Unifying Variational and Asymptotic Derivations of SPN: A Nonclassical Perspective

A unified theoretical interpretation of simplified spherical harmonics (SP$_N$) equations is developed by establishing a connection between variational and asymptotic derivations of transport approximations. Starting from the nonclassical linear Boltzmann equation posed in extended phase space, the nonclassical spherical harmonic approximation (NSHA) is analyzed under diffusion scaling to derive a hierarchy of asymptotic SP$_N$ equations whose coefficients depend on moments of the free-path distribution.It is shown that the SP$_N$ equations obtained from the variational formulation of Mishra \textit{et al.} arise as the exponential free-path specialization of this asymptotic framework. In this limit, the NSHA-derived equations reduce exactly to the classical SP$_N$ system, providing an explanation for the previously observed equivalence between variational SP$_N$ and generalized SP$_N$ formulations. When the free-path distribution departs from exponential form, the asymptotic NSHA formulation yields modified transport coefficients that cannot be recovered within the classical variational framework.To illustrate the implications of this coefficient dependence, numerical benchmarks are presented in which classical and nonclassical SP$_N$ solutions are compared for media characterized by prescribed non-exponential free-path statistics. These calculations verify coincidence in the exponential limit and demonstrate systematic divergence as the free-path distribution departs from exponential behavior.These results show that SP$_N$ theory is fundamentally asymptotic and that the variational SP$_N$ formulation represents a realization of a broader transport hierarchy rather than an independent approximation. This interpretation clarifies the role of free-path statistics in determining SP$_N$ transport coefficients and establishes NSHA as a unified framework for extending diffusion-based transport models beyond the classical regime.