Inhomogeneity growth in two-component fermionic systems

The dynamics of fermionic many-body systems is investigated in the framework of Boltzmann–Langevin (BL) stochastic one-body approaches. Within the recently introduced Boltzmann–Langevin one-body (BLOB) model, we examine the interplay between mean-field effects and two-body correlations, of stochastic nature, for nuclear matter at moderate temperature and in several density conditions, corresponding to stable or mechanically unstable situations. Numerical results are compared with analytic expectations for the fluctuation amplitude of isoscalar and isovector densities, probing the link to the properties of the employed effective interaction; namely, symmetry energy (for isovector modes) and incompressibility (for isoscalar modes). For unstable systems, clusterization is observed. The associated features are compared with analytical results for the typical length and timescales characterizing the growth of unstable modes in nuclear matter and for the isotopic variance of the emerging fragments. We show that the BLOB model is generally better suited than simplified approaches previously introduced to solve the BL equation, and it is therefore more advantageous in applications to open systems, such as heavy-ion collisions.

Figure 1

(left) Mean-value and (right) variance of the number of effective in-medium collisions per nucleon and per unit time $\delta t=1\mathrm{fm}/c$ for SMF (full black circles) and BLOB, as a function of density. The different open symbols correspond to BLOB calculations with different numbers of test particles. The dashed lines in the left panel represent fits to mean-value spectra and are replotted identically in the right panel for comparison. The crosses in the right panel correspond to the SMF variance (black dots) multiplied by $N_{\mathrm{test}}$. An asy-stiff parametrization is used for the symmetry energy (the characteristics of the interactions are defined in Sec. 4).

Figure 2

The effective symmetry energy, scaled by $N_{\mathrm{test}}$, as extracted from SMF simulations, using Eq. (22), for nuclear matter at temperature $T=3$ MeV and at several density values. The different symbols correspond to the three edge sizes $l$ of the cells over which the isovector variance is evaluated. Two parametrizations of the symmetry energy are considered in the calculations: asy-soft (left panel) and asy-stiff (right panel). In both panels, the dot-dashed line represents the analytical expression of the symmetry energy. For the asy-stiff case (right panel), a SMF calculation where the collision term is suppressed is also shown for the edge size $l=2.05$ fm (full violet line). Error bars indicate the standard deviation of the $F_{\mathrm{eff}}^{\mathrm{v}}/N_{\mathrm{test}}$ distribution obtained considering several time instants within a large interval at equilibrium.

Figure 3

Isovector variance as a function of time calculated, in a single event, with (left) SMF and (right) BLOB, for different values of the system density, corresponding to the different colors and ranging from $0.02{\mathrm{fm}}^{-3}$ (blue line) to $0.2{\mathrm{fm}}^{-3}$ (red line). Cells of different size $l$ are considered in the calculations: (top) $l=1$ fm, (middle) $l=2.05$ fm, and (bottom) $l=3.55$ fm. An asy-stiff form of the symmetry energy is considered.

Figure 4

The isovector variance, at equilibrium, as a function of the system density, calculated with SMF (full circles) and BLOB (open circles), for two parametrizations of the symmetry energy: (left) asy-soft and (right) asy-stif. Cells of different size $l$ are considered in the calculations: (top) $l=1$ fm, (middle) $l=2.05$ fm, and (bottom) $l=3.55$ fm. For one case (asystiff, $l=2.05$ fm) other calculations are shown: SMF with the collision term suppressed (full violet line), and BLOB calculations for different constant values of ${\sigma }_{\mathrm{NN}}$ (green lines). Error bars indicate the standard deviation of the isovector variance distribution obtained considering several time instants within a large interval at equilibrium.

Figure 5

(left) The isovector variance, at equilibrium, as a function of the number of test particles, $N_{\mathrm{test}}$, as obtained in SMF simulations (open circles) and in BLOB calculations (open squares), using a constant ${\sigma }_{\mathrm{NN}}=50$ mb, for nuclear matter at density ${\rho }^0=0.08{\mathrm{fm}}^{-3}$ and temperature $T=3$ MeV. BLOB calculations are also shown for ${\sigma }_{\mathrm{NN}}=100$ mb (green full circles). Cells of different size $l$ are considered, as indicated in the figure. Dashed lines represent a fit of SMF simulations, assuming a trend proportional to the inverse of the test particle number. (right) Time evolution of the isovector variance, as obtained in a single event, for BLOB calculations with constant ${\sigma }_{\mathrm{NN}}=100$ mb and $l=2.0\mathrm{fm}$. The different curves correspond to different number of test particles employed in the simulations. The asy-stiff form of the symmetry energy is considered.

Figure 6

(a) The isoscalar mean-field potential as a function of the density. The dots indicate the two density values considered in SMF and BLOB simulations. (b) The Landau parameter $F_0$ as a function of the density. (c) Real (for $F_0>0$) and imaginary (for $F_0<0$) roots of the dispersion relation; the spinodal region corresponds to $F_0<-1$, the region for $-1<F_0<0$ corresponds to Landau damping and positive values of $F_0$ define stable modes. Blue dots indicate the $\gamma$ values corresponding to the density ${\rho }^0=0.053{\mathrm{fm}}^{-3}$. The $\gamma$ value corresponding to ${\rho }^0=0.14{\mathrm{fm}}^{-3}$ is beyond the interval shown in the figure (in the direction of the green arrow). (d) The chemical potential, divided by the Fermi energy, as a function of temperature $T$ for the two density values considered in the calculations: ${\rho }^0=0.053{\mathrm{fm}}^{-3}$ (blue line) and ${\rho }^0=0.14{\mathrm{fm}}^{-3}$ (green line). The dots correspond to the temperature ($T=3$ MeV) considered in the simulations.

Figure 7

BLOB calculation. Response intensity ${\tilde{\sigma }}_k^2\left(t\right)={\sigma }_k^2\left(t\right)/{\sigma }_k^2(t=0)$, averaged over 100 dynamical paths, as a function of the wave number $k$. The different curves correspond to different time instants, as indicated on the figure. (left) Results for ${\rho }^0=0.053{\mathrm{fm}}^{-3}$ (spinodal). (right) Results for ${\rho }^0=0.14{\mathrm{fm}}^{-3}$ (Landau damping).

Figure 8

Early time evolution of the response intensity ${\tilde{\sigma }}_k^2\left(t\right)$ for several modes. $n=j$ stands for all $k$ modes within $2\pi (j-1)/L\le k<2\pi j/L$. (a) BLOB calculations for ${\rho }^0=0.053{\mathrm{fm}}^{-3}$ (spinodal). (b) SMF calculations for ${\rho }^0=0.053{\mathrm{fm}}^{-3}$. (c) BLOB calculations for ${\rho }^0=0.14{\mathrm{fm}}^{-3}$ (Landau damping).

Figure 9

BLOB calculations for nuclear matter at ${\rho }_0=0.053{\mathrm{fm}}^{-3}$ and temperature $T=3$ MeV: The growth rate ${\mathrm{\Gamma }}_k$, as extracted from the simulations [see Eq. (35)], is plotted as a function of the wave number $k$ (full line with dots). Uncertainties are evaluated from the linear-response fit. The dashed lines correspond to analytical predictions, obtained by solving Eq. (32) with different values of the width $\sigma$ of the Gaussian smearing factor. For comparison, a calculation in one dimension within the approach of Ref. [33] is also shown (green diamonds).

Figure 10

(left) Density-landscape at several time instants for nuclear matter in a periodic box at ${\rho }^0=0.053{\mathrm{fm}}^{-3}$ and $T=3$ MeV and (right) in a hot open nuclear system formed in a head-on ${}^{136}\mathrm{Xe}+{}^{124}\mathrm{Sn}$ collision at $32A\mathrm{MeV}$. Arrows indicate the beam direction. Both simulations employ the BLOB approach with the same mean-field properties. The big arrow proposes an analogy between nuclear matter and the open system in correspondence with the rise of the spinodal instability.

Figure 11

BLOB simulation of head-on ${}^{136}\mathrm{Xe}+{}^{124}\mathrm{Sn}$ collisions at $32A\mathrm{MeV}$. (upper row) Isotopic distribution (full dots) in potential ripples containing $N^{\prime }$ neutrons and $Z^{\prime }$ protons for cluster-forming configurations with masses around $A^{\prime }=15$ (left) and $A^{\prime }=24$ (right) at $t=130$ fm/$c$. The lines correspond to analytic distributions [see Eq. (36)], corresponding to the density extracted from different portions of potential ripples (see text). (middle row) The same as in the upper row, but at $t=200$ fm/$c$. (bottom) Average isospin content measured in potential ripples at two different times, as indicated on the figure, as a function of ${\rho }_{\mathrm{well}}$. Larger density values correspond to prefragments of larger mass. The dot-dashed line with an arrow indicates the asymmetry of the projectile-target composite system.

Figure 12

(a) Schematic representation of the definition of test-particle agglomerates in their initial ($A,B$) and final ($C,D$) states in momentum space in a $h^3$ volume. (b) Convergence of a binary nucleon-nucleon collision configuration towards a situation where Pauli blocking is strictly satisfied. The path in the plane determined by the occupancy numbers, $f_C$ and $f_D$, of the final states collects the sequence of modulations in phase space of the test-particle clouds where the occupancy of the destination regions is iteratively optimized. (c) Examples of possible shape modulations for the packets associated with nucleons in the final state.

Figure 13

(top) Energy distribution for ${\rho }^0=0.08$ and (bottom) $0.16{\mathrm{fm}}^{-3}$, as obtained in SMF (dashed line) and BLOB (full line) calculations at $t=80\mathrm{fm}/c$. The colored band connects the Fermi–Dirac distributions corresponding to $T=3$ and $T=5$ MeV.