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1. Fast spectral separation method for kinetic equation with anisotropic non-stationary collision operator retaining micro-model fidelity

   Zhao, Yue; Lei, Huan (August 2025, arXivorg)

   In this paper, we present a generalized, data-driven collisional operator for one-component plasmas, learned from molecular dynamics simulations, to extend the collisional kinetic model beyond the weakly coupled regime. The proposed operator features an anisotropic, non-stationary collision kernel that accounts for particle correlations typically neglected in classical Landau formulations. To enable efficient numerical evaluation, we develop a fast spectral separation method that represents the kernel as a low-rank tensor product of univariate basis functions. This formulation admits an O(N log N) algorithm via fast Fourier transforms and preserves key physical properties, including discrete conservation laws and the H-theorem, through a structure-preserving central difference discretization. Numerical experiments demonstrate that the proposed model accurately captures plasma dynamics in the moderately coupled regime beyond the standard Landau model while maintaining high computational efficiency and structure-preserving properties. 

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2. Taylor dispersion and phase mixing in the non‐cutoff Boltzmann equation on the whole space

   https://doi.org/10.1112/plms.12616  

   Bedrossian, Jacob; Coti_Zelati, Michele; Dolce, Michele (July 2024, Proceedings of the London Mathematical Society)

   Abstract In this paper we describe the long‐time behavior of the non‐cutoff Boltzmann equation with soft potentials near a global Maxwellian background on the whole space in the weakly collisional limit (that is, infinite Knudsen number ). Specifically, we prove that for initial data sufficiently small (independent of the Knudsen number), the solution displays several dynamics caused by the phase mixing/dispersive effects of the transport operator and its interplay with the singular collision operator. For ‐wavenumbers with , one sees anenhanced dissipationeffect wherein the characteristic decay time‐scale is accelerated to , where is the singularity of the kernel ( being the Landau collision operator, which is also included in our analysis); for , one seesTaylor dispersion, wherein the decay time‐scale is accelerated to . Additionally, we prove almost uniform phase mixing estimates. For macroscopic quantities such as the density , these bounds imply almost uniform‐in‐ decay of in due to phase mixing and dispersive decay. 

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3. Global existence for an isotropic modification of the Boltzmann equation

   https://doi.org/10.1016/j.jfa.2024.110423  

   Snelson, Stanley (June 2024, Journal of Functional Analysis)

   Motivated by the open problem of large-data global existence for the non-cutoff Boltzmann equation, we introduce a model equation that in some sense disregards the anisotropy of the Boltzmann collision kernel. We refer to this model equation as isotropic Boltzmann by analogy with the isotropic Landau equation introduced by Krieger and Strain (2012) [35]. The collision operator of our isotropic Boltzmann model converges to the isotropic Landau collision operator under a scaling limit that is analogous to the grazing collisions limit connecting (true) Boltzmann with (true) Landau. Our main result is global existence for the isotropic Boltzmann equation in the space homogeneous case, for certain parts of the “very soft potentials” regime in which global existence is unknown for the space homogeneous Boltzmann equation. The proof strategy is inspired by the work of Gualdani and Guillen (2022) [22] on isotropic Landau, and makes use of recent progress on weighted fractional Hardy inequalities. 

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4. Foundations of magnetohydrodynamics

   https://doi.org/10.1063/5.0274784  

   LeVan, Jarett; Baalrud, Scott_D (July 2025, Physics of Plasmas)

   In this tutorial, a derivation of magnetohydrodynamics (MHD) valid beyond the usual ideal gas approximation is presented. Non-equilibrium thermodynamics is used to obtain conservation equations and linear constitutive relations. When coupled with Maxwell's equations, this provides closed fluid equations in terms of material properties of the plasma, described by the equation of state and transport coefficients. These properties are connected to microscopic dynamics using the Irving–Kirkwood procedure and Green–Kubo relations. Symmetry arguments and the Onsager–Casimir relations allow one to vastly simplify the number of independent coefficients. Importantly, expressions for current density, heat flux, and stress (conventionally Ohm's law, Fourier's law, and Newton's law) take different forms in systems with a non-ideal equation of state. The traditional form of the MHD equations, which is usually obtained from a Chapman–Enskog solution of the Boltzmann equation, corresponds to the ideal gas limit of the general equations. 

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5. Participation Factors for Nonlinear Autonomous Dynamical Systems in the Koopman Operator Framework

   https://doi.org/10.1109/TAC.2025.3574309  

   Takamichi, Kenji; Susuki, Yoshihiko; Netto, Marcos (May 2025, IEEE Transactions on Automatic Control)

   We devise a novel formulation and propose the concept of modal participation factors to nonlinear dynamical systems. The original definition of modal participation factors (or simply participation factors) provides a simple yet effective metric. It finds use in theory and practice, quantifying the interplay between states and modes of oscillation in a linear time-invariant (LTI) system. In this paper, with the Koopman operator framework, we present the results of participation factors for nonlinear dynamical systems with an asymptotically stable equilibrium point or limit cycle. We show that participation factors are defined for the entire domain of attraction, beyond the vicinity of an attractor, where the original definition of participation factors for LTI systems is a special case. Finally, we develop a numerical method to estimate participation factors using time series data from the underlying nonlinear dynamical system. The numerical method can be implemented by leveraging a well-established numerical scheme in the Koopman operator framework called dynamic mode decomposition. 

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