Introduction

Plasmas consist of both positive ions and negative electrons and although these particles carry electric charge, the overall charge of a plasma is very close to zero, and on macroscopic scales the charge densities of ions and electrons are almost equal. This does not mean that every small region of plasma is perfectly neutral; rather it means that over larger volumes, the densities of positive and negative charges balance each other. The term quasineutrality captures this near-neutrality: a fundamental property of plasmas maintained by the collective behavior of charged particles which tend to screen electric fields created within the plasma through the process of Debye Shielding (Fig. 1). 

When deriving macroscopic fluid models, such as magnetohydrodynamics (MHD), quasineutrality is imposed as an exact condition: ion and electron densities are set equal everywhere in the plasma. This assumption greatly simplifies the governing equations and eliminates fast temporal effects, allowing MHD to capture the large-scale plasma dynamics relevant for fusion devices, astrophysics, and space physics, without the need to resolve small spatial and temporal scales.  

However, the imposition of quasineutrality occurs after the fluid equations have been derived from kinetic theory by taking fluid moments of the Vlasov equation in the collisionless case or the Boltzmann equation in the collisional case. Therefore, there is no underlying kinetic dynamical model that enforces quasineutrality from first principles. As a result, we lack intuition about how quasineutrality is realized at the kinetic level, and how to model kinetic effects in a plasma that is assumed quasineutral.  

This gives rise to a conceptual gap: while quasineutrality is the foundation of fluid models like MHD, the kinetic theory of plasmas, based on the Vlasov equation coupled with Maxwell’s or Poisson’s equations, does not admit quasineutrality automatically. Instead, it fully resolves electrostatic and electromagnetic dynamics, including charge separation and fast oscillations such as Langmuir waves.  

And this leads to a deep question: How does a quasineutral plasma actually behave at the kinetic level?  

Answering this is essential for building a consistent bridge between kinetic theory, which describes microscopic particle dynamics, and fluid theory, which assumes quasineutrality from the outset. Without such a bridge, our understanding of quasineutral plasmas remains incomplete.  

Our recent efforts to fill this gap 

In collaboration with colleagues from the University of Texas at Austin (J. W. Burby and P. J. Morrison), Université Côte d'Azur (E. Tassi), and the University of Ioannina (G. N. Throumoulopoulos), we have recently taken steps toward filling this gap. By applying Hamiltonian methods and the Dirac theory of constraints to collisionless electrostatic plasma models such as the Vlasov-Poisson and Vlasov-Ampère systems, we developed new frameworks that explore quasineutral limits and impose quasineutrality directly at the kinetic level. These approaches make it possible to construct Hamiltonian dynamical systems that enforce charge neutrality while still retaining detailed kinetic information, providing, for the first time, a consistent Hamiltonian kinetic formulation for quasineutral, collisionless plasma dynamics.  

In one line of our work [1], we applied slow manifold reduction and the theory of Poisson-Dirac submanifolds to derive a Hamiltonian formulation of the quasineutral limit of the Vlasov-Poisson system. This framework removes fast Langmuir oscillations and shows that maintaining quasineutrality requires the bulk plasma flow to remain incompressible. Within this setting, the electric field emerges from the balance of plasma stresses that would otherwise lead to compression. The resulting Hamiltonian structure unifies well-known Poisson brackets from both incompressible fluid dynamics and collisionless kinetic theory, providing a consistent dynamical description of quasineutral plasmas.   

In a second approach [2], we used the Dirac theory of constraints to impose quasineutrality directly on the Vlasov–Poisson and Vlasov–Ampère systems, which describe electrostatic plasma dynamics. By constructing generalized Dirac brackets for these models, we obtained constrained systems that conserve charge density and enforce quasineutrality through new advection terms in the Vlasov equations. In this formulation, the electric field is eliminated from the dynamics and replaced by effective generalized-force terms. And this work goes a step further by providing numerical experiments comparing Vlasov-Poisson and quasineutral Vlasov dynamics using semi-Lagrangian method. By these experiments we showed that enforcing quasineutrality significantly alters plasma dynamics at the kinetic level (Fig. 2) while by identifying the generalized forces which should act on the plasma parcels to maintain quasineutrality we have a systematic way to assess the validity of the quasineutral approximation across different kinetic scales.