The accurate quantification and analysis of information are critical in many scientific disciplines, from data science and information theory to quantum mechanics and statistical physics. Traditionally, Shannon entropy has long served as the foundational measure of information, providing a scalar summary of uncertainty within a probability distribution [1]. While Shannon entropy’s elegance and simplicity have ensured its widespread adoption, it also bears significant limitations, particularly when dealing with complex, non-Gaussian distributions or systems characterized by high-dimensional data.

A key limitation of Shannon entropy is its reduction of the functional space of distributions to a single scalar value, which can lead to identical entropy values for distinct distributions, thereby potentially obscuring their unique characteristics. This issue is particularly pronounced in analyzing non-Gaussian distributions, common in fields such as finance, genomics, and signal processing, where Shannon entropy often fails to capture intricate dependencies and tail behaviors critical to understanding information dynamics.

Moreover, Shannon entropy’s reliance on a scalar summary frequently decouples it from other important statistical properties, such as variance, which may offer more direct insights into data spread and reliability [2]. In certain cases, particularly involving nonlinear mappings, Shannon entropy can paradoxically suggest a reduction in information even when measurements enhance understanding of a variable, as it primarily emphasizes distribution flatness at the expense of other significant features [3].

To address these challenges, various extensions of entropy have been proposed, including Rényi entropy [4] and Tsallis entropy [5]. While these measures have their merits, they remain fundamentally scalar-centric and do not fully address the need to capture the functional characteristics of information distribution across a distribution’s entire support.

In response to these limitations, we propose derangetropy, a novel conceptual framework that departs from traditional approaches to information measurement. Unlike existing measures, derangetropy is not merely an extension of entropy but represents a new framework that captures the dynamics of information across the entire support of probability distributions. By incorporating self-referential and periodic properties, derangetropy offers a richer understanding of information dynamics, particularly in systems where information evolve cyclically or through feedback mechanisms such as artificial neural networks.

This paper establishes the foundation of derangetropy by exploring its theoretical framework and properties. We begin in Section 2 by introducing the mathematical definition and key properties of derangetropy. Section 3 analyzes the information dynamics and equilibrium within this framework. In Section 4, we discuss the self-referential and self-similar nature of derangetropy. Section 5 provides a combinatorial perspective on derangetropy, connecting it to fundamental principles of combinatorial analysis. Finally, Section 6 summarizes the findings and suggests potential avenues for future research, including applications in information theory and beyond.

The derangetropy functional, denoted by ${\rho }_f\left(x\right)$, is distinguished by its self-referential nature, whereby ${\rho }_f\left(x\right)$ itself is a valid PDF. This implies that ${\rho }_f\left(x\right)$ not only encapsulates the informational content of the underlying distribution $f\left(x\right)$ but also recursively uncovers the hierarchical structure of its own information content. This recursive and hierarchical relationship is formalized as: where ${\rho }_f^{\left(n\right)}\left(x\right)$ represents the nth iteration of the derangetropy functional, and is the associated CDF. The initial conditions are set by where $F\left(x\right)$ denotes the CDF of the original distribution. This recursive structure ensures that each subsequent layer ${\rho }_f^{\left(n\right)}\left(x\right)$ integrates the informational content of all preceding layers, thereby constructing a comprehensive hierarchical representation of information.

To elucidate this self-referential property, we consider the first and second layers of the derangetropy functional for the uniform distribution over the interval $(0,1)$, as illustrated in Figure 3. The first layer, ${\rho }^{\left(1\right)}\left(x\right)$, reflects the basic informational structure of the uniform distribution. In contrast, the second layer, ${\rho }^{\left(2\right)}\left(x\right)$, introduces additional complexity, manifesting in pronounced peaks and troughs. This evolution underscores the recursive amplification inherent in the derangetropy functional, progressively unveiling deeper informational layers with each iteration.

Further insights into the self-referential nature of the derangetropy functional can be obtained by examining its behavior for the arcsin distribution, as shown in Figure 4. The first layer captures the strong boundary effects characteristic of the arcsin distribution, resulting in a pronounced concave upward profile near the interval’s endpoints. As recursion advances to the second layer, these boundary effects are amplified, giving rise to more pronounced oscillations and a complex informational structure. The transition in concavity observed between the first and second layers signifies a shift from a simple, boundary-focused representation to a more intricate depiction of the distribution’s information content.

The third layer introduces a bell-shaped curve, indicative of a stabilization process within the recursive structure. The previously amplified boundary effects become harmonized, redistributing the information more uniformly across the distribution. This bell-shaped curve suggests a tendency towards centralization and equilibrium within the distribution as recursion progresses. The smoothing of oscillations in the third layer reflects the derangetropy functional’s inherent ability to guide the system towards an informational equilibrium, where initially sharp boundary effects are moderated and the information distribution becomes more balanced and Gaussian-like.

An intriguing and nontrivial observation emerges when comparing the second- and third-level derangetropies of the arcsin distribution with the first- and second-level derangetropies of the uniform distribution. Notably, the second-level derangetropy of the arcsin distribution closely resembles the first-level derangetropy of the uniform distribution, while the third-level derangetropy of the arcsin distribution mirrors the second-level derangetropy of the uniform distribution. This resemblance reveals a deep structural connection between these distributions under the framework of the derangetropy functional.

The transformation $X={sin}^2\left({\displaystyle \frac{\pi }{2}}U\right)$ such that $U\sim Uniform(0,1)$, which maps the uniform distribution to the arcsin distribution, is pivotal in explaining this connection. The sine function within the derangetropy functional plays a crucial role in this transformation. When applied to the arcsin distribution, whose CDF is given by ${\mathcal{G}}_{\mathrm{Arcsin}}\left(x\right)={\displaystyle \frac{2}{\pi }}arcsin\left(\sqrt{x}\right)$, the sine function simplifies as follows:This expression directly mirrors the transformation $X={sin}^2\left({\displaystyle \frac{\pi }{2}}U\right)$, establishing a structural similarity between the arcsin and uniform distributions. The recursive nature of the derangetropy functional, which inherently tends to smooth and centralize information, reflects this underlying transformation. The arcsin distribution, characterized by its strong boundary effects, requires additional recursive layers to attain a similar informational structure to that of the uniform distribution. This explains why the second-level derangetropy of the arcsin distribution mirrors the first-level derangetropy of the uniform distribution, and why the third-level derangetropy of the arcsin distribution resembles the second-level derangetropy of the uniform distribution. Through its recursive application, the derangetropy functional reveals the deep structural relationships between these distributions.

Lastly, we note that as n approaches infinity, the recursive process drives the distribution towards a degenerate distribution centered at the median. Mathematically, this means that the $nth$ level derangetropy ${\rho }_f^{\left(n\right)}\left(x\right)$ converges in distribution to a Dirac delta function $\delta (x-m)$, where all the probability mass is concentrated at the median $x=m$; i.e., This convergence behavior is a direct consequence of the derangetropy functional’s design, which inherently favors the centralization of information. The recursive smoothing effect ensures that, with each iteration, the distribution becomes more focused around the central point, ultimately leading to a situation where all mass is concentrated at the median. It is worth mentioning that the derangetropy functional exhibits self-similar patterns, particularly in the earlier stages of recursion. However, as the recursion deepens, the structure becomes more centralized about median and less fractal-like, leading to increasingly negative scaling exponents.

The concept of derangetropy is deeply rooted in combinatorial principles, particularly the notion of derangements. In combinatorics, a derangement is a permutation of a set where no element appears in its original position, representing a state of maximal disorder. Extending this concept to continuous probability distributions, the derangetropy functional ${\rho }_f\left(x\right)$ quantifies the local interplay between order and disorder at each point in the distribution.

A key mathematical tool in deriving the properties of derangetropy is Euler’s reflection formula, given by: where $\Gamma \left(z\right)$ is the Gamma function, a continuous analogue of the factorial function. This formula establishes a fundamental connection between the sine function and the combinatorial structures underlying the derangetropy functional. The derangetropy functional, initially defined by formula (1) can be reformulated using Euler’s reflection formula as:

This reformulation highlights the recursive self-referential nature of the distribution, where the distribution refers to its own values at each point x, thereby influencing the overall informational dynamics of the system. The terms $F{\left(x\right)}^{F\left(x\right)}$ and ${(1-F\left(x\right))}^{1-F\left(x\right)}$ introduce a self-weighting modulation mechanism that amplifies disorder non-linearly in regions of high probability, while regions with large $1-F\left(x\right)$ values experience a similar recursive influence. In continuous distributions, the combinatorial complexity is encapsulated by the Gamma functions $\Gamma \left(F\right(x\left)\right)$ and $\Gamma (1-F(x\left)\right)$. These functions serve as continuous analogues of the derangement process, quantifying the degree of disorder at each point by representing the combinatorial complexity associated with arranging elements less than x or greater than x, respectively. The interplay between these Gamma functions at each point and the self-weighing mechanism determine the local derangetropy and reflects the underlying combinatorial structure of the distribution.

Moreover, the system governed by the derangetropy functional continuously transitions between states of order and disorder. This dynamic conversion is modulated by the oscillatory and structural informational energies inherent within the distribution, reflecting the probabilistic laws that govern the system’s behavior. Equilibrium is achieved when the total informational energy $E_{\mathrm{Total}}\left(x\right)$ is minimized across the distribution, representing a balance between order and disorder. Deviations from equilibrium increase derangetropy, indicating higher disorder at specific points x within the distribution, a behavior crucial for understanding extreme events or rare occurrences.

Consequently, the derangetropy functional also captures both local variability and global stability within the system. At any given point x, the local derangetropy is determined by whether x aligns with its expected order. Aggregating these local measures provides a global assessment of the system’s informational content, consistent with the principles of complex systems, where global properties emerge from local interactions. This dual capability makes derangetropy a powerful tool for analyzing both micro and macro structures within a wide range of probabilistic systems.

This paper introduced derangetropy, a functional measure designed to capture the dynamic nature of information within probability distributions. By offering a functional representation rather than a scalar summary, derangetropy provides deeper insights into the interplay between order and disorder across a distribution’s entire support. The measure’s self-referential and periodic properties facilitate a rigorous analysis of information stability and evolution, enhancing our understanding of complex systems.

The versatility of derangetropy was demonstrated through its mathematical formulation and empirical analysis across various distributions. This novel approach extends beyond the capabilities of traditional entropy measures, allowing for a more detailed examination of information dynamics.

Looking forward, several promising research directions emerge. In the realm of deep neural networks, derangetropy could regulate information flow, preventing overconcentration or dispersion and thereby enhancing network robustness and generalization. Incorporating equilibrium relations into the backpropagation process may also introduce more efficient training methods.

In quantum information theory, derangetropy could provide new tools for analyzing information dynamics in quantum systems, offering insights where traditional measures fall short. Additionally, applying derangetropy within statistical mechanics and thermodynamics could deepen our understanding of complex systems at both equilibrium and nonequilibrium states.

In summary, derangetropy represents a significant conceptual and practical advancement in information theory, opening new avenues for theoretical exploration and practical application. Future research will likely expand its utility, revealing further insights into the dynamics of information and its potential to enhance the design and function of deep neural networks and other complex systems.