Conditions for predicting quasistationary states by rearrangement formula

Predicting the long-lasting quasistationary state for a given initial state is one of central issues in Hamiltonian systems having long-range interaction. A recently proposed method is based on the Vlasov description and uniformly redistributes the initial distribution along contours of the asymptotic effective Hamiltonian, which is defined by the obtained quasistationary state and is determined self-consistently. The method, to which we refer as the rearrangement formula, was suggested to give precise prediction under limited situations. Restricting initial states consisting of a spatially homogeneous part and small perturbation, we numerically reveal two conditions that the rearrangement formula prefers: One is a no Landau damping condition for the unperturbed homogeneous part, and the other comes from the Casimir invariants. Mechanisms of these conditions are discussed. Clarifying these conditions, we validate to use the rearrangement formula as the response theory for an external field, and we shed light on improving the theory as a nonequilibrium statistical mechanics.

Figure 1

Schematic picture for the rearrangement formula (19) in the HMF model. The left panel shows $\mu$ space, and dotted and solid lines represent contours of $f_{\mathrm{I}}$ and $f_{\mathrm{A}}$ respectively. The right panel shows $\theta$ dependence of distribution functions, where the angle variable $\theta$ is defined by the asymptotic effective Hamiltonian ${\mathcal{H}}_{\mathrm{A}}$, and the red solid line marked by A corresponds to the red solid contour A of the left panel. Along the contour, the initial state $f_{\mathrm{I}}$ depends on the angle $\theta$ as described by the red dotted line in the right panel. The blue solid contour marked by B is another example. For the two examples A and B, the positive $p$ axis corresponds to $\theta =0$. In this schematic picture $f_{\mathrm{I}}$ is assumed to be spatially homogeneous and a decreasing function of energy, though the rearrangement formula is also applicable to a spatially inhomogeneous $f_{\mathrm{I}}$.

Figure 2

Asymptotic values of $M_x$ for the initial distributions (37). The red squares are for Case 1, and the green circles are for Case 2. The open and filled symbols are computed with the grid sizes $G=256$ and 512 respectively. $\epsilon =0.1$. The red solid and the green dashed lines are from the approximated theory (29) for Case 1 and Case 2 respectively.

Figure 3

${\epsilon }_2$ dependence of $M_x$ with the fixed ${\epsilon }_1=0.1$ for the initial state (40). The red solid line represents the full theory (21), the orange dashed line is approximated theory (29), and points are from numerics. The sizes of grid are $G=128$ (purple triangles), 256 (light blue circles), and 512 (blue squares).

Figure 4

${\epsilon }_1$ dependence of $M_x$ with the fixed ${\epsilon }_2=0.01$ for the initial state (40). The solid red line represents the full theory (21), the orange dashed line is the approximated theory (29), and filled points are from numerics. The grid sizes are $G=256$ (light blue circles) and 512 (blue squares). Open symbols represent relative errors with the full theory: $R$ for $G=256$ (light blue circles) and 512 (blue squares), and $-R$ for $G=256$ (pink triangles) and 512 (purple inverse triangles). The black solid line is a guide for eyes, and has slope 2.