Double Lynden-Bell structure of low-energy quasistationary distributions in the Hamiltonian mean-field model

In the Hamiltonian mean-field model, we study the core-halo structure of low-energy quasistationary states under unsteady water-bag type initial conditions. The core-halo structure results in the superposition of two independent Lynden-Bell distributions. We examine the completeness of the Lynden-Bell relaxation and the relaxation between these two Lynden-Bell distributions.

Figure 1

$\widehat{\eta }=0.15$ configuration of four data $M_0=0.53,M_{\min },0.72,\text{and}0.78$ on the $(M_0,\widehat{E})$ plane to be used in Figs. 3 and 4. Here, $M_{\min }\sim 0.6556$ is the magnetization of the initial water-bag distribution Eq. (15) at the minimum $\widehat{E}$. The blue curve and the red line represent the initial water-bag states $\widehat{E}\left(M_0\right)$ and the energy per particle ${\widehat{E}}_{{\varepsilon }_F}$ of the Vlasov stationary water-bag state $f_{{\varepsilon }_F}$ for $\eta =1500$, respectively.

Figure 2

Residue of the Lynden-Bell entropy of the Lynden-Bell equilibrium $S_{\mathrm{eq}}$ for given $E$ and $\eta =1500$ against that of the simulation result $S_{\mathrm{sim}}$ as a function of the residual energy per particle ${\widehat{E}}_{\mathrm{res}}$. Blue (upper) and purple (lower) plotted dots represent, respectively, the cases of $M_0=0.66\text{–}0.78$ (with $0.01$ increments) and $M_0=0.41\text{–}0.65$ (with $0.02$ increments) using the distributions averaged over 10 runs.

Figure 3

Deviation of the simulation resultant $f\left(\varepsilon \right)$ (red dots) averaged over 20 runs from the single Lynden-Bell equilibrium (gray curve) for $M_0=M_{\min },0.72,\text{and}0.78$ (from top to bottom).

Figure 4

$M_0=0.53,M_{\min },\text{and}0.78$ (from top to bottom) double Lynden-Bell theoretical semipredictions of simulation resultant $f\left(\varepsilon \right)$ (red dots) averaged over 20 runs. Green (upper) and cyan (lower) solid curves represent the full and halo part of the double Lynden-Bell distribution, respectively. The former is constrained to satisfy the three conservation laws and the self-consistency condition Eq. (2) and by adjusting values of the Lynden-Bell entropy, stationary magnetization $M_s$, and ${\eta }_c$ by hand. Gray dashed curves represent the Lynden-Bell equilibrium for given $E$ and $\eta =1500$.

Figure 5

$M_0=0.72$ theoretical result for $f\left(\varepsilon \right)$ obtained by solving the on-shell entropy maximization condition for $M_s=0.6345$, and the simulation resultant $f\left(\varepsilon \right)$ averaged over 20 runs (red dots). The full and halo part of the former are drawn as green (upper) and cyan (lower) curves, respectively.

Figure 6

$M_0=0.72$ contour curves of $S$ (purple, lower tangent) and $M$ (blue, upper tangent) on the $({\widehat{N}}_c,{\widehat{E}}_c)$ and $({\widehat{N}}_c,{\widehat{\eta }}_c)$ planes for $M_s=0.6345$ at the simulation resultants ${\widehat{\eta }}_c$ and ${\widehat{E}}_c$, respectively. The red dot is the simulation resultant point. $S$ and $M$ are shown with equal contour intervals.