Modified Kalman Filtering Derived from Non-Maxwellian Distribution Functions in Open Systems

Abstract

Kalman filtering is exact for linear dynamics with Gaussian state and observation statistics, but a mean-covariance representation cannot retain finite non-Gaussian structure generated by source-driven kinetic evolution. We formulate a modified filtering theory for open plasma systems in which the additional state structure is derived from a non-Maxwellian velocity-space distribution (NMDF) rather than introduced as an empirical residual family. A kinetic manifold defines $f_s(\mathbf X_s,\mathbf v)$, while a fixed diagnostic map $H_D$ generates the measurement PDF $p_{D,s}$; the projected kinetic equation determines the state-prediction dynamics. The posterior evolves continuously and is corrected by Bayes' rule, with positivity-constrained relative-entropy projection when required. The Gaussian posterior with affine dynamics and a linear Gaussian observation model recovers the Kalman-Bucy and discrete Kalman limits. The first explicit non-Gaussian closure is a five-coordinate INMDF from exact five-moment inversion, with Kappa retained as a broad-tail alternative. Using seven Alcator C-Mod Langmuir-probe ion-saturation-current PDFs, six kinetic manifolds are propagated through the same source statistics, noise model, normalization, and probe response. Because the published histograms lack time ordering, recursive tracking is not tested. In universal leave-one-condition-out prediction, double-INMDF ranks first in all four held-out divertor conditions, with mean error 0.0989 versus 0.1089 for MDF. Within-region calibration gives nearly identical divertor errors for first- and double-INMDF, 0.0936 and 0.0937, while two-Maxwellian gives the smallest midplane error, 0.0718. Predictions remain conditional on published source and noise controls, but show that a frozen source-to-kinetic response generalizes to an unseen current PDF and that the preferred response is region dependent.