Correlating isothermal compressibility to nucleon fluctuations in the inner crust of neutron stars

The question of how and which physical observables or thermodynamic parameters can best predict the onset of a possible phase transition in the inner crust of neutron stars remains largely unresolved. Using semiclassical Monte Carlo simulations, we investigate the isothermal compressibility and density fluctuations in a region of relevance to the dynamics of the inner crust. We show that the isothermal compressibility serves as a robust observable to characterize the transition from the nonuniform crust to the uniform core for proton fractions over 0.2. Moreover, we show explicitly how the two-component isothermal compressibility, computed using the Kirkwood-Buff theory, is directly connected to the fluctuations in the number density, recorded in the grand canonical ensemble by monitoring the number of particles in a small volume located at the center of the simulation box. That is, we compute mean-square particle fluctuations and compare them against the isothermal compressibility for different proton fractions. Although our results show that the mean-square particle fluctuations are proportional to the isothermal compressibility, the lack of a perfect correlation is attributed to the relatively small number of particles included in the simulations. The nonunity slope observed in the dimensionless isothermal compressibility—total nucleon fluctuation variance relationship suggests that the inner crust of neutron stars is composed of anisotropic and inhomogeneous matter.

Figure 1

Schematic illustration of a configuration confined to a simulation box of length $L$. For large systems, one could in principle implement a grand canonical formulation by adopting the central sphere of radius $R$ as the system of interest and letting the rest of the box act as the particle reservoir. The orange arrows show the possible trajectories of some nucleons during the MC simulation.

Figure 2

Snapshots of thermalized nucleon configurations for a variety of baryon densities, at proton fractions of (a) 0.1, (b) 0.2, and (c) 0.4, and a temperature of $T=0.5$ and $1.5\mathrm{MeV}$. Red and green solid circles depict protons and neutrons, respectively. For space limitations, we do not show nucleon configurations at the intermediate temperature range of $T=0.75–1.25$.

Figure 3

$\rho T{\kappa }_T$ versus (a)–(c) $T$ and (d)–(f) $\rho$ for different proton fractions of 0.1, 0.2, and 0.4, respectively. The various lines are added to guide the eye.

Figure 4

(a) An example for the number of nucleons inside the central sphere as a function of $s$ for $\rho =0.03{\mathrm{fm}}^{-3}$ and $y_p=0.1$ at $T=1$ MeV. (b) Variance in the neutron (${\sigma }_n^2$, filled symbols) and proton (${\sigma }_p^2$, empty symbols) number versus temperature for different proton fractions of 0.1 (left panel), 0.2 (middle panel), and 0.4 (right panel). The various lines are added to guide the eye.

Figure 5

(a) Mean-square nucleon fluctuations versus isothermal compressibility for various temperatures at a proton fraction of 0.1, 0.2, and 0.4, respectively; solid lines are to guide the eye. (b) $L^3{V}_{sph}^{-1}{\langle N\rangle }^{-1}{\sigma }_t^2$ versus ${\rho }_{sph}T{\kappa }_T$ for all studied conditions. The gray dashed line stands for a slope of unity.