Asymptotic Derivation of Nonclassical SPN Equations from the Spherical Harmonic Approximation

A systematic asymptotic connection between the nonclassical spherical harmonic approximation (NSHA) and the simplified \(P_N\) (SP\(_N\)) equations is developed for linear particle transport with nonclassical free-path distributions. Starting from the nonclassical transport equation posed in extended phase space, the NSHA moment system is analyzed under diffusion scaling and asymptotically reduced to macroscopic transport equations. This analysis shows that the diffusion-limit reduction of NSHA(\(N\)) yields a hierarchy of coupled second-order equations constituting a nonclassical SP\(_N\) system, with transport coefficients expressed explicitly in terms of moments of the free-path distribution.The derivation is carried out in detail for \(N=1\) and \(N=3\), recovering the nonclassical diffusion (SP\(_1\)) and SP\(_3\) equations through a systematic asymptotic procedure. The resulting equations retain full sensitivity to the underlying free-path statistics and reduce exactly to the classical SP\(_N\) equations in the exponential free-path limit. The general SP\(_N\) structure is then established for arbitrary odd order \(N\), and consistency with the explicit SP\(_1\) and SP\(_3\) limits is demonstrated.The analysis is further extended to anisotropic scattering by retaining the full angular dependence of the scattering kernel within the NSHA framework. Corresponding anisotropic SP\(_1\) and SP\(_3\) equations are derived, and the general anisotropic SP\(_N\) form is obtained. In all cases, the macroscopic equations arise directly from asymptotic elimination of the free-path variable, without phenomenological closure assumptions. These results establish NSHA as a unified theoretical foundation for nonclassical SP\(_N\) models and clarify their connection to diffusion-based transport approximations in media with nonexponential free-path statistics.