On the structure of the two-stream instability -- complex G-Hamiltonian structure and Krein collisions between positive- and negative-action modes

TL;DR

This paper reveals that the two-stream instability's band structure in plasma physics is governed by a complex G-Hamiltonian framework, where Krein collisions between modes with different actions lead to instability boundaries.

Contribution

It demonstrates that the instability diagram arises from the Hamiltonian nature of the system, specifically through a complex G-Hamiltonian structure and Krein collisions, providing a new theoretical understanding.

Findings

The instability boundaries correspond to Krein collisions between modes with different signatures.

The Hamiltonian structure explains the band structure of the instability diagram.

Destabilization occurs when positive- and negative-action modes resonate.

Abstract

The two-stream instability is probably the most important elementary example of collective instabilities in plasma physics and beam-plasma systems. For a warm plasma with two charged particle species based on a 1D warm-fluid model, the instability diagram of the two-stream instability exhibits an interesting band structure that has not been explained. We show that the band structure for this instability is the consequence of the Hamiltonian nature of the warm two-fluid system. Interestingly, the Hamiltonian nature manifests as a complex G-Hamiltonian structure in wave-number space, which directly determines the instability diagram. Specifically, it is shown that the boundaries between the stable and unstable regions are locations for Krein collisions between eigenmodes with different Krein signatures. In terms of physics, this rigorously implies that the system is destabilized when a positive-action mode resonates with a negative-action mode, and that this is the only mechanism by which the system can be destabilized. It is anticipated that this physical mechanism of destabilization is valid for other collective instabilities in conservative systems in plasma physics, accelerator physics, and fluid dynamics systems, which admit infinite-dimensional Hamiltonian structures.