The spinodal amplification of density fluctuations is treated perturbatively within dissipative fluid dynamics for the purpose of elucidating the prospects for this mechanism to cause a phase separation to occur during a relativistic nuclear collision. The present study includes not only viscosity but also heat conduction (whose effect on the growth rates is of comparable magnitude but opposite), as well as a gradient term in the local pressure, and the corresponding dispersion relation for collective modes in bulk matter is derived from relativistic fluid dynamics. A suitable two-phase equation of state is obtained by interpolation between a hadronic gas and a quark-gluon plasma, while the transport coefficients are approximated by simple parametrizations that are suitable at any degree of net baryon density. We calculate the degree of spinodal amplification occurring along specific dynamical phase trajectories characteristic of nuclear collision at various energies. The results bring out the important fact that the prospects for spinodal phase separation to occur can be greatly enhanced by careful tuning of the collision energy to ensure that the thermodynamic conditions associated with the maximum compression lie inside the region of spinodal instability.
High-energy nuclear collisions produce quark-gluon plasmas that expand and hadronize. If the associated phase transition is of first order then the hadronization should proceed through a spinodal phase separation. We explore here the possibility of identifying the associated clumping by analysis of suitable N-particle momentum correlations.
Extending a previously developed two-phase equation of state, we simulate head-on relativistic lead-lead collisions with fluid dynamics, augmented with a finite-range term, and study the effects of the phase structure on the evolution of the baryon density. For collision energies that bring the bulk of the system into the mechanically unstable spinodal region of the phase diagram, the density irregularities are being amplified significantly. The resulting density clumping may be exploited as a signal of the phase transition, possibly through an enhanced production of composite particles.
We recently introduced a fluid-dynamical model for simulating relativistic nuclear collisions in the presence of a first-order phase transition and made explorative studies of head-on lead-lead collisions. We give here a more detailed account of this novel theoretical tool and carry out more exhaustive studies of the phase-separation dynamics. Extracting the density enhancement caused by the spinodal instabilities, the associated clump size distribution, and the resulting transverse flow velocity, we examine the sensitivity of these quantities to the strength of the gradient term that promotes…
A density dependent relativistic mean-field model is determined to reproduce the components of the nucleon self-energy at low densities. This model is used to investigate spinodal instabilities in isospin asymmetric nuclear matter at finite temperatures. The inhomogeneous density distributions in the spinodal region are investigated through calculations in a cubic Wigner-Seitz cell. Compared to results obtained in phenomenological calculations the spinodal region is large, i.e. the spinodal region at zero temperature can reach densities above 0.12 fm$^{-3}$. The predicted spinodal region is co…
The coordinate and momentum space configurations of the net baryon number in heavy ion collisions that undergo spinodal decomposition, due to a first-order phase transition, are investigated using state-of-the-art machine-learning methods. Coordinate space clumping, which appears in the spinodal decomposition, leaves strong characteristic imprints on the spatial net density distribution in nearly every event which can be detected by modern machine learning techniques. On the other hand, the corresponding features in the momentum distributions cannot clearly be detected, by the same machine lea…
As an extension of our previous work, the growth of density fluctuations in the spinodal region of charge asymmetric nuclear matter is investigated in the basis of the stochastic mean-field approach in the non-relativistic framework. A complete treatment of density correlation functions are presented by including collective modes and non-collective modes as well.
Early growth of density fluctuations of nuclear matter in spinodal region is investigated employing the stochastic mean-field approach. In contrast to the earlier treatments in which only collective modes were included in the calculations, in the present work non-collective modes are also included, thus providing a complete treatment of the density correlation functions. Calculations are carried out for symmetric matter in non-relativistic framework using a semi-classical approximation.
The Vlasov-Nordheim equation is solved numerically on a lattice for nuclear matter in two dimensions. We discuss the reliability of the model at normal density and then study the response of the system to small perturbations. We find deterministic chaos inside the spinodal zone where fragment formation occurs. We discuss in detail the dynamical features of this phenomenon in order to clarify the mechanisms leading to nuclear disassembly in heavy--ion collisions.
The phase diagram of nuclear matter is quite rich - it shows such phenomena as phase-transitions, formation of condensates, clustering, etc. From the analysis of the spinodal instability, one can learn about the region of liquid-gas coexistence in nuclear matter at low densities and finite isospin asymmetries. In a recent paper, we have shown that asymmetric nuclear matter at sub-nuclear densities should undergo only one type of instability. The associated order parameter is dominated by the isoscalar density and so the transition is of liquid-gas type. The instability goes in the direction of…
The expansion of strongly interacting matter formed in high-energy nuclear collisions drives the system through the region of phase coexistence. The present study examines the associated spinodal instability and finds that the degree of amplification may be sufficient to raise the prospect of using the spinodal pattern formation as a diagnostic tool for probing the hadronization phase transition.
Quantal effects on growth of spinodal instabilities in charge asymmetric nuclear matter are investigated in the framework of a stochastic mean field approach. Due to quantal effects, in both symmetric and asymmetric matter, dominant unstable modes shift towards longer wavelengths and modes with wave numbers larger than the Fermi momentum are strongly suppressed. As a result of quantum statistical effects, in particular at lower temperatures, magnitude of density fluctuations grows larger than those calculated in semi-classical approximation.
We investigate the impact of pairing correlations on the behavior of unstable asymmetric nuclear matter at low temperature. We focus on the relative role of the pairing interaction, coupling nucleons of the same type (neutrons or protons), with respect to the symmetry potential, which enhances the neutron-proton attraction, along the clusterization process driven by spinodal instabilities. It is found that, especially at the transition temperature from the normal to the superfluid phase, pairing effects may induce significant variations in the isotopic content of the clusterized matter. This a…
We demonstrate, by numerical simulations, that the dynamics of nuclear matter mean field inside the spinodal region is chaotic. Spontaneous symmetry-breaking - no explicit fluctuating term is considered - occurs leading to wild unpredictable density fluctuations. A proper recipe to calculate an average Lyapunov exponent in this multidimensional phase space is introduced. The latter is calculated for different values of the density in order to characterize in a quantitative way the chaotic and regular regions. It is argued that the mean field chaoticity can be the main mechanism of the nuclear …
Spinodal instabilities and early growth of baryon density fluctuations in symmetric nuclear matter are investigated in the basis of stochastic extension of relativistic mean-field approach in the semi-classical approximation. Calculations are compared with the results of non-relativistic calculations based on Skyrme-type effective interactions under similar conditions. A qualitative difference appears in the unstable response of the system: the system exhibits most unstable behavior at higher baryon densities around $ρ_{b}=0.4 ~ρ_{0}$ in the relativistic approach while most unstable behavior o…
We analyze the spinodal instabilities of spin polarized asymmetric nuclear matter at zero temperature for several configurations of the neutron and proton spins. The calculations are performed with the Brueckner--Hartree--Fock (BHF) approach using the Argonne V18 nucleon-nucleon potential plus a three-nucleon force of Urbana type. An analytical parametrization of the energy density, which reproduces with good accuracy the BHF results, is employed to determine the spinodal instability region. We find that, independently of the of the orientation of the neutron and proton spins, the spinodal ins…
Asymmetric nuclear matter at sub-saturation densities is shown to present only one type of instabilities. The associated order parameter is dominated by the isoscalar density and so the transition is of liquid-gas type. The instability goes in the direction of a restoration of the isospin symmetry leading to a fractionation phenomenon. These conclusions are model independent since they can be related to the general form of the asymmetry energy. They are illustrated using density functional approaches.
Based on IBUU simulations calibrated by previous efforts of the transport model evaluation project, we have studied the specific shear viscosity $η/s$ of nuclear matter in the spinodal region using the Green-Kubo method. With the momentum-independent mean-field potential which reproduces reasonably well empirical nuclear matter properties and nuclear phase diagram, we have generated dynamically stable and thermalized nuclear cluster systems in a box with the periodic boundary condition. Extensive results of the $η/s$ at different average densities and temperatures in uniform and non-uniform sy…
Density fluctuations of expanding nuclear matter are studied within a mean-field model in which fluctuations are generated by an external stochastic field. Fluctuations develop about a mean one-body phase-space density corresponding to a hydrodinamic motion that describes a slow expansion of the system. A fluctuation-dissipation relation suitable for a uniformly expanding medium is obtained and used to constrain the strength of the stochastic field. The distribution of the liquid domains in the spinodal decomposition is derived. Comparison of the related distribution of the fragment size with …
Nuclei undergo a phase transition in nuclear reactions according to a caloric curve determined by the amount of entropy. Here, the generation of entropy is studied in relation to the size of the nuclear system.
We study the spinodal instabilities of asymmetric nuclear matter at finite temperature within the microscopic Brueckner--Hartree--Fock (BHF) approximation using the realistic Argonne V18 nucleon-nucleon potential plus a three-body force of Urbana type. Our results are compared with those obtained with the Skyrme force SLy230a and the relativistic mean field models NL3 and TW. We find that BHF predicts a larger spinodal region. This result is a direct consequence of the fact that our Brueckner calculation predicts a larger critical temperature and saturation density of symmetric nuclear matter …
We investigate the dynamical properties of asymmetric nuclear matter at low density. The occurrence of new instabilities, that lead the system to a dynamical fragment formation, is illustrated, discussing in particular the charge symmetry dependence of the structure of the most important unstable modes. We observe that instabilities are reduced by charge asymmetry, leading to larger size and time scales in the fragmentation process. Configurations with less asymmetric fragments surrounded by a more asymmetric gas are favoured. Interesting variances with respect to a pure thermodynamical predic…
The diabatic approach to collective nuclear motion is reformulated in the local-density approximation in order to treat the normal modes of a spherical nuclear droplet analytically. In a first application the adiabatic isoscalar modes are studied and results for the eigenvalues of compressional (bulk) and pure surface modes are presented as function of density and temperature inside the droplet, as well as for different mass numbers and for soft and stiff equations of state. We find that the region of bulk instabilities (spinodal regime) is substantially smaller for nuclear droplets than for i…
Based on an effective QCD Lagrangian we discuss the properties of charge density fluctuations in the vicinity of chiral phase transition. We explore thermodynamics in the presence of spinodal phase separation. We show that appearance of spinodal decomposition in a non-equilibrium first order phase transition results in divergence of the charge density fluctuations related with the electric charge and baryon number conservation. Consequently, divergent fluctuations at the chiral phase transition are not only attributed to the critical end point but are also there along the first order phase tra…
Using transport theory to model central Au + Au collisions in the energy region of 20 - 110 MeV/u, at impact parameters b <= 5 fm, we predict a measurable impact of spinoidal instability as the collective expansion sets in with energy. Two transport models are employed, the pBUU model, solving a Boltzmann-Uehling-Uhlenbeck equation, and the Brownian Motion (BM) model, solving a set of Langevin equations to describe the motion of individual nucleons in a noisy nuclear medium. We find without ambiguity, for the first time, that a combination of delayed equilibration, onset of collective expansio…
We study the effect of strong magnetic fields, of the order of $10^{18}$-$10^{19}$ G, on the instability region of nuclear matter at subsaturation densities. Relativistic nuclear models both with constant couplings and with density dependent parameters are considered. It is shown that a strong magnetic field can have large effects on the instability regions giving rise to bands of instability and wider unstable regions. As a consequence we predict larger transition densities at the inner edge of the crust of compact stars with strong magnetic field. The direction of instability gives rise to a…
The thermodynamics of a first order chiral phase transition is considered in the presence of spinodal phase separation within the Nambu-Jona-Lasinio (NJL) model. The properties of the basic thermodynamic observables in the coexistence phase are discussed for zero and non-zero quark masses. We focus on observables that probe the chiral phase transition. In particular, the behavior of the specific heat and entropy as well as charge fluctuations are calculated and analyzed. We show that the specific heat and charge susceptibilities diverge at the isothermal spinodal lines. We determine the scalin…
Spinodal instability in nuclear matter with light cluster degrees of freedom
INFN, Laboratori Nazionali del Sud（意大利国家核物理研究所，南实验室）；Department of Physics and Astronomy “Ettore Majorana”, University of Catania（卡塔尼亚大学埃托雷·马约拉纳物理与天文学系）
Effects of viscosity and thermal conductivity on the dynamics of first-order phase transitions are studied. The nuclear gas-liquid and hadron-quark transitions in heavy-ion collisions are considered. We demonstrate that at non-zero thermal conductivity, $κ\neq 0$, onset of spinodal instabilities occurs on an isothermal spinodal line, whereas for $κ=0$ instabilities take place at lower temperatures, on an adiabatic spinodal.
This paper focuses on the isospin properties of the asymmetric nuclear-matter liquid-gas phase transition analyzed in a mean-field approach, using Skyrme effective interactions. We compare two different mechanisms of phase separation for low-density matter: equilibrium and spinodal decomposition. The isospin properties of the phases are deduced from the free-energy curvature, which contains information both on the average isospin content and on the system fluctuations. Some implications on experimentally accessible isospin observables are presented.
Results on dissipative isoscalar modes of a hot and dilute nuclear droplet are presented. As compared to the adiabatic limit (part I), realistic dissipation yields a substantial reduction of the growth rates for all unstable modes, while the area of spinodal instability in the ($\varrho$,T)-plane remains unchanged. The qualitative features of multifragmentation through spinodal decomposition as obtained in the adiabatic limit are not significantly affected by dissipation.
Using a recently developed method for the inclusion of fluctuation in the BUU dynamics, we study the self-consistent propagation of inherent thermal noise of unstable nuclear matter. The large time behaviour of the evolving system exhibits synergism between fluctuation and non-linearities in a universal manner which manifest in the appearance of macroscopic structure in the average description.
Self consistent quantum approaches are used to study the instabilities of finite nuclear systems. The frequencies of multipole density fluctuations are determined as a function of dilution and temperature, for several isotopes. The spinodal region of the phase diagrams is determined and it appears that instabilities are reduced by finite size effects. The role of surface and volume instabilities is discussed. It is indicated that the important chemical effects associated with mechanical disruption may lead to isospin fractionation.
We demonstrate that the instabilities of asymmetric nuclear matter at sub-saturation densities do not present two types of instabilities as usually discussed but a unique one. The associated order parameter is everywhere dominated by the isoscalar density and so the transition is of liquid-gas type even in the so-called chemical instability region.
Within the covariant density functional theory of nuclear matter we build equations of state of $Δ$-admixed compact stars. Uncertainties in the interaction of $Δ(1232)$ resonance states with nuclear matter, due to lack of experimental data, are accounted for by varying the coupling constants to scalar and vector mesonic fields. We find that, over a wide range of the parameter space allowed by nuclear physics experiments and astrophysical observations, cold catalyzed star matter exhibits a first order phase transition which persists also at finite temperature and out of $β$-equilibrium in the n…
If the deconfinement phase transformation of strongly interacting matter is of first-order and the expanding chromodynamic matter created in a high-energy nuclear collision enters the corresponding region of phase coexistence, a spinodal phase separation might occur. The matter would then condense into a number of separate blobs, each having a particular net strangeness that would remain approximately conserved during the further evolution. We investigate the effect that such `strangeness trapping' may have on strangeness-related hadronic observables. The kaon multiplicity fluctuations are sig…
The thermodynamics of a first-order chiral phase transition is considered in the presence of spinodal phase separation using the Nambu-Jona-Lasinio model in the mean field approximation. We focus on the behavior of conserved charge fluctuations. We show that in non-equilibrium the specific heat and charge susceptibilities diverge as the system crosses the isothermal spinodal lines.
The density fluctuations of nuclear matter are studied within a mean-field model in wich fluctuations are generated by an external stochastic field. The constraints imposed on the random force by the fluctuation-dissipation theorem are analyzed. It is shown that in the proximity of the borders of the spinodal region the assumption of a white-noise stochastic field can be reliably used. The domain distribution of the liquid phase in the spinodal decomposition of nuclear matter is derived. The related distribution of fragment sizes compares favourably with the experimental fragment distribution …
We consider the modification of the Cahn-Hilliard equation when a time delay process through a memory function is taken into account. We then study the process of spinodal decomposition in fast phase transitions associated with a conserved order parameter. Finite-time memory effects are seen to affect the dynamics of phase transition at short times and have the effect of delaying, in a significant way, the process of rapid growth of the order parameter that follows a quench into the spinodal region. These effects are important in several systems characterized by fast processes, like nonequilib…
We investigate the speed of sound in nuclear matter at finite temperature and density~(chemical potential) in the nonlinear Walecka model. The numerical results suggest that the behaviors of sound speed are closely related to the the nuclear liquid-gas (LG) phase transition and the associated spinodal structure. The adiabatic sound speed is nonzero at the critical endpoint (CEP) in the mean field approximation. We further derive the boundary of vanishing sound velocity in the temperature-density phase diagram, and point out the region where the sound wave equation is broken. The distinction be…
Isotopic fluctuations in fragment formation are investigated in a quasi-analytical description of the spinodal decomposition scenario. By exploiting the fluctuation-dissipation relations the covariance matrix of density fluctuations is derived as a function of the wave vector for nuclear matter at given values of density, charge asymmetry, temperature, and of the time that the system spends in the instability region. Then density fluctuations in ordinary space are implemented with a Fourier transform performed in a finite cubic lattice. Inside this box, domains with different density coexist, …
We examine the dynamics of statistical fluctuations in nuclear matter. Linear response functions for the average phase space density are derived within Landau theory. Properties of the stochastic forces are deduced from the quantal fluctuation dissipation theorem, with a suitable generalization to unstable modes. Sizable quantum effects are found both inside and outside the spinodal regime.
We study the fully nonlinear time evolution of a holographic system possessing a first order phase transition. The initial state is chosen in the spinodal region of the phase diagram, and includes an inhomogeneous perturbation in one of the field theory directions. The final state of the time evolution shows a clear phase separation in the form of domain formation. The results indicate the existence of a very rich class of inhomogeneous black hole solutions.
We discuss the features of instabilities in asymmetric nuclear matter, in particular the relation between the nature of fluctuations, the types of instabilities and the properties of the interaction. We show a chemical instability appears as an instability against isoscalar-like fluctuations. Then starting from phenomenological hadronic field theory (QHD), including exchange terms, we discuss the symmetry energy and the relation to the dynamical response inside the spinodal region.
In the present work we include the isovector-scalar $δ$-meson in the quark-meson coupling model (QMC) and study the properties of asymmetric nuclear within QMC without and with the $δ$-meson. Recent constraints set by isospin diffusion on the slope parameter of the nuclear symmetry energy at saturation density are used to adjust the model parameters. The thermodynamical spinodal surfaces are obtained and the instability region at subsaturation densities within QMC and QMC$δ$ models are compared with mean-field relativistic models. The distillation effect in the QMC model is discussed.
The density and temperature dependence of the nuclear symmetry free energy is investigated using microscopic two- and three-body nuclear potentials constructed from chiral effective field theory. The nuclear force models and many-body methods are benchmarked to properties of isospin-symmetric nuclear matter in the vicinity of the saturation density as well as the virial expansion of the neutron matter equation of state at low fugacities. The free energy per particle of isospin-asymmetric nuclear matter is calculated assuming a quadratic dependence of the interaction contributions on the isospi…
Using the test-particle method to solve the transport equation derived from the Nambu-Jona-Lasino (NJL) model, we study how phase separation occurs in an expanding quark matter like that in a heavy ion collision. To test our method, we first investigate the growth rates of unstable modes of quark matter in a static cubic box and find them to agree with the analytical results that were previously obtained using the linear response theory. In this case, we also find the higher-order scaled density moments to increase with time and saturate at values significantly larger than one, which correspon…
Dynamics of dilute nuclear matter with light clusters and in-medium effects
Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Catania（意大利国家核物理研究所（INFN）卡塔尼亚分部）；INFN, Laboratori Nazionali del Sud（意大利国家核物理研究所（INFN）南方实验室）；Dipartimento di Fisica e Astronomia（物理与天文学系）
Nuclear thermodynamics studies evidenced the existence of a first order phase transition, namely of the liquid-gas type, without paying attention to the isospin degree of freedom. On the other hand, only few results with the introduction of the isospin variable have been so far obtained. Moreover above all a key question remains. It concerns the origin of the dynamics of the phase transition: spinodal instabilities or not with possible consequences and new signatures related to the introduction of the isospin variable.
We construct a simple two-phase equation of state intended to resemble that of compressed baryon-rich matter and then introduce a gradient term in the compressional energy density to take account of fintie-range effects in non-uniform configurations. With this model we study the interface between the two coexisting phases and obtain estimates for the associated interface tension. Subsequently, we incorporate the finite-range equation of state into ideal or viscous fluid dynamics and derive the collective dispersion relation for the mechanically unstable modes of bulk matter in the spinodal reg…
A canonical ensemble model is used to describe a caloric curve of nuclear liquid-gas phase transition. Allowing a discontinuity in the freeze out density from one spinodal density to another for a given initial temperature, the nuclear liquid-gas phase transition can be described as first order. Averaging over various freeze out densities of all the possible initial temperatures for a given total reaction energy, the first order characteristics of liquid-gas phase transition is smeared out to a smooth transition. Two experiments, one at low beam energy and one at high beam energy show differen…
Asymmetric nuclear matter is investigated in the low density region below the nuclear saturation density. Microscopic calculations based on the Dirac Brueckner Hartree-Fock (DBHF) approach with realistic nucleon-nucleon potentials are used to adjust a low density functional. This functional is constructed on a density expansion of the relativistic mean field theory which allows a clear interpretation of the role of the mesons to the equation of state. It is shown that a correction term should be added to the functional in order to take into account the effects beyond the mean field. Two functi…
The existence of phase transitions from liquid to gas phases in asymmetric nuclear matter (ANM) is related with the instability regions which are limited by the spinodals. In this work we investigate the instabilities in ANM described within relativistic mean field hadron models, both with constant and density dependent couplings at zero and finite temperatures. In calculating the proton and neutron chemical potentials we have used an expansion in terms of Bessel functions that is convenient at low densities. The role of the isovector scalar $δ$-meson is also investigated in the framework of r…
A multi-task learning (MTL) framework, called gradient kernel ridge regression, for nuclear masses and separation energies is developed by introducing gradient kernel functions to the kernel ridge regression (KRR) approach. By taking the WS4 mass model as an example, the gradient KRR network is trained with the mass model residuals, i.e., deviations between experimental and theoretical values of masses and one-nucleon separation energies, to improve the accuracy of theoretical predictions. Significant improvements are achieved by the gradient KRR approach in both the interpolation and the extr…
We perform a three-dimensional, finite temperature Skyrme-Hartree-Fock study of inhomogeneous nuclear matter to determine the critical density and temperature for the phase transition between the pasta phase and homogeneous matter and its properties. We employ four different parametrizations of the Skyrme nuclear energy-density functional, SkM$^*$, SLy4, NRAPR and SQMC700, which span a range of saturation-density symmetry energy behaviours constrained by a variety of nuclear experimental probes. For each of these interactions we calculate free energy, pressure, entropy and chemical potentials …
The critical properties for the transition to warm, asymmetric, non-homogeneous nuclear matter are analysed within a thermodynamical spinodal approach for a set of well calibrated equations of state. It is shown that even though different equations of state are constrained by the same experimental, theoretical and observational data, and the properties of symmetric nuclear matter are similar within the models, the properties of very asymmetric nuclear matter, such as the one found inside of neutron stars, differ a lot for various models. Some models predict larger transition densities to homog…
