# Generalized Derangetropy Functionals for Modeling Cyclical Information Flow

## TL;DR

This paper introduces a new mathematical framework for modeling how information flows in systems with cycles and feedback, using generalized derangetropy operators.

## Contribution

The novelty lies in the generalized derangetropy operators that act on probability densities to model cyclical and feedback-driven information flow.

## Key findings

- The operators induce a spectral diffusion process governed by the heat equation.
- The framework converges toward a Gaussian characteristic function, offering insights into long-term information dynamics.
- Applications include artificial neural networks, communication theory, and non-equilibrium statistical mechanics.

## Abstract

This paper introduces a functional framework for modeling cyclical and feedback-driven information flow using a generalized family of derangetropy operators. In contrast to scalar entropy measures such as Shannon entropy, these operators act directly on probability densities, providing a topographical representation of information across the support of the distribution. The proposed framework captures periodic and self-referential aspects of information evolution through functional transformations governed by nonlinear differential equations. When applied recursively, these operators induce a spectral diffusion process governed by the heat equation, with convergence toward a Gaussian characteristic function. This convergence result establishes an analytical foundation for describing the long-term dynamics of information under cyclic modulation. The framework thus offers new tools for analyzing the temporal evolution of information in systems characterized by periodic structure, stochastic feedback, and delayed interaction, with potential applications in artificial neural networks, communication theory, and non-equilibrium statistical mechanics.