Phase-Structure Analysis of a One-Dimensional Distribution Model with Competing Smoothing and Localization

This work presents a minimal structural model for analyzing shape transitions in one‑dimensional probability distributions. The model introduces two control parameters—Δf representing smoothing strength and γT representing localization strength—and studies how their competition generates three characteristic regimes: Gaussian, non‑Gaussian, and localized profiles. A Landau‑type structural potential with nonlinear parameter dependence is used to describe the deformation of distribution shapes. Numerical evaluation shows that the phase boundaries in the (Δf, γT) plane form smooth nonlinear curves, arising from the interplay between global nonlinear response and state‑dependent feedback. The model further analyzes skewness and kurtosis as statistical indicators of shape transitions, revealing a non‑monotonic behavior of kurtosis in the strongly localized regime. This behavior reflects the simultaneous sharpening of the central peak and thickening of the tails, a feature commonly observed in nonlinear distribution systems. Conceptual correspondences with nonlinear optical propagation (NLS‑type spreading vs. self‑focusing) and reaction–diffusion systems (diffusion vs. local growth) are discussed to clarify the structural similarity of “spreading vs. concentrating” dynamics across different fields. Because the formulation does not rely on any specific physical system, the model functions as a minimal and general framework for studying phase‑like transitions in distribution shapes. It provides a compact reference structure for systems in which smoothing and localization mechanisms coexist, including nonlinear diffusion, pattern‑forming systems, and generalized gradient‑flow dynamics. The work aims to clarify the basic structural features of such competing processes and to provide a foundation for future extensions to higher‑dimensional distributions and more general nonlinear coefficient forms.