervik and Seaborg [Ner], concentrating on intermediate-energy experiments, and, later on, Robb Grover [Rob], focusing on relativistic energies, reported that a nucleus undergoing a violent collision does not only disintegrate into light particles but also into intermediate-mass fragments 1 (IMF), lighter than ordinary fission products but heavier than alpha particles. Still exploiting spallation reactions, pioneering experiments with particle accelerators [Kau, War, Hir, And, Bar, Kor, Kot, Hsi, Avd, Vio] further focused on IMF formation. Suggestive results on single events were provided by heavy projectiles impinging on photoemulsions [Jak].

\floatbox

[\capbeside\thisfloatsetupcapbesideposition=left,top,capbesidewidth=.45]figure[\FBwidth]

Figure 2.1: A schematic distinction among different energy regimes for heavy-ion collisions, in terms of De Broglie wavelength as a function of incident energy per nucleon of a projectile. For comparison, the radii of Pb and Ca are indicated. Two-body nucleon-nucleon collisions are suppressed at low energies, for large , they are recovered at Fermi energy and become dominant at relativistic energies, for small , leading to the participant-spectator picture.

The phenomenological picture became even richer when, almost contemporaneously, new measurements with heavy-ion beams from Fermi-energy to relativistic domains showed that nuclear systems can produce a sumptuous multifragmentation process in many IMF [Huf, Col, WCI]. Heavy-ion collisions could actually probe a large variety of perturbations acting on a nuclear system as a function of the incident energy. At least three regimes could be schematically distinguished, as indicated in Fig.2.1. Pauli-blocking factors of final orbitals largely suppress two-body nucleon-nucleon collisions at low energy (from the Coulomb barrier to around 15 MeV per nucleon). Those latter should on the other hand be included at Fermi energy (from about 15 to 200 MeV per nucleon) and become dominant in the participant-spectator regime (from about 200 MeV per nucleon to relativistic energies). Of course, transitions between these domains are smooth.

The conditions at Fermi energy are clearly peculiar with respect to higher energies because of a dominant mechanical contribution. It’s interesting to mention that the earliest models to describe fragment formation in such conditions were in the spirit of fission, or the liquid-drop model in general, and often similarities with normal classical liquids were searched. In this respect, Frölich [Fro] in 1973 and Griffin in 1976 [Gri] proposed a very similar scenario to the fragmentation of liquid jets observed by Rayleigh in 1882 [Ray]. The picture of the Rayleigh instability could describe efficiently the mechanism of peripheral collisions where a third fragment comes from the neck rupture [Mon]. The Rayleigh instability was proposed again also in more recent speculations about the possibility that nuclei could explode like a bubble [Mor] and, despite such mechanism does not seem to occur, it might actually offer insights for some violent out-of-equilibrium processes. Also the possibility of successive asymmetric fission events was advocated [Mor], or the extension of the saddle-point interpretation to highly excited systems [Lop], and resulted even successful in describing experimental data, as far as no observables could constrain the chronology of the process.

Still in parallel to early experiments, possible analogies with phase transitions, which are general phenomena occurring in interacting many-body systems [Sch, Hoc, Kim, Ste, Mor], were searched, and inspired improvements on the theoretical description along the avenue of statistical approaches [Bot, Bot, Bon, Rad, Gro], eventually coupled with hydrodynamic descriptions to take into account the entrance channel. In particular, due to the analogies between the nuclear forces and the Van-der-Waals interaction, the nuclear matter equation of state (EOS) foresees the possibility [Jac, Mul] of experiencing liquid-gas phase transitions connected to the appearing of a mixed vapour phase. In this spirit, early experiments on heavy-ion collisions were addressed to tracking signals of caloric curves [Poc, Sch, Nat], a subject that has been intensively discussed till recently [Zhe, Bor].

As soon as more accurate measurements confirmed that the nucleus can disassemble simultaneously (rather than sequentially) into fragments, and that the associated density profile describes dilute nuclear matter, the -order phase transition picture was taken as the underlying phenomenology [Gul, Cho, Gro].

EOS properties and finite-size effects become the research terrain for several years of experiments with heavy-ion beams in the Fermi-energy range where event-by-event correlation measurements and large angular acceptances were exploited widely [Sou], leading to new findings on the nuclear multifragmentation phenomenon [Bow, Dag, Bor, Mor].

In particular, by exploiting event-by-event correlations, more sophisticated thermodynamic analyses could trace first-order phase transitions in finite systems as characterised by negative specific heat and bimodal behaviour of the distribution of the order parameter [Bin, Cho, Cho]. The latter physically corresponds to the simultaneous presence of different classes of physical states for the same value of the system conditions that trigger the transition (like the temperature, for instance), and is a stronger observable for a first-order phase transition.

In fact, under suitable conditions, a bimodal character of experimental observables, such as the distribution of the heaviest cluster produced in each collision event [Bon], or the asymmetry between the charges of the two heaviest reaction products [Pic] has been interpreted as a signal of the finite-size counterpart of the liquid-gas phase transition in nuclear matter, which is associated to a finite latent heat [Gul]: the energy (and density) of the system and the size of the largest fragments are order parameters which rule the evolution of the system from a configuration where a large cluster survives at low excitation energy to a configuration where the system disassembles in small fragments at high-excitation. In this framework, liquid-gas coexistence related to density fluctuations trigger the multifragmentation scenario, rather than, like in fission, surface energy. For instance, the transition from a compact system to fission is one additional phase transition process triggered by the Coulomb field, without being associated to any latent heat [Leh].

The rich topic outlined above has been historically mostly addressed to from the point of view of statistical models, both on the theory side, and from the corresponding tracing of global experimental observables. Even though huge progress resulted from such approach, still many questions remained unanswered for long time and are still at the moment topic of research. We may mention three of them, among many others.

In spallation, data on kinematics and correlations were so far sufficient for supporting a general phenomenological framework relying on the fission-evaporation picture, but they were incomplete for also explaining the mechanism of production of light nuclides (see § 4).

In heavy-ion collisions at Fermi and intermediate energies the bimodal behaviour of fragment-production observables have been related to first-order phase transition in a statistical framework, but it was then argued that also dynamical approaches were producing similar signatures without necessarily requiring the reaching of thermodynamical equilibrium [LeF]. In this respect, the thermodynamics of heavy-ion collisions could not be disconnected from microscopic approaches, suited for handling fluctuations, when searching for solid EOS observables. All conceptual ingredients for microscopic approaches had already been prepared long in advance, but only recently they can be exploited in predictive models (see § 6.2).

In exotic mechanisms at the threshold between multifragmentation and fission, like neck fragmentation producing ternary or even quaternary events, approaches which are extension of fission models (like chains of binary fissions) can not be compatible with the kinematics of the mechanism which have been observed to be rather rapid (see § 6.3).

A fourth natural question would be whether it is possible to answer the three questions above within a common framework. On the experimental side, the increasing accuracy of correlation measurements could then bring the study of the origin of IMF formation, limited to a statistical picture of the exit channel, to the study of a process as a function of time, and also extended to out-of-equilibrium intervals of time. Those latter characterise the beginning of the process, they entirely characterise some specific channels and they allow to connect different mechanisms to the same entrance channel. They also allow to understand how the process of fragment formation evolves microscopically, in presence of instabilities or in presence of a threshold between different exit channels. The study of instability growing in the nuclear system, described as a Fermi liquid, progressed on the theoretical side in parallel with the research on the EOS, since the first experimental evidence of phase-transition behaviours. It has then been further accelerated by the discovery of the first signals of spinodal decomposition [Bor]. In this spirit, nuclear many-body dynamics and transport approaches emerged as an alternative powerful line of investigation where equilibrium assumptions could be dropped and which keeps the equivalence to the statistical description at equilibrium. Transport approaches have been firstly addressed to probe the reaction mechanisms associated to the occurrence of phase transitions [Aic, Lip, Mor, Col].

More recently, isospin effects related to the phase transition phenomenology [Cho, Duc], as well as the study of currents of neutrons and protons related to transport effects [Bar], extended the research to the related isospin physics [sym], further emphasising the need of dynamical approaches to search for features which exceed the statistical picture.

Chapter 3 Modelling instabilities

How can we describe the interplay between mean-field and many-body correlation effects in unstable nuclear matter? In this situation, where clusterisation is observed, the fluctuation of one-body densities determine both mechanical properties of fragments, like excitation, size or density, and how neutrons and protons are shared among emerging fragments and the surrounding environment. A theory which can handle these conditions can also be adapted to draw a microscopic description of heavy-ion collisions as a function of time. This is the ultimate purpose of this work. We briefly review in this chapter the reason why a Boltzmann-Langevin approach is well suited for this application.

Main sources for this chapter: this chapter exploits a publication in preparation [Nap]

3.1 Large-amplitude fluctuations in fermionic systems

The stochastic microscopic description of non-equilibrium large-amplitude dynamics of many-fermion systems is a long-standing challenge for theoretical modelling. Nowadays it is experiencing a strong acceleration due to unprecedented computing resources. Even more importantly, these microscopic descriptions are stimulating increasing interest since they have come within the experimental reach in many areas of physics at the same time, from heavy-ion collisions [WCI, sym] to solid-state physics (examples are metal clusters [Cal, Fen] or electrons in nanosystems [ChG]), to ultracold atomic gases [Dal, Blo] and, more indirectly, to some areas of astrophysics [Hor, Seb, Sch].

Large-amplitude collective motion of fermionic systems [Rin, Mar] is efficiently described with the time-dependent Hartree-Fock (TDHF) framework, or time-dependent local density approximation (TDLDA), in condense-matter applications [Yab, Rei], as far as the variance of the involved observables is small. However, when the system experiences a violent dynamics, fluctuations and bifurcations might have to be expected, determining the shortcoming of the TDHF approximation. Under these conditions, large fluctuations would tend spontaneously to drive the system far away from the mean-field TDHF evolution along many different directions. Depending on the degree of excitation, a large variety of solutions have been developed to include additional correlations into the mean-field representation, ranging from the description of small- to large-amplitude collective motion till addressing chaotic dynamics.

In this chapter we focus on fermionic systems in nuclear matter, but some concepts could have a larger interdisciplinary interest. In the nuclear context, rich discussions on these approaches have been collected in dedicated reviews, e.g. [Sim]. In this work we focus on those approaches which can be extended to the description of very-large-amplitude dynamics, and which can eventually handle bifurcating dynamical paths, i.e. the possibility that the system can oscillate between very different exit channels. This is for instance the case of threshold conditions between nuclear fusion and multifragmentation which may be encountered in heavy-ion collisions at Fermi energies. In this work we focus on stochastic approaches of nuclear dynamics based on mean-field models, and we skip discussions on molecular dynamics (either non-antisimmetrised [Aic, Aic, Pei, Pei, Har, Mar, Chi, Pap] or antisimmetrised [Fel, Fel, Ono]). Since early investigations, it was evident that fluctuations could be handled naturally by describing the propagation of several stochastic mean-field trajectories simultaneously, along a bundle of different dynamical paths. Intuitively, as schematically illustrated in Fig. 3.1, each trajectory is a possible “story” of the system tracked by the evolution of a given observable as a function of time.

Figure 3.1: Schematic illustration of mean-field trajectories developing from an initial state in absence (left) and in presence (right) of large-amplitude fluctuation. In the latter case, trajectories split apart in a pattern of bifurcating mean-field subensembles. Each subensemble of mean fields is characterised by small fluctuations within an envelope around a corresponding mean trajectory (sketched by a dashed line).

As far as the bunch of dynamical trajectories could be reduced to correlated channels (as typically in a low-energy framework), it is sufficient to adopt a scheme of coherent-state propagation in the small-amplitude limit as within the time-dependent generator coordinate method (TDGCM) framework [Rei, Gou], or a variational approach à la Balian-Vénéroni [Bal, Bal, Sim]. Beyond the single-particle picture, to handle large-amplitude regimes at low energy it is convenient to propagate non-correlated states [Lac]; even further beyond this picture, when dealing with a rather excited system, several degrees of freedom are explored in a large-amplitude dynamics. In particular, two-body nucleon-nucleon collisions are only partially suppressed by Pauli blocking and they should be taken into account. Accordingly, especially in presence of mean-field instabilities, collective motion may be driven to a chaotic regime; in a nuclear system, this would result in a highly non-linear process, like the formation of nuclear fragments. This is the case where it is not sufficient to describe larger spread widths of observables or dissipative behaviour, like in the extended TDHF (ETDHF) framework [Won, Won, Lac], but it is also required to follow bifurcation paths which deviate from the mean trajectory. In this situation, where the ensemble of mean-field trajectories to be handled is very large, it was proposed to adopt a coherent description in a statistical framework, leading to the original formulation of stochastic TDHF (STDHF) [Rei]. A more recent reformulation of such approach restricts the incoherent correlations to two-fermion collisional correlations [Sur, Sla].

Already in the earliest works, it was suggested to further reduce the quantal description to its semiclassical analogue, the Boltzmann Langevin Equation (BLE) [Ayi], in terms of semiclassical distribution functions. Among the various ways followed in deriving the BLE (Density matrix formalism [Ayi], truncation of the Bogolioubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy for density matrices [Bal], Green-function techniques [Rei]), it was also advocated that BLE is a semiclassical analogue of the STDHF approach when mean-field trajectories are reorganised in subensembles [Rei, Abe]. The semiclassical transport treatment defined by the BLE has the minimum requirement of keeping the Fermi statistics for the distribution functions; it is a kinetic equation of Boltzmann-Uehling-Uhlenbeck or Boltzmann-Nordheim-Vlasov type (BUU/BNV) [Ber, Cas, Bon] supplemented by a fluctuation contribution which is incorporated explicitly in the collision term and acts as a continuous source.

We introduce thereafter approaches to solve the BLE in periodic nuclear matter. We test the methods in periodic nuclear matter so as to be able to compare numerical results with analytic expectations and draw conclusions on the range of observables that the model could yield as well as its limitations. In particular, we focus on two fluctuation mechanisms which relate the properties of the effective interaction to the phenomenology observed in several violent nuclear processes. These mechanisms are driven by isoscalar fluctuations, which, when mean-field instabilities occur, lead to the disassembly of the nuclear system (according to the dispersion relation of the unstable modes) and isovector fluctuations, which should ideally reflect the strength of the nuclear symmetry energy.

3.2 Handling N-body correlations

3.2.1 General framework

It is usual to describe the evolution of an -body system by replacing the Liouville-von Neumann equation, which involves variables, by the equivalent BBGKY hierarchy of equations. Custom approximations can be applied by truncating or reducing the complexity of the hierarchy so as to obtain as many unknown variables as the number of equations: this is the avenue for constructing beyond-mean-field approximations. We recall that the kinetic equations for the one-body density matrix, including the BLE, can be obtained from a second-order truncation of the BBGKY hierarchy, for a two-body interaction  [Bal, Cas, Bon], i.e. considering the first two lines of the following chain of coupled equations, which represent the BBGKY hierarchy:

(3.1)

where is a partial trace involving the many-body density matrix and are kinetic energy operators.

The explicit inclusion of correlations beyond order k=2 would be necessary to describe high-coupling regimes [Lac], with also the inclusion of interactions beyond two bodies to include additional nuclear-structure features [Sch], or cluster correlations. On the other hand, a second-order scheme is well suited for our purpose of studying large-amplitude dynamics of many fermions, where we neglect structure effects.

3.2.2 Recovering effects beyond the second-order truncation through a stochastic treatment

The passages and approximations leading from the second order scheme to the usual form of the BLE are well documented in the literature [Ayi]. Let us simply spot the main manipulations which allow to obtain the highly non-linear character of the BLE [Ayi]. This scheme imposes to consider that for an interval of time there exist a subensemble of mean field trajectories for which fluctuations are small with respect to the mean trajectory ; fluctuations propagate keeping the mean unchanged (see Fig. 3.1 for illustration). This is equivalent to imposing that and , i.e. the probability to find two particles at two configuration points, are not decorrelated at all times, while fluctuations act keeping the mean unchanged. In this case, the two-body density matrix recovers some correlations of the upper orders of the BBGKY sequence and we can write at a time :

(3.2)
(3.3)
(3.4)

where is the Møller wave operator [Ree, Sur] describing the diffusion of a particle with respect to another particle in the nuclear medium, related to a diffusion matrix , which is on its turn related to the nucleon-nucleon differential cross section ; in other words, the first term of the RHS of eq.3.2 contains collision correlations while the second term expresses a fluctuation of vanishing first moment, which may be expressed as a fluctuating collision term, introducing a fluctuation around the collision integral [Abe]. We may remark that setting would impose a full decorrelation between and , and remove all orders . This would reduce to the (quantum) Boltzmann kinetic equation, which corresponds to a second-order truncation of the hierarchy without a fluctuation contribution.

Finally, the description associated with one single mean-field trajectory in an interval of time yields the following form, similar to STDHF:

(3.5)

where the residual terms and represent an average collision contribution and a continuous source of fluctuation seeds, respectively. Beyond the interval of time the fluctuating term replaces the mean field by a set of new mean-field subensembles , which may eventually yield bifurcations:

(3.6)

In the same spirit, in the similar framework of the quantal version of the Stochastic mean field theory in the first BBGKY order, from the analysis of correlation momenta, a demonstration was given that the handling of several mean fields has the effect of recovering in an approximate form part of the contribution from the orders of the hierarchy which have been cut off, leading to a simplified (rather than a truncated) BBGKY theory [Lac]. Such argumentation should also apply to Eq. (3.5). Indeed the fluctuations introduced in the theory may account for the unknown and neglected many-body correlations. Moreover, with respect to the quantal version of SMF of ref. [Lac], here we consider two-body effects explicitly, through the inclusion of the collision integral. It may be noted that Eq. (3.5) transforms into an ETDHF theory if the fluctuating term is suppressed. Through a Wigner transform we can then replace Eq. (3.5) by a corresponding set of semiclassical BL mean field trajectories

(3.7)

where the evolution of a statistical ensemble of Slater determinants is replaced by the evolution of an ensemble of distribution functions , which at equilibrium correspond to a correct Fermi statistics. is the effective Hamiltonian acting on . The residual average and fluctuating contributions of Eq. (3.5) are replaced by Uehling-Uhlenbeck (UU) analogue terms [Ueh]. is related to the mean number of transitions within a single phase-space cell . While conserving single-particle energies, acts as a Markovian contribution expressed through its correlation [Col]

(3.8)

where is a diffusion coefficient.

3.3 Obtaining the BLOB description

From Eq.(2) and from the procedure detailed in ref. [Ayi] we assume that the fluctuating term in Eq. (3.7) should involve the same contributions composing the average collision term , i.e. the transition and the Pauli-blocking terms. This implies that also should be expressed in terms of one-body distribution functions. This latter possibility can be exploited by replacing the residual terms by a similar UU-like term which respects the Fermi statistics both for the occupancy mean value and for the occupancy variance. In this case, the occupancy variance at equilibrium should be equal to in a phase-space cell and correspond to the movement of extended portions of phase space which have the size of a nucleon, i.e. the residual term should carry nucleon-nucleon correlations. A natural solution to satisfy such requirement is to rewrite the residual contribution in the form of a similar rescaled UU collision term where a single binary collision involves extended phase-space portions of equal isospin , to simulate wave packets, and Pauli-blocking factors act on the corresponding final states , , also treated as extended phase-space portions. The choice of defining each phase-space portion , , and so that its isospin content is either or is necessary to preserve the Fermi statistics for both neutrons and protons, and it imposes that blocking factors are defined accordingly in phase space cells for the given isospin species. The above conditions lead to the Boltzmann-Langevin One Body (BLOB) equations [Nap ]:

(3.9)

where is the degeneracy factor, integrations are over momenta and scattering angles , and is the transition rate, in terms of relative velocity between the two colliding phase-space portions and differential nucleon-nucleon cross section

(3.10)

and contains the products of occupancies and vacancies of initial and final states over their full phase-space extensions.

(3.11)

Fluctuations and bifurcations, due to higher order correlations than the =2 truncation are created through a stochastic approach which exploits the correlations carried in Eq. (3.9).

3.4 Implementation of Boltzmann-Langevin approaches

The stochastic term in Eq. (3.7) can be kept separate and treated as a stochastic force related to an external potential . Such approach was followed in an early implementation by Suraud and Belkacem [Sur, Bel], where a fluctuating term is obtained from exploiting the quadrupole and octupole momenta of the local momentum distribution in configuration space. Again, this strategy is used in the Brownian One Body (BOB) model [Gua], where a fluctuating term is prepared by associating a Brownian force to a stochastic one-body potential, or in the stochastic mean field (SMF) model [Col], where the fluctuating term corresponds to kinetic equilibrium fluctuations of a Fermi gas. While in the first approach fluctuation seeds were injected at the beginning of the dynamical process, as undulations in the spacial density landscape, in the second approach they could be injected at successive intervals of time . Nevertheless, in both treatments, fluctuations are implemented only in the coordinate space, i.e. they are projected on the spacial density.

The difference between Eq. (3.9) and usual stochastic mean-field approaches is that those latter build fluctuations from introducing a well adapted external force or a distribution of initial conditions which should be accurately prepared in advance. On the contrary, Eq. (3.9) introduces fluctuations in full phase space intermittently, at any time, letting them develop spontaneously and continuously during the whole dynamical process.

Table 3.1 collects the main numerical implementations of Boltzmann-Langevin approaches which led, so far, to a transport model; the list is not exhaustive because it does not consider approaches which derived from those given in the list, and some details are described thereafter.

Model formal. coll. Pauli seeds from: time phase sp.
STDHF [Rei] quant. no yes m.f. subens. full
Bauer   [Bau] semicl. yes no collisions full
Suraud  [Sur] semicl. yes yes q/o momenta projected
BOB    [Gua] semicl. yes yes ondulation projected
SMF     [Col] semicl. yes yes external field projected
BLOB [Nap ] semicl. yes yes collisions full

extended phase-space portions moved at once in one binary-collision event.
two test particles moved in one binary-collision event (standard UU).
see remarks in ref. [Cha] and § 3.5.
exploiting mean-field subensembles.

Table 3.1: Boltzmann-Langevin numerical implementations handling large amplitudes (other implementations exist but, to our knowledge, they are derived from those reported in this table). The table indicates, for each model, the formalism, the presence of a collision term, whether Pauli blocking is fully satisfied, the type of fluctuation seeds, the time when fluctuation seeds are injected (either intermittently, at time intervals or at the beginning only), and whether fluctuations act on full phase space or if they are projected on a suitable subspace.

3.5 Exploiting correlations in BLOB and handling metrics

In the BLOB framework, eq. (3.9) is exploited to generate stochastic dynamical paths in phase space. The system is sampled through the usual test-particle method, often adopted for the numerical resolution of transport equations [Ber], with the difference that, in the case of the BLOB implementation, the phase-space portions and involved in single two body collisions are not two individual test particles but rather agglomerates of test particles of equal isospin, where is the number of test particles per nucleon used in the simulations. The initial states and are constructed by agglomeration around two phase-space sites, which are sorted at random, inside a phase space cell of volume h, according to the method proposed in ref. [Nap ] and further improved in ref. [Nap]. At successive intervals of time, by scanning all phase space in search of collisions, all test-particle agglomerates are redefined accordingly in cells, so as to continuously restore nucleon-nucleon correlations. Since test particles could be sorted again in new agglomerates to attempt new collisions in the same interval of time, the nucleon-nucleon cross section contained in the transition rate should be divided by to give the scaled cross section used in eq. (3.10):

(3.12)

The metrics of the test particle agglomerates is defined in such a way that the packet width in coordinate space is the closest to , where corresponds to the screened cross section prescription proposed by Danielewicz [Dan, Cou], which was found to describe recent experimental data  [Lop]. In this way, the spatial extension of the packet decreases as the nucleon density increases.

Boltzmann-Langevin solutions where an ensemble of test particles are moved in one bunch and the nucleon-nucleon cross section is scaled by where already followed in the early approach by Bauer and Bertsch [Bau], or in more recent implementations [Mal]. There is however a very fundamental difference: in the Bauer-and-Bertsch approach the Pauli-blocking term is not applied to the involved portions of phase space which are actually interested by the scattering at a given time , as imposed by Eq. (3.11), but it is applied only to the centroids of the two colliding packets. Such approximation makes the Pauli blocking satisfied only approximately, with the drawback of loosing the Fermi statistics [Cha]. In the direction of BLOB, to prevent the above problem, a first practical solution was proposed in ref. [Riz]. These augmentations also complete the survey of table 3.1.

Figure 3.2: (a) Definition of test-particle agglomerates in their initial (,) and final (,) states in momentum space in a volume. (b) Convergence of a binary nucleon-nucleon collision configuration towards a situation where Pauli blocking is strictly satisfied. The path collects the sequence of modulations in phase-space of the test-particle clouds where the occupancy of the destination regions is iteratively optimised. (c) Examples of possible shape modulations for a final state.

In BLOB, special attention is paid to the metrics when defining the test-particle agglomeration: the agglomerates are searched requiring that they are the most compact configuration in the phase space metrics which does neither violate Pauli blocking in the initial and in the final states, nor energy conservation in the scattering. For this purpose, when a collision is successful, its configuration is further optimised by modifying the shape and the width of the initial and final states [Nap]. Fig. 3.2 illustrates the paths of a collision configuration which by a procedure of successive modulations is brought to a situation which respects Pauli blocking strictly. The resulting occupancy functions of the modulated final-state density profiles should possibly approach unity. If such modulation procedure results unsuccessful, the collision is rejected. The rate of rejections due to unsuccessful modulation of the collision configuration is close to zero in open systems (collisions) so that the correlation between attempted and effective collision number is identical if a UU or a BLOB collision term is applied to the same mean-field, provided that the same nucleon-nucleon cross section is used. On the other hand, in uniform nuclear matter at equiliubrium, where only nucleons close to the Fermi surface can be involved in two-body collisions, the occurrence of such rejections becomes not negligible when the temperature considered is very low compared to the Fermi momentum. In this case, the perfect correspondence between attempted and effective collision rates in BUU (or SMF) and BLOB is lost (see § 4.3.1 for further insight).

A remarkable advantage of the renormalised form of the residual contribution in Eq. (3.9) is to connect directly the fluctuation variance to the local properties of the system, regardless the test-particle number. Such aspect has a general relevance because it makes their phenomenology independent from many aspects of the numerical implementation. The dependence on persists on the other hand in the mean-field representation, therefore when the fluctuation amplitude is small the global fluctuation phenomenology may suffer from noise effects produced by the use of a finite number of test particles in the numerical implementation of the transport equation. This remark should be taken in mind for the study of fluctuations of relatively small amplitude, like isovector fluctuations. The interplay between physical and numerical fluctuations will be carefully investigated in the following.

Chapter 4 Inhomogeneity growth in two-component fermionic systems

The advantage of one-body approaches with collisional correlations is to sample aspects of the behaviour of Fermi liquids [Lif, Pin] and to allow including them in the description of heavy-ion collisions [Pet ]. In this chapter, we focus on the fluctuation phenomenology in nuclear matter at moderate temperature and in several density conditions.

In particular we check how Eq. (3.9) handles the fluctuation variance of isoscalar and isovector one-body densities in equilibrated nuclear matter, in comparison with analytical expectations for fermionic systems interacting through effective forces. We aim to demonstrate that the implementation of Eq. (3.9) is better suited than approximate methods, like SMF, to sample physical observables related to inhomogeneity development (which is equivalent to fragment formation in finite open systems).

We focus on dynamical fluctuations, skipping discussions on the possible forms of the nuclear interaction and its isospin dependence, for which schematic descriptions are employed. We refer readers interested in the details of the nuclear interaction and its isospin dependence to the widespread literature on the topic, se e.g. [Bar, Li, Dan, Dan, sym]. The purpose of this chapter is demonstrating that by solving the BLE in full phase space it is possible to describe the dispersion relation successfully and to enhance the isovector fluctuations with respect to the standard BUU. Such approach can therefore be applied to nuclear processes like heavy-ion collisions ensuring that observables related to the form of the nuclear potential and to the associated instabilities can be described efficiently.

Main sources for this chapter: this chapter exploits a publication in preparation [Nap] and some recent talks [Nap].

4.1 Application to nuclear matter

Thereafter, both BLOB and SMF models are prepared as relying on a strictly identical implementation of the mean field, so that they differ only for the residual contribution. This study has a more general relevance, being intended to compare a BL approach where fluctuations are included as projected on the density landscape and a BL approach where fluctuations are introduced in full phase space. We employ SMF and BLOB to represent the first and the second strategy, respectively.

A simplified Skyrme-like (SKM) effective interaction [Gua, Bar], where momentum dependent terms are omitted, is employed in the propagation of the one-body distribution function, corresponding to the following definition of the potential energy per nucleon:

(4.1)

with , being the saturation density and . This parameterization, with MeV, MeV and , corresponds to a soft isoscalar equation of state with a compressibility modulus  MeV. An additional term as a function of the density-gradient introduces a finite range of the nuclear interaction and accounts for some contribution from the zero-point motion of nucleons [Gua]. is related to various properties of the interaction range, like the surface energy of ground-state nuclei (the best fit imposing a value of MeV fm), the surface tension (light-fragment emission, in comparison to available data is better described for a smaller range given by MeV fm in BLOB), and the ultraviolet cutoff in the dispersion relation for wavelengths in the spinodal instability (larger spectrum for a smaller range). Either a linear (asy-stiff) or a quadratic (asy-soft) density dependence [Col] of the potential part of the symmetry energy coefficient, , is obtained by setting, respectively:

asy-stiff (4.2)
asy-soft (4.3)

If not otherwise specified, in this work the collision term involves an isospin- and energy-dependent free nucleon-nucleon cross section with an upper cutoff at  mb.

To simulate nuclear-matter properties, we fix the system density and the temperature . We prepare the system in a cubic periodic box of edge size  fm, and we subdivide it in a lattice of cubic cells of edge size where we calculate density variances. For the sake of simplicity, we consider symmetric nuclear matter, i.e. with equal number of neutrons and protons. We initially define the system imposing a uniform-matter effective field depending only on the density considered, and a corresponding effective Hamiltonian . Accordingly, the phase-space distribution function , not depending on configuration space (because the system is homogeneous), is the Fermi-Dirac equilibrium distribution at the given temperature , for a chemical potential .

4.2 Fluctuations in nuclear matter and the Boltzmann-Langevin equation

Either from the stochastic fluctuating residual term of the BLOB treatment or from an external stochastic force of the SMF approach we introduce a small disturbance in uniform matter

(4.4)

which lets a fluctuation develop in time around the mean trajectory .

By considering neutron and proton distributions functions, we can further decompose fluctuations in isoscalar modes and isovector modes

(4.5)
(4.6)

The time evolution of both those modes is obtained by applying the BL equation (3.7) to the phase-space fluctuations. For symmetric matter, and retaining only first order terms in , one obtains:

(4.7)

where the index stands either for isoscalar () or isovector () modes, and is an external stochastic force (SMF) or a fluctuating stochastic field (BLOB). We dropped the average collision term because we consider small temperatures.

To build our stochastic descriptions we assumed that, at least locally, fluctuation have small amplitude around their mean trajectory so that . When the system is described as a periodic box, collective modes are associated to plane waves of wave number . In this case, by expanding on plane waves expressed in Fourier components, we can study the evolution in time of phase space density fluctuations

(4.8)

and undulations in the density landscape

(4.9)

Rewritten in Fourier components, and substituting , Eq. (4.7) takes the form

(4.10)

where and are Fourier components of the potential and of the stochastic fluctuating field , respectively.

When only stable modes can propagate, the response of the system to the action of the stochastic fluctuating field determines the equilibrium variance associated with the fluctuation . The inverse Fourier transform of gives the equilibrium variance of spacial density correlations

(4.11)

in a cell of volume at temperature . At equilibrium, when the level density for a degeneracy can be defined, these variances are related to the curvature of the free energy density through the fluctuation–dissipation theorem so that

(4.12)

where , and for an average extending over all modes.

On the other hand, for unstable modes, the diffusion coefficient , or rather its projection on a given unstable mode , , determines the following evolution for the intensity of response for the wave number  [Col, Col]:

(4.13)

where both the initial fluctuation seeds and the fluctuation continuously introduced by the collisional correlations contribute to an exponential amplification of the disturbance, characterised by the growth time .

In the following, starting from Eq. (4.10), we select two very instructive situations which are isovector modes in uniform matter and, successively, isoscalar fluctuations in unstable matter. Translated into a violent nuclear-collision scenario, the first situation defines how isospin distributes among different phases and portions of the system, and the second situation coincides with the process of separation of those portions of the system into fragments.

4.3 Isovector fluctuations in nuclear matter

Isovector effects in nuclear processes may arise from different mechanisms [Bar, DiT], like the interplay of isospin and density gradients in the reaction dynamics, or nuclear cluster formation, or the decay scheme of a compound nucleus. Alternately, in systems undergoing a nuclear liquid-gas phase transition, a role is played also by isospin distillation [Cho, Bar], a mechanism which consists in producing a less symmetric nucleon fraction in the more volatile phase of the system along the direction of phase separation in a space, as an effect of the potential term in the symmetry energy [Duc, Col].

Thus, it is particularly interesting to analyse the developing of isovector fluctuations in two-component nuclear matter. Those latter correspond to phase-space density modes where neutrons and protons oscillate out of phase. In processes where fragments arise rapidly, like in first-order phase transitions, isovector fluctuations contribute in determining the isotopic properties of the low- and high-density fractions which compose the mixed phase.

Selecting isovector modes () in Eq. (4.10), the phase-space density corresponds to . In order to isolate the isovector behaviour, we prepare nuclear matter in stable conditions. To keep nuclear matter uniform (no inhomogeneities will arise in configuration space), we keep only the isovector contribution in the potential, in absence of isoscalar terms, and we rely on the quantity

(4.14)

where is the uniform-matter density and is the potential term in the symmetry energy. Following the procedure of ref. [Col], is obtained from the above quantity by introducing an interaction range through a Gaussian smearing of width , and by taking the Fourier transform; its derivative with respect to yields

(4.15)

Substituting in Eq. (4.12) we obtain a relation between the isovector variance and the symmetry free energy

(4.16)

where can be assimilated to an effective symmetry free energy which, at zero temperature and neglecting surface effects, coincides with the symmetry energy, . In conventional BUU calculations however, the smearing effect of the test particles introduces a corresponding scaling factor [Col], so that

(4.17)

Such scaling actually reduces drastically the isovector fluctuation variance produced by the UU collision term. In this paragraph we investigate how the collision term used in the BLOB approach differs from employing a UU treatment. Since the former is not an average contribution and it acts independently of the number of test particles, we expect a larger isovector fluctuation variance.

Figure 4.1: Energy spectra for and  fm. The coloured bands indicate the variation of a Fermi-Dirac distribution from to  MeV, the first value corresponding to the initial conditions. Spectra evaluated at  fm/c, through the SMF and BLOB approaches are overlapped. An asy-stiff parameterisation is used.

To prepare a transport calculation, the system is sampled for several values of and the potential, restricted to the only isovector contribution, is tested for a stiff and a soft density dependence of the symmetry energy.

The system is initialised with a Fermi-Dirac distribution at a temperature  MeV. As shown in Fig. 4.1, both SMF and BLOB transport dynamics succeed to preserve the initial distribution quite efficiently as a function of time, even though a flattening of the spectrum, due to the fact that the Fermi statistics is not perfetly preserved, around an effective , should be accounted for. This temperature modification depends on the parameters of the calculation and is larger for larger densities.

Figure 4.2: Numerical solution of Eq. (4.16) for SMF, including the normalisation by for asy-stiff and asy-soft forms of the symmetry energy and for different sizes of the cells where the isovector variance is evaluated. The temperature correction is applied as equal to calculated at saturation density, or extracted for each density bin. For the asy-stiff case a calculation relying only on a collisionless dynamics is added.

From a set of calculations for different densities ranging from to  fm we obtain a numerical solution of Eq. (4.16) for SMF. We use the average equilibrium temperature  MeV, calculated at saturation density, for all other densities or, alternatively an equilibrium temperature extracted for each density bin from the slope of the Fermi-Dirac distribution evolved in time. The isovector variance has been calculated in cells of edge size , and  fm and multiplied by , in order to extract and to compare it with the symmetry energy . The comparison, shown in Fig. 4.2, is satisfactory and it is the closest in shape to for larger cells than  fm but, however, the large scaling factor had to be taken into account. The better agreement in larger cells reflects the decreasing importance of surface effects, which should allow recovering the (volume) symmetry energy. We notice that an equivalent calculation where the collision term is suppressed yields identical distributions; such collisionless calculation corresponds to switching off the collision term either in SMF or, equivalently, in BLOB, since the mean-field is implemented identically. The need of scaling by , to recover the expected fluctuation value, reflects the fact that isovector fluctuations are not correctly implemented in SMF, and the fluctuations which arise in the system are related to the use of a finite number of test particles. Indeed, in SMF, much attention is paid to a good reproduction of isoscalar fluctuations and amplification of mean-field unstable modes, by introducing an appropriate external field [Col]. On the other hand, explicit fluctuation terms are not injected in the isovector channel. In this case one just obtains the fluctuations related to the use of a finite number of test particles, which, as far as the Fermi statistics is preserved, amount to the physical ones divided by .

Figure 4.3: Mean-value and variance of the number of effective collisions per nucleon for SMF and BLOB calculated at equilibrium (the average collision rate is constant in time) as a function of density. Dashed lines are quadratic fits to the mean-value specra; the fit to mean-values is repeated over the variance spectrum to show for the BLOB calculation the perfect correspondence, and for the SMF calculation the correspondence with the variance scaled by Ntest. Calculation with an increased number are presented for BLOB to show the independence on . An asy-stiff parameterisation is used.

This fluctuation reduction in SMF is illustrated in Fig. 4.3. When studying the mean and variance of the number of effective collisions per nucleon in SMF, a negligible variance is found even if the mean grows with density. In practice, within the UU description, larger densities provide a larger number of collision candidates, so that, even if also the difficulty in relocating collision partners increases due to the Pauli blocking, the resulting number of effective collisions per nucleon grows significantly with density. Though, the corresponding collision variance does not follow such trend, keeping a dependence on density reduced by a factor with respect to the mean. This completes the study of ref.[Col] concerning SMF. Now we move to discuss also BLOB calculations. Fig. 4.3 indicates that, despite the use of the same nucleon-nucleon cross section (which produces equal rates of attempted collisions per nucleon for all the employed approaches, not shown), the number of effective collisions per nucleon differs in the two models due to the different treatment of the Pauli blocking, which is more severe in BLOB, owing to the nucleon wave packet extension (for instance, 98% is the Pauli rejection rate in BLOB at  fm). However, the study presented in Fig. 4.3 confirms that the variance of the number of effective collisions per nucleon in BLOB is large and exactly equals the mean value, according to the Poisson statistics [Bur].

Figure 4.4: Isovector variance as a function of time for different values of the system density for an asy-stiff form of the symmetry energy, for SMF and for BLOB, evaluated in cells of different size . BLOB systematically shows larger values and a longer convergence time for large densities. Both calculation converge to the same variance at the small-density limit.
Figure 4.5: Convergence value of isovector variance as a function of system density for asy-stiff and asy-soft forms of the symmetry energy evaluated in cells of different size , for BLOB and SMF. For one case (asy-stiff,  fm) other calculations are added: a collisionless calculation is identical to SMF; various BLOB calculations employing a constant with different values exhibits a dependence on the nucleon-nucleon cross section.

Fig. 4.4 shows that the isovector variance in BLOB results larger than in SMF and, in general, larger than in a corresponding collisionless calculation. Such difference is therefore the effect of the treatment of collisional correlations in BLOB, which displays a dependence with the system density. In particular, the low-density limit of the spectrum corresponds to a situation where the collision rate is vanishing. In this case, the BLOB procedure is practically ineffective (see § 4.3.1) and all approaches converge to the same isovector variance, just related to the finite number of test particles employed. At larger density than saturation ( fm) BLOB displays a longer paths to convergence which is due to the difficulty of relocating large portions of phase space in binary collisions without violating Pauli blocking. Fig. 4.5 condenses and extends the information of Fig. 4.4 by displaying the density evolution of the isovector variance attained at equilibrium as evaluated in cells of different size , for asy-stiff and asy-soft forms of the symmetry energy. The SMF data correspond to those analysed in Fig. 4.2. The BLOB spectra progressively deviate from SMF data for increasing density. Such deviation increases for larger cell sizes indicating that the isovector fluctuations are better built in large volumes [Riz]. This is related to the variety of configurations, concerning shape and extension of the nucleon wave packet, which occur in the implementation of the fluctuating collision integral. This introduces a smearing of fluctuations on a scale comparable to the wave packet extension in phase space. However, the gain in isovector variance exhibited by the BLOB approach, indicates that the dependence on is partially reduced with respect to the SMF scheme.

4.3.1 Interference between mean field, collisional correlations and numerical noise

According to Eq. (3.9), the BLOB approach should introduce and revive isovector fluctuations continuously. However, the procedure has chances to work only if there are no other antagonist sources which destroy isovector correlations. The agglomeration procedure employed in BLOB is actually able to construct agglomerates of test particles of the same isospin species and which are located around local density maxima in random selected phase-space cells: this technique should preserve at least partially the isovector correlations in the system, contrarily to the usual BUU technique which smears them out. This advance with respect to BUU is however far from being sufficient because the greatest smearing effect comes from the mean field itself which, even in absence of any explicit fluctuation seed, is actually affected by its own numerical noise, due to the use of a finite number of test particles in the numerical resolution of the transport equations ; such spurious contribution imposes the dependence of on  [Col]. If this latter may be negligible with respect to the large isoscalar fluctuations introduced by the BLOB stochastic collision term (in presence of instabilities), it becomes a highly interfering contribution for the weaker isovector modes. As discussed in ref. [Rei], the use of a finite number of test particles, i.e. the approximate mapping of the one-body distribution function, induces a numerical noise that may even cause deviation from the fermionic statistics, towards a classical behaviour of the system. This effect is more pronounced when the collision integral is neglected. Indeed the latter contains explicit Pauli-blocking factors and helps restoring the fermionic behaviour. The numerical noise leads, on a short time scale, to fluctuations corresponding to the expected value, but reduced by (as far as the Fermi statistics is still preserved). In other words, the numerical noise induces an effective diffusion coefficient and an effective relaxation time . depends on the test-particle number (becoming infinite when the test-particle number goes to infinity). If a small number of test particles is considered, and two-body collisions are not so frequent, then will prevail over and over . For this reason, though in principle the fluctuation equilibrium value should not depend on the nucleon-nucleon cross section, our results are sensitive to the cross section amplitude and to the number of test particles employed.

Figure 4.6: Study of dependence of isovector variance for asy-stiff forms of the symmetry energy  fm, evaluated in cells of different size , for a collisionless approach, SMF with constant =50 mb, BLOB with constant =50 and 100 mb. Left. dependence of the mean value at equilibrium as a function of for various models compared to a low. Right. Time dependence showing the tendency to lose dependence for BLOB with constant =100 mb.

Two ways can be tested to overcome this problem: either the collision term should be considerably enhanced, or fluctuations generated by the action of test particles should be controlled.

The first solution can be achieved by simply multiplying the nucleon-nucleon cross section by a large factor, with the drawback of then handling incorrect collision rates which would be a severe concern when describing out-of-equilibrium processes, like heavy-ion collisions. Some tests in the first direction are proposed in Fig. 4.5, by employing a constant with progressively larger values, showing that the isovector variance grows with the collision rate as expected. As shown in Fig. 4.5, we observe that the fluctuation variance built by BLOB may deviate significantly, up to a factor 10, from the SMF results, especially when considering larger cells () to evaluate the one-body density.

The second solution would consist in employing the largest possible number of test particles per nucleon. In this case, the collisionless transport model would ideally correspond to the Vlasov approach and, when collisional correlations are introduced, interferences with spurious stochastic sources can be highly reduced. As far as numerical complexity can be handled, Fig. 4.6, left, illustrates such situation: SMF and collisionless calculations show the same behaviour and no dependence on . On the other hand, BLOB calculations show a dependence, which deviates more and more, for large test particle numbers, from the SMF results, especially in the largest cells, where the BLOB fluctautions are better entertained. Fig. 4.6, right, illustrates that small, progressively increasing values of , are related to a systematically decreasing isovector variance, which is still completely dominated by the noise. Only when the number of test particles per nucleon becomes very large, the isovector variance loses its dependence on and exhibits a clear tendency to grow in time towards a larger value, signing that isovector correlations are not only preserved, but they are also revived. However, since we are interested in physical conditions at low temperature, the number of nucleon-nucleon collisions is extremely low and insufficient to rapidly introduce a pattern of isovector correlations: the isovector variance shows in fact a very gentle growth.

In conclusion, the BLOB fluctuation source term works well in conditions where the collision rate is large enough, as compared to the spurious dissipative terms associated with the finite number of test particles and to the mean-field propagation. These conditions are likely reached in the first, non equilibrated stages of heavy ion collisions at Fermi and intermediate energies, but not necessarily for equilibrated nuclear matter at low temperature. In the latter case, the variance associated with the fluctuating collision integral can be recovered by artificially increasing the n-n cross section employed in the calculations.

4.4 Isoscalar fluctuations in mechanically stable and unstable nuclear matter

If fluctuation seeds are introduced in homogeneous neutral nuclear matter at low temperature, Landau zero-sound [Lan] collective modes should stand out and propagate in the system. We analyse in the present section whether, as aimed, the BLOB approach is able to develop isoscalar fluctuations of correct amplitude spontaneously, and not from an external contribution, in nuclear matter when the system is placed in a dynamically unstable region of the equation of state [Bel], like the spinodal zone, where a density rise is related to a pressure fall. In this circumstance, as soon as fluctuation seeds are generated, unstable zero-sound waves should be amplified in time. In the opposite situation, in conditions of mechanical stability, undamped stable zero-sound waves propagate. When unstable modes succeed to get amplified, inhomogeneities develop and eventually lead to mottling patterns at later times. This mechanism has been intensively investigated [Cho] foremost because in open dissipative systems, like heavy-ion collisions, it corresponds to a catastrophic process which can lead to the formation of nuclear fragments [Tab, Bor]. When the temperature is significant, two-body collision rates become prominent and these mean-field dominated zero-sound waves are absorbed and taken over by hydrodynamical first-sound collective modes. Since our approach exploits two-body collisions to introduce fluctuations in a self-consistent mean field, we expect the possible occurrence of a zero-to-first-sound transition which, at variance with other Fermi liquids [Abe], should be even smeared out due to the small values taken by the Landau parameter in nuclear matter. It was found that, depending on how the system is prepared and on the type of collective motion, such transition should arise in a range of temperature from 4 to 5 MeV and occur as late as 200 fm/c [Lar, Kol]. In practice, zero-sound modes associated to wave vectors characterise the system as long as the corresponding phase velocity exceeds the velocity of a particle on the Fermi surface or, equivalently, as long as the corresponding frequency is much higher then the two-body collision frequency . These premises imply that after defining a homogeneous initial configuration at a suited finite and not so large temperature, we should study early intervals of time to extract properties of the response function which can be compared with zero-sound conditions.

Figure 4.7: Two choices for the system density are illustrated in relation with corresponding quantities. (a) Nuclear potential. (b) Landau parameter. (c) real () and imaginary () roots of the dispersion relation; the spinodal region corresponds to , the region for corresponds to Landau damping and positive values of define stable modes. The arrow indicates that the point for  fm is situated at a very negative value of the ordinate. (d) Temperature effects on the dispersion relation for the two system densities.

For the numerical approach we keep the previous scheme for the definition of the box metrics; the isoscalar density variance is calculated over cells of edge size fm. We now use the full parametrisation of the energy potential per nucleon Eq. (4.4), where we use a stiff density dependence of (the same parametrisation was analysed in ref. [Col]). A value of MeV fm is chosen for the surface term. Nuclear matter is isospin symmetric and is initially uniform and prepared at a temperature MeV and a densities equal to and  fm. Fig. 4.7a illustrates the potential values related to these choices. test particles per nucleon are employed. The collision term involves the usual isospin- and energy-dependent free nucleon-nucleon cross section with an upper cutoff at mb.

4.4.1 Sampling zero-sound propagation

The early growth of fluctuations in nuclear matter can be described in a linear-response approximation [Col] as far as deviations from the average dynamical path are small. In Eq. (4.10), by selecting isoscalar modes (, we drop the index in the following), and setting residual contributions equal to zero, we obtain a linearised Vlasov equation in terms of frequencies to describe nuclear matter with isoscalar contributions:

(4.18)

Different wave numbers are decoupled, each linked to a collective solution given by the Fourier-transformed equation of motion. By applying the self-consistency condition

(4.19)

we obtain the dispersion relation for the propagation of density waves in Fermi liquids at :

(4.20)

where and are pair solutions due to the invariance . As well documented in the literature, at , eigenmodes depend on states near the Fermi level. The momentum integral should therefore be restricted to the Fermi surface so that , being the Fermi energy, and angular and energy dependencies can be decoupled so that the dispersion relation reduces to an expression where solutions correspond to sound velocities

(4.21)

in units of Fermi velocity . In this case, introducing the Landau parameter

(4.22)

linked to the number of levels at Fermi energy , the dispersion relation takes the form of the Lindhard function [Kal]

(4.23)

where the dependence on has been removed by the introduction of the sound speed , equal for all waves. For the two selected system densities, Fig. 4.7b illustrates the Landau parameter and Fig. 4.7c presents the roots of the dispersion relation.

Such expression, derived for zero temperature, is not consistent with the incorporation of temperature effects through a short-mean-free-path two-body dissipative mechanism. Nevertheless, it was proposed [Yan] that the inclusion of a temperature dependence is still possible in a semiclassical picture when supposing that a moving boundary of the system is involved in the dissipation of energy from collective to microscopic degrees of freedom. Such so-called wall-dissipation model [Blo] can be applied to a Fermi gas at finite but small temperature by identifying the moving boundary with the Fermi surface of the system. This argumentation results in including the ratio between the chemical potential at a temperature and the Fermi energy , which carries the temperature dependence

(4.24)

as illustrated in Fig. 4.7d for the two selected densities.

As a further modification, we consider that zero-sound conditions also present a strong dependence on the interaction range. This latter can be included in the dispersion relation by applying a Gaussian smearing factor of the mean-field potential , which is related to the nuclear interaction range in configuration space [Col, Kol].

(4.25)

From Eq. (4.24) and Eq. (4.25), the dispersion relation, Eq. (4.23), will involve an effective Landau parameter,

(4.26)

Mechanically unstable conditions are experienced when the evolution of local density and pressure implies that the incompressibility is negative, so that this situation is reflected by an effective Landau parameter smaller than :

(4.27)

and it corresponds to imaginary solutions of the dispersion relation [Pom]. By replacing , the relation yielding imaginary solutions can be put in the form:

(4.28)

where . The growth rate is obtained from the solutions of the dispersion relation