Conditions for predicting quasistationary states by rearrangement formula

TL;DR

This paper investigates the conditions under which the rearrangement formula accurately predicts quasistationary states in Hamiltonian systems with long-range interactions, highlighting the roles of Landau damping and Casimir invariants.

Contribution

The study identifies specific conditions that determine the validity of the rearrangement formula for predicting quasistationary states, enhancing its theoretical foundation.

Findings

Rearrangement formula works well when no Landau damping occurs.

Casimir invariants influence the applicability of the formula.

Conditions for the formula's validity are clarified and discussed.

Abstract

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 spatially homogeneous part and small perturbation, we numerically reveal two conditions that the rearrangement formula prefers: One is no Landau damping condition for unperturbed homogeneous part, and the other comes from the Casimir invariants. Mechanisms of these conditions are discussed. Clarifying these conditions, we inform validity 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.