Description
The fundamental closure of the BBGKY hierarchy is based on largeness of the plasma parameter, g, the number of particles in a Debeye sphere. Closure yields the Hamiltonian [1] Vlasov-Maxwell (VM) equations and/or the metriplectic [2] VM equations with the Landau-Lenard-Balescu (LLB) collision operator. Both models are used with arbitrary initial conditions, with the latter model having a formal H-theorem to thermal equilibrium. Here we address two issues: i) Because g depends on a temperature, for any initial condition, what is meant by this approximation? ii) The LLB collision operator derivation uses the Bogoliubov (B) assumption on the decay of correlations. Given that structures can appear in phase space, to what degree is this assumption valid. With regard to issue i) we can consider alternative closure procedures. However, issue ii) is problematic given the lack of rigor and numerical testing, which is prohibitive because of high dimensionality. A 1D electrostatic plasma model that can bring computations within reach has been proposed [3]. This model replaces particles by disks that exhibits 1-D features at short distances, but retains 3-D features at large distances. The dynamics of this model is investigated analytically, implementation for numerical validation of the B assumption is formulated, and recent results described.
[1] J Marsden, et al., Contemp. Math. 28, 115 (1984).
[2] P J Morrison, and M. Updike, Phys. Rev. E 09, 045202 (2024).
[3] F Pegoraro et al., Rev. Mod. Plasma Phys. 8, 36 (2024)