Fractional Differential Entropy (FDE)

• Fractional Differential Entropy (FDE) is a generalized entropy measure that incorporates fractional powers of the logarithm, capturing nonlocal, heavy-tailed, and multiscale behaviors.
• It distinguishes between a genuinely fractionalized information functional and the use of standard Shannon entropy applied to densities generated by fractional dynamics.
• FDE’s versatile framework underpins diverse applications, from anomalous diffusion and quantum harmonic oscillators to decision-making models in finance and hydraulic modeling.

Searching arXiv for the cited FDE-related papers to ground the article in current literature. Fractional Differential Entropy (FDE) denotes two closely related but non-identical constructions in recent arXiv literature. In one line of work, it is the continuous analogue of Ubriaco’s fractional entropy, defined by

$$H^{\alpha}(f)=\int f(x)\,(-\log f(x))^{\alpha}\,dx,\qquad 0<\alpha<1,$$

with the Shannon differential entropy recovered in the limit $\alpha\to 1$ (Paul et al., 3 Jul 2025). In another line of work, the term refers to the ordinary Shannon differential entropy

$S(t)=-\int \rho(x,t)\ln \rho(x,t)\,dx$

evaluated on probability densities generated by fractional diffusion or fractional quantum dynamics, so that the fractional aspect enters through the governing equation rather than through a modified entropy functional (Aghamohammadi et al., 2013, Boumali et al., 2024). This terminological bifurcation is central to the subject: some papers study a genuinely fractionalized information functional, whereas others study standard information measures on fractional-state densities.

1. Formal origins and terminological scope

The Ubriaco construction begins from Abe’s representation of Shannon entropy and replaces the ordinary derivative with a left Riemann–Liouville fractional derivative ${}_{-\infty}^{RL}D_t^\alpha$, using the analytic continuation property $${}_{-\infty}D_t^\alpha e^{\beta t}=\beta^\alpha e^{\beta t}$$. In the discrete setting this yields

$$H_U^\alpha(p)=\sum_{i=1}^n p_i(-\log p_i)^\alpha,\qquad 0\le \alpha\le 1,$$

which reduces to Shannon entropy at $\alpha=1$ and tends to $1$ as $\alpha\to 0^+$ (Paul et al., 3 Jul 2025). The same discrete form is used in portfolio decision models under risk, where $\alpha$ is interpreted as a behavioral parameter: $\alpha\to 1$0 corresponds to risk-tolerant or adventurous attitudes, whereas $\alpha\to 1$1 corresponds to risk-averse or conservative attitudes (Paul et al., 3 Jul 2025).

The continuous analogue, termed Fractional Differential Entropy in the 2025 literature, is

$\alpha\to 1$2

for a nonnegative absolutely continuous random variable with density $\alpha\to 1$3 (Paul et al., 3 Jul 2025). By contrast, the diffusion literature uses the term for the time-dependent Shannon differential entropy of solutions to fractional PDEs, and the fractional quantum harmonic oscillator literature computes

$\alpha\to 1$4

without altering the entropy kernel itself (Aghamohammadi et al., 2013, Boumali et al., 2024).

A persistent misconception is therefore that FDE always denotes a new non-Shannon functional. The cited literature shows otherwise. In the fractional diffusion and fractional quantum settings, the entropy functional remains Shannon’s; only the density is fractional in origin (Aghamohammadi et al., 2013, Boumali et al., 2024).

2. Continuous Ubriaco-type FDE

For the continuous formulation, the basic object is the expectation of a fractional power of the negative log-likelihood:

$\alpha\to 1$5

The formulation in (Paul et al., 3 Jul 2025) focuses on densities satisfying $\alpha\to 1$6 on their support, so that $\alpha\to 1$7 remains real-valued. This requirement induces parameter restrictions or support truncations for standard families, such as $\alpha\to 1$8 for uniform laws, $\alpha\to 1$9 for exponential laws, and $S(t)=-\int \rho(x,t)\ln \rho(x,t)\,dx$0 for normal laws.

Two dynamic variants are introduced but not further developed in that paper. The past FDE is

$S(t)=-\int \rho(x,t)\ln \rho(x,t)\,dx$1

and the residual FDE is

$S(t)=-\int \rho(x,t)\ln \rho(x,t)\,dx$2

where $S(t)=-\int \rho(x,t)\ln \rho(x,t)\,dx$3 is the cdf and $S(t)=-\int \rho(x,t)\ln \rho(x,t)\,dx$4 (Paul et al., 3 Jul 2025).

The limiting behavior is explicit. As $S(t)=-\int \rho(x,t)\ln \rho(x,t)\,dx$5,

$S(t)=-\int \rho(x,t)\ln \rho(x,t)\,dx$6

so the construction recovers Shannon’s differential entropy. As $S(t)=-\int \rho(x,t)\ln \rho(x,t)\,dx$7,

$S(t)=-\int \rho(x,t)\ln \rho(x,t)\,dx$8

Thus the map $S(t)=-\int \rho(x,t)\ln \rho(x,t)\,dx$9 continuously deforms from ${}_{-\infty}^{RL}D_t^\alpha$0 at ${}_{-\infty}^{RL}D_t^\alpha$1 to Shannon’s differential entropy at ${}_{-\infty}^{RL}D_t^\alpha$2 (Paul et al., 3 Jul 2025).

The continuous finance paper explicitly notes a direct analogue,

${}_{-\infty}^{RL}D_t^\alpha$3

but does not use it in the empirical portfolio study, which remains discrete and bin-based (Paul et al., 3 Jul 2025).

3. Structural properties and analytic evaluations

For ${}_{-\infty}^{RL}D_t^\alpha$4 and ${}_{-\infty}^{RL}D_t^\alpha$5, the integrand

${}_{-\infty}^{RL}D_t^\alpha$6

is nonnegative and concave in ${}_{-\infty}^{RL}D_t^\alpha$7, and ${}_{-\infty}^{RL}D_t^\alpha$8 is accordingly concave on the convex set of densities bounded by ${}_{-\infty}^{RL}D_t^\alpha$9 (Paul et al., 3 Jul 2025). The pointwise maximum of $${}_{-\infty}D_t^\alpha e^{\beta t}=\beta^\alpha e^{\beta t}$$0 occurs at

$${}_{-\infty}D_t^\alpha e^{\beta t}=\beta^\alpha e^{\beta t}$$1

so values of the density near $${}_{-\infty}D_t^\alpha e^{\beta t}=\beta^\alpha e^{\beta t}$$2 contribute առավելally to the local fractional information density (Paul et al., 3 Jul 2025).

Differentiation with respect to $${}_{-\infty}D_t^\alpha e^{\beta t}=\beta^\alpha e^{\beta t}$$3 gives

$${}_{-\infty}D_t^\alpha e^{\beta t}=\beta^\alpha e^{\beta t}$$4

This yields a dichotomy. If $${}_{-\infty}D_t^\alpha e^{\beta t}=\beta^\alpha e^{\beta t}$$5, equivalently $${}_{-\infty}D_t^\alpha e^{\beta t}=\beta^\alpha e^{\beta t}$$6, then $${}_{-\infty}D_t^\alpha e^{\beta t}=\beta^\alpha e^{\beta t}$$7 and $${}_{-\infty}D_t^\alpha e^{\beta t}=\beta^\alpha e^{\beta t}$$8 decreases in $${}_{-\infty}D_t^\alpha e^{\beta t}=\beta^\alpha e^{\beta t}$$9. If $$H_U^\alpha(p)=\sum_{i=1}^n p_i(-\log p_i)^\alpha,\qquad 0\le \alpha\le 1,$$0, equivalently $$H_U^\alpha(p)=\sum_{i=1}^n p_i(-\log p_i)^\alpha,\qquad 0\le \alpha\le 1,$$1, then $$H_U^\alpha(p)=\sum_{i=1}^n p_i(-\log p_i)^\alpha,\qquad 0\le \alpha\le 1,$$2 increases in $$H_U^\alpha(p)=\sum_{i=1}^n p_i(-\log p_i)^\alpha,\qquad 0\le \alpha\le 1,$$3 (Paul et al., 3 Jul 2025). The discrete decision-theoretic paper states the parallel theorem for $$H_U^\alpha(p)=\sum_{i=1}^n p_i(-\log p_i)^\alpha,\qquad 0\le \alpha\le 1,$$4 and relates it to whether probabilities lie below or above $$H_U^\alpha(p)=\sum_{i=1}^n p_i(-\log p_i)^\alpha,\qquad 0\le \alpha\le 1,$$5 (Paul et al., 3 Jul 2025).

The transformation laws are asymmetric. Translation invariance holds:

$$H_U^\alpha(p)=\sum_{i=1}^n p_i(-\log p_i)^\alpha,\qquad 0\le \alpha\le 1,$$6

Scaling does not simplify to a linear law:

$$H_U^\alpha(p)=\sum_{i=1}^n p_i(-\log p_i)^\alpha,\qquad 0\le \alpha\le 1,$$7

Accordingly, FDE is translation invariant but not scale invariant (Paul et al., 3 Jul 2025).

For independent variables, the joint entropy is subadditive:

$$H_U^\alpha(p)=\sum_{i=1}^n p_i(-\log p_i)^\alpha,\qquad 0\le \alpha\le 1,$$8

with equality only at $$H_U^\alpha(p)=\sum_{i=1}^n p_i(-\log p_i)^\alpha,\qquad 0\le \alpha\le 1,$$9. A general chain rule is not established in the cited work (Paul et al., 3 Jul 2025).

Several bounds relate FDE to Shannon’s differential entropy. Under $\alpha=1$0,

$\alpha=1$1

For bounded support $\alpha=1$2, the paper also gives

$\alpha=1$3

together with further log-based upper and lower bounds (Paul et al., 3 Jul 2025).

Analytical evaluations are available for several standard families. For a uniform law on $\alpha=1$4 with $\alpha=1$5,

$\alpha=1$6

For an exponential law with rate $\alpha=1$7,

$\alpha=1$8

For a normal law with $\alpha=1$9,

$1$0

The same paper also reports formulas or numerical evaluations for gamma, Pareto II, triangular, Cauchy, Beta, Weibull, and generalized Pareto families (Paul et al., 3 Jul 2025).

4. FDE as Shannon entropy in fractional diffusion

In the diffusion setting, the entropy is the Shannon differential entropy of a time-dependent density:

$1$1

possibly regularized by the relative entropy with respect to a stationary density $1$2,

$1$3

Near stationarity,

$1$4

so it is minus the Kullback–Leibler divergence and is nonpositive (Aghamohammadi et al., 2013).

For linear, time-translationally invariant systems that admit a stationary density, entropy saturates exponentially:

$1$5

with $1$6, where $1$7 is the leading nonzero spectral mode (Aghamohammadi et al., 2013). On compact manifolds this produces explicit rates, such as $1$8 on the circle of radius $1$9 and $\alpha\to 0^+$0 on the two-sphere of radius $\alpha\to 0^+$1.

When no stationary density exists, the asymptotic law is logarithmic:

$\alpha\to 0^+$2

where $\alpha\to 0^+$3 arises from the scaling transformation $\alpha\to 0^+$4, $\alpha\to 0^+$5 (Aghamohammadi et al., 2013). For the general linear fractional equation

$\alpha\to 0^+$6

the coefficient is

$\alpha\to 0^+$7

In isotropic $\alpha\to 0^+$8 dimensions with a single spatial fractional order $\alpha\to 0^+$9, this becomes

$\alpha$0

This yields several standard specializations. For ordinary diffusion, $\alpha$1. For time-fractional subdiffusion,

$\alpha$2

For space-fractional diffusion,

$\alpha$3

the entropy growth law is

$\alpha$4

The one-dimensional Cauchy case $\alpha$5 has slope $\alpha$6 even though the variance diverges (Aghamohammadi et al., 2013).

A major significance of this formulation is methodological rather than terminological. Variance can fail for heavy-tailed fractional diffusions, whereas entropy remains finite and continues to encode the diffusion-speed coefficient $\alpha$7 (Aghamohammadi et al., 2013). This suggests a robust role for Shannon entropy as a transport diagnostic in anomalous regimes.

5. Fractional quantum harmonic oscillator and related information measures

For the one-dimensional fractional quantum harmonic oscillator, the governing equation is Laskin’s space-fractional Schrödinger equation with harmonic potential,

$\alpha$8

with

$\alpha$9

Although the title refers to the Riesz–Feller fractional derivative, the explicit operator definitions and computations are carried out for the symmetric Riesz case $\alpha\to 1$00, where the Fourier symbol is $\alpha\to 1$01 (Boumali et al., 2024).

The stationary problem is solved in momentum space. The ground-state momentum wavefunction is

$\alpha\to 1$02

and higher states are generated algebraically as

$\alpha\to 1$03

where $\alpha\to 1$04 are Riesz–Feller Hermite “polynomials” that reduce to ordinary Hermite polynomials as $\alpha\to 1$05 (Boumali et al., 2024). Position-space wavefunctions are obtained by inverse Fourier transform,

$\alpha\to 1$06

and are normalized numerically.

In this setting, the entropy functional is not fractionalized. The paper defines

$\alpha\to 1$07

and explicitly states that it “does not introduce a distinct ‘fractional’ differential entropy with altered kernels or fractional operators in the definition” (Boumali et al., 2024). The same applies to Fisher information,

$\alpha\to 1$08

which is computed in standard form.

The numerical study reports $\alpha\to 1$09 and $\alpha\to 1$10 for $\alpha\to 1$11, along with the entropy density $\alpha\to 1$12, the Fisher information density $\alpha\to 1$13, the Fisher–Shannon product

$\alpha\to 1$14

and the LMC complexity

$\alpha\to 1$15

(Boumali et al., 2024). The paper gives qualitative interpretations: smaller $\alpha\to 1$16 produces heavier-tailed and more nonlocal densities, typically associated with larger uncertainty and reduced sharpness, while the limit $\alpha\to 1$17 recovers the standard harmonic oscillator structure.

6. Decision-theoretic and hydraulic applications

In decision theory under risk, the discrete Ubriaco entropy

$\alpha\to 1$18

is used to define two risk measures. The first is Expected Utility–Fractional Entropy,

$\alpha\to 1$19

and the second is Expected Utility–Fractional Entropy and Variance,

$\alpha\to 1$20

Actions are ranked by minimizing $\alpha\to 1$21 (Paul et al., 3 Jul 2025). The PSI 20 application computes log-returns, bins them into $\alpha\to 1$22 intervals, estimates empirical pmfs, evaluates the risk measures, and then trains a feedforward ANN with scaled conjugate gradient using a $\alpha\to 1$23 train/validation/test split and $\alpha\to 1$24 bootstrap runs (Paul et al., 3 Jul 2025). The same paper notes that the continuous analogue of Ubriaco’s entropy is available but not employed. It also records a numerical inconsistency: although natural logarithms are stated, the hypothetical portfolio table matches base-$\alpha\to 1$25 logarithms (Paul et al., 3 Jul 2025).

In hydraulic modeling, continuous FDE is used as the objective in a maximum-entropy principle. For one-dimensional vertical velocity in wide open channels, the problem is to maximize

$\alpha\to 1$26

subject to normalization and a mean-velocity constraint, with $\alpha\to 1$27. Setting $\alpha\to 1$28 yields an explicit pdf and, after series approximation and cdf matching with $\alpha\to 1$29, the velocity profile

$\alpha\to 1$30

where

$\alpha\to 1$31

Regression coefficients $\alpha\to 1$32 range from $\alpha\to 1$33 to $\alpha\to 1$34, and the model is compared against the Chiu, SL, KG, and KT entropy-based formulations (Paul et al., 3 Jul 2025).

A closely related construction is used for the vertical distribution of suspended sediment concentration. The normalized concentration $\alpha\to 1$35 is treated as a random variable on $\alpha\to 1$36, with zero surface concentration and a type I monotone profile. Maximization of

$\alpha\to 1$37

under normalization and mean-concentration constraints, again with $\alpha\to 1$38, produces an explicit pdf and the normalized concentration profile

$\alpha\to 1$39

with

$\alpha\to 1$40

Across experimental and field datasets, reported $\alpha\to 1$41 values lie between $\alpha\to 1$42 and $\alpha\to 1$43, and comparisons are made with Tsallis-, Shannon-, Rouse-, Rényi-, and Fractional Wang-based models (Paul et al., 8 Jul 2025).

Taken together, these applications show two distinct operational roles for FDE. It functions either as a tunable uncertainty measure used directly in decision scores and ranking models, or as a variational objective whose maximizer supplies tractable distributional closures for complex open-channel flows. This suggests that the main unifying theme of FDE is not a single canonical formula, but a shared emphasis on fractional weighting of information content and on entropy-based characterization of systems with nonlocality, heavy tails, or multiscale structure.