Two-Stream Instability and Bernstein-Greene-Kruskal Mode Formation in Coulomb One Component Plasma

TL;DR

This work investigates the Two-Stream Instability (TSI) in strongly coupled plasmas using fully kinetic 3D molecular dynamics with long-range Coulomb interactions. It demonstrates the nonlinear saturation of TSI via the emergence of a Bernstein–Greene–Kruskal (BGK) mode, and shows that BGK vortices can dynamically destabilize, leading to energy decay, with a distinctive spectral hump indicating a coherent mode. The study reveals that long-range forces are essential for TSI in this regime, as shielding suppresses the instability, and that the onset depends on a threshold streaming velocity near $v_0$, with geometry and coupling strength modulating the nonlinear evolution. Spectral analyses corroborate the BGK interpretation through Doppler-shifted dispersion and hole-merger dynamics, highlighting the rich kinetic behavior beyond traditional fluid or Vlasov-PIC descriptions. These insights advance understanding of kinetic instabilities in strongly coupled plasmas and may inform experimental diagnostics and plasma control strategies in both laboratory and astrophysical contexts.

Abstract

We investigate the Two-Stream Instability in a strongly coupled plasma using classical molecular dynamics simulations with long-range Coulomb interactions between particles. The nonlinear evolution of the instability is identified by the emergence of a Bernstein-Greene-Kruskal (BGK) mode. Our simulations capture key microscopic effects, such as inter-particle correlations, collisional dynamics, and coherent wave-particle interactions-features often absent in traditional fluid and kinetic models, including Particle-In-Cell and Vlasov approaches. In the linear regime, the instability grows rapidly and saturates within a few tens of plasma periods. As the system transitions into the nonlinear saturation phase, a single BGK mode emerges. This mode (or phase-space hole) becomes dynamically unstable in the nonlinear regime, characterized by a continuous decay of electrostatic energy over time. An energy budget analysis reveals a bump in an otherwise thermal spectrum, indicating the excitation of a coherent mode, further confirmed through a numerical rendering of the dispersion relation. The pairwise interaction plays a crucial role: pronounced instability and BGK mode formation occur with long-range Coulomb forces, while such structures are suppressed under shielded Coulomb interactions. We observe the emergence of a single BGK mode across all coupling strengths in the fluid regime, provided the streaming velocity exceeds a critical threshold.

Figures (8)

• Figure 1: Phase-space evolution of the TSI for $\Gamma = 0.1$ in a long-range Coulomb OCP. Each dot represents a single negatively charged particle. Insets display the corresponding velocity distribution at each time snapshot. A full animation of the molecular dynamics simulation, showing the emergence of a BGK mode excited by TSI in Coulomb OCP, is available at the provided URL.
• Figure 2: Real-space evolution of two interpenetrating particle streams (blue and yellow) in the MD simulation for $\Gamma = 0.1$. The streaming particles exhibit space-charge bunching at $t = 15,\ 20,\ \text{and}\ 25\ \omega_{pd}^{-1}$
• Figure 3: Evolution of the growth of TSI using perturbed kinetic energy $(|E_k|)$ (solid line) defined by Eq. \ref{['Linear_KE_eqn']}, and spectral energy $(|P_{k_x^{max}}|)$ (dash-dotted line) of the maximally growing mode. The inset shows TSI's linearly fitted growth rate in exponential regime $\gamma_e$ obtained from MD simulations.
• Figure 4: Effect of streaming velocity on the TSI for $\Gamma = 0.1$. Top panel with $v_0 =2\ v_{th}$, and bottom panel with $v_0 = 3\ v_{th}$. The instability is more pronouncedly observed at $v_0 = 2\ v_{th}$, and completely vanishes for $v_0 = 3\ v_{th}$.
• Figure 5: Effect of box size on the TSI for $\Gamma = 0.1$. Top panel with $L_z = 0.25 L_x$, middle panel with $L_z = 0.50 L_x$ and bottom panel with $L_z = 0.75 L_x$. However, in each panel, $L_x = L_y$. The increase in box dimension leads to a stabilized void formation in TSI. The bottom panel of Fig. \ref{['Figure_6']} shows the TSI for $\Gamma = 10$ when $L_x = L_y =L_z$.