---
title: Fractional Differential Entropy (FDE)
url: https://www.emergentmind.com/topics/fractional-differential-entropy-fde
type: topic
---

# Fractional Differential Entropy (FDE)

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 $\alpha\to 1$ [2507.02323]. 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 [1305.5407] [2409.11916]. 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 $\alpha\to 0^+$ [2507.02323]. The same discrete form is used in portfolio decision models under risk, where $\alpha$ is interpreted as a behavioral parameter: $\alpha\approx 1$ corresponds to risk-tolerant or adventurous attitudes, whereas $\alpha\approx 0$ corresponds to risk-averse or conservative attitudes [2507.02683].

The continuous analogue, termed Fractional Differential Entropy in the 2025 literature, is
$$
H^\alpha(f)=\int_0^\infty f(x)\,(-\log f(x))^\alpha\,dx,
$$
for a nonnegative absolutely continuous random variable with density $f$ [2507.02323]. 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
$$
S_{n,\alpha}(x)=-\int \rho_{n,\alpha}(x)\ln \rho_{n,\alpha}(x)\,dx
$$
without altering the entropy kernel itself [1305.5407] [2409.11916].

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 [1305.5407] [2409.11916].

## 2. Continuous Ubriaco-type FDE

For the continuous formulation, the basic object is the expectation of a fractional power of the negative log-likelihood:
$$
H^\alpha(f)=E\!\left[(-\log f(X))^\alpha\right].
$$
The formulation in [2507.02323] focuses on densities satisfying $0<f(x)\le 1$ on their support, so that $(-\log f(x))^\alpha$ remains real-valued. This requirement induces parameter restrictions or support truncations for standard families, such as $B-A\ge 1$ for uniform laws, $\lambda\le 1$ for exponential laws, and $\sigma\ge 1/\sqrt{2\pi}$ for normal laws.

Two dynamic variants are introduced but not further developed in that paper. The past FDE is
$$
\bar H^\alpha(f;t)=\int_0^t \frac{f(x)}{F(t)}\left[-\log\!\left(\frac{f(x)}{F(t)}\right)\right]^\alpha dx,
$$
and the residual FDE is
$$
H^\alpha(f;t)=\int_t^\infty \frac{f(x)}{\bar F(t)}\left[-\log\!\left(\frac{f(x)}{\bar F(t)}\right)\right]^\alpha dx,
$$
where $F(t)$ is the cdf and $\bar F(t)=1-F(t)$ [2507.02323].

The limiting behavior is explicit. As $\alpha\to 1$,
$$
\lim_{\alpha\to 1}H^\alpha(f)=\int f(x)(-\log f(x))\,dx=H_S(f),
$$
so the construction recovers Shannon’s differential entropy. As $\alpha\to 0^+$,
$$
\lim_{\alpha\to 0^+}H^\alpha(f)=\int f(x)\,dx=1.
$$
Thus the map $\alpha\mapsto H^\alpha(f)$ continuously deforms from $1$ at $\alpha=0^+$ to Shannon’s differential entropy at $\alpha=1$ [2507.02323].

The continuous finance paper explicitly notes a direct analogue,
$$
H_{U,\mathrm{diff}}^\alpha(f)=\int_{\mathbb{R}^d} f(x)\,(-\log f(x))^\alpha dx,
$$
but does not use it in the empirical portfolio study, which remains discrete and bin-based [2507.02683].

## 3. Structural properties and analytic evaluations

For $0<f\le 1$ and $0<\alpha\le 1$, the integrand
$$
h^\alpha(f)=f(-\log f)^\alpha
$$
is nonnegative and concave in $f$, and $H^\alpha(f)$ is accordingly concave on the convex set of densities bounded by $1$ [2507.02323]. The pointwise maximum of $h^\alpha(f)$ occurs at
$$
f=e^{-\alpha},
$$
so values of the density near $e^{-\alpha}$ contribute առավելally to the local fractional information density [2507.02323].

Differentiation with respect to $\alpha$ gives
$$
\frac{d}{d\alpha}H^\alpha(f)=\int f(x)\,I(x)^\alpha\log I(x)\,dx,\qquad I(x)=-\log f(x).
$$
This yields a dichotomy. If $I(x)\in(0,1)$, equivalently $f(x)\in(e^{-1},1)$, then $\log I(x)<0$ and $H^\alpha(f)$ decreases in $\alpha$. If $I(x)\ge 1$, equivalently $f(x)\le e^{-1}$, then $H^\alpha(f)$ increases in $\alpha$ [2507.02323]. The discrete decision-theoretic paper states the parallel theorem for $h_U^\alpha(p)=p(-\log p)^\alpha$ and relates it to whether probabilities lie below or above $1/e$ [2507.02683].

The transformation laws are asymmetric. Translation invariance holds:
$$
H^\alpha(f_{X+c})=H^\alpha(f_X).
$$
Scaling does not simplify to a linear law:
$$
H^\alpha(f_{mX})=\int f_X(u)\,[ -\log f_X(u)+\log m]^\alpha\,du.
$$
Accordingly, FDE is translation invariant but not scale invariant [2507.02323].

For independent variables, the joint entropy is subadditive:
$$
H^\alpha(f_{X,Y})\le H^\alpha(f_X)+H^\alpha(f_Y),\qquad 0<\alpha\le 1,
$$
with equality only at $\alpha=1$. A general chain rule is not established in the cited work [2507.02323].

Several bounds relate FDE to Shannon’s differential entropy. Under $0<f\le 1$,
$$
H^\alpha(f)\le [H_S(f)]^\alpha.
$$
For bounded support $[0,b]$, the paper also gives
$$
H^\alpha(f)\le b^{1-\alpha}[H_S(f)]^\alpha,
$$
together with further log-based upper and lower bounds [2507.02323].

Analytical evaluations are available for several standard families. For a uniform law on $[A,B]$ with $B-A\ge 1$,
$$
H^\alpha(f)=[\log(B-A)]^\alpha.
$$
For an exponential law with rate $\lambda\le 1$,
$$
H^\alpha(f)=\int_0^\infty e^{-y}\,[\log(1/\lambda)+y]^\alpha\,dy
=\frac{1}{\lambda}\Gamma(1+\alpha,\log(1/\lambda)).
$$
For a normal law with $\sigma\ge 1/\sqrt{2\pi}$,
$$
H^\alpha(f)=\frac{1}{\sqrt{\pi}}\int_0^\infty e^{-t}t^{-1/2}\big(t+\log(\sqrt{2\pi}\sigma)\big)^\alpha dt.
$$
The same paper also reports formulas or numerical evaluations for gamma, Pareto II, triangular, Cauchy, Beta, Weibull, and generalized Pareto families [2507.02323].

## 4. FDE as Shannon entropy in fractional diffusion

In the diffusion setting, the entropy is the Shannon differential entropy of a time-dependent density:
$$
S(t)=-\int \rho(x,t)\ln \rho(x,t)\,dx,
$$
possibly regularized by the relative entropy with respect to a stationary density $\rho_\infty$,
$$
S_{\rho_\infty}(t)=-\int \rho(x,t)\ln\!\left[\frac{\rho(x,t)}{\rho_\infty(x)}\right]dx.
$$
Near stationarity,
$$
S_{\rho_\infty}(t)= -\int \frac{[\rho(x,t)-\rho_\infty(x)]^2}{\rho_\infty(x)}\,dx+\text{higher orders},
$$
so it is minus the Kullback–Leibler divergence and is nonpositive [1305.5407].

For linear, time-translationally invariant systems that admit a stationary density, entropy saturates exponentially:
$$
S(t)=S(\rho_\infty)+Ce^{-\lambda t}+o(e^{-\lambda t}),
$$
with $\lambda=-2\operatorname{Re}(E_1)>0$, where $E_1$ is the leading nonzero spectral mode [1305.5407]. On compact manifolds this produces explicit rates, such as $\lambda=2/a^2$ on the circle of radius $a$ and $\lambda=4/a^2$ on the two-sphere of radius $a$.

When no stationary density exists, the asymptotic law is logarithmic:
$$
S(t)=c\ln t+S_0+o(1),
$$
where $c=\operatorname{tr}(R)$ arises from the scaling transformation $t\to e^a t$, $x\to e^{aR}x$ [1305.5407]. For the general linear fractional equation
$$
D_t^{\beta_0}\rho=\sum_j a_j D_{x_j}^{\beta_j}\rho,
$$
the coefficient is
$$
c=\sum_j \frac{\beta_0}{\beta_j}.
$$
In isotropic $d$ dimensions with a single spatial fractional order $\alpha$, this becomes
$$
c=d\frac{\beta_0}{\alpha}.
$$

This yields several standard specializations. For ordinary diffusion, $c=d/2$. For time-fractional subdiffusion,
$$
S(t)=S_0+\frac{d\beta}{2}\ln t+o(1).
$$
For space-fractional diffusion,
$$
\partial_t\rho=-D(-\Delta)^{\alpha/2}\rho,\qquad 0<\alpha\le 2,
$$
the entropy growth law is
$$
S(t)=S_0+\frac{d}{\alpha}\ln(Dt)+o(1).
$$
The one-dimensional Cauchy case $\alpha=1$ has slope $c=1$ even though the variance diverges [1305.5407].

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 $c$ [1305.5407]. 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,
$$
i\hbar \frac{\partial \Psi(x,t)}{\partial t}
=
\left[
D_\alpha(-\hbar^2\Delta)^{\alpha/2}+\frac{1}{2}m\omega^2x^2
\right]\Psi(x,t),\qquad 1<\alpha\le 2,
$$
with
$$
D_\alpha=\left(\frac{1}{2m}\right)^{\alpha/2}.
$$
Although the title refers to the Riesz–Feller fractional derivative, the explicit operator definitions and computations are carried out for the symmetric Riesz case $\theta=0$, where the Fourier symbol is $|p|^\alpha$ [2409.11916].

The stationary problem is solved in momentum space. The ground-state momentum wavefunction is
$$
\phi_0^{(\alpha)}(k)=\exp\!\left(-\frac{2|k|^{\frac{\alpha}{2}+1}}{\alpha+2}\right),
$$
and higher states are generated algebraically as
$$
\phi_n(k)=i^n\,\widetilde H_n(k)\,\phi_0^{(\alpha)}(k),
$$
where $\widetilde H_n(k)$ are Riesz–Feller Hermite “polynomials” that reduce to ordinary Hermite polynomials as $\alpha\to 2$ [2409.11916]. Position-space wavefunctions are obtained by inverse Fourier transform,
$$
\psi_n(x)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}e^{ikx}\phi_n(k)\,dk,
\qquad
\rho_n(x)=|\psi_n(x)|^2,
$$
and are normalized numerically.

In this setting, the entropy functional is not fractionalized. The paper defines
$$
S_{n,\alpha}(x)=-\int_{-\infty}^{+\infty}\rho_{n,\alpha}(x)\ln \rho_{n,\alpha}(x)\,dx,
$$
and explicitly states that it “does not introduce a distinct ‘fractional’ differential entropy with altered kernels or fractional operators in the definition” [2409.11916]. The same applies to Fisher information,
$$
F_{n,\alpha}(x)=\int_{-\infty}^{+\infty}\frac{[\partial_x\rho_{n,\alpha}(x)]^2}{\rho_{n,\alpha}(x)}\,dx,
$$
which is computed in standard form.

The numerical study reports $S_x(\alpha)$ and $F_x(\alpha)$ for $n=0,1,2,3$, along with the entropy density $-\rho\ln\rho$, the Fisher information density $(\partial_x\rho)^2/\rho$, the Fisher–Shannon product
$$
P=\frac{1}{2\pi e}e^{\frac{2}{3}S}F,
$$
and the LMC complexity
$$
C=e^S\int \rho^2(x)\,dx
$$
[2409.11916]. The paper gives qualitative interpretations: smaller $\alpha$ produces heavier-tailed and more nonlocal densities, typically associated with larger uncertainty and reduced sharpness, while the limit $\alpha\to 2$ recovers the standard harmonic oscillator structure.

## 6. Decision-theoretic and hydraulic applications

In decision theory under risk, the discrete Ubriaco entropy
$$
H_U^\alpha(p)=\sum_x p(x)(-\log p(x))^\alpha,\qquad \alpha\in[0,1],
$$
is used to define two risk measures. The first is Expected Utility–Fractional Entropy,
$$
R_{\mathrm{EU-FE}}(y)
=
\lambda H_y^\alpha(\theta)
-
(1-\lambda)
\frac{E[u(X(y,\theta))]}{\max_{y'\in Y}|E[u(X(y',\theta))]|},
$$
and the second is Expected Utility–Fractional Entropy and Variance,
$$
R_{\mathrm{EU-FEV}}(y)
=
\frac{\lambda}{2}
\left[
H_y^\alpha(\theta)+
\frac{\mathrm{Var}[X(y,\theta)]}{\max_{y'\in Y}\mathrm{Var}[X(y',\theta)]}
\right]
-
(1-\lambda)
\frac{E[u(X(y,\theta))]}{\max_{y'\in Y}|E[u(X(y',\theta))]|}.
$$
Actions are ranked by minimizing $R(y)$ [2507.02683]. The PSI 20 application computes log-returns, bins them into $J=15$ intervals, estimates empirical pmfs, evaluates the risk measures, and then trains a feedforward ANN with scaled conjugate gradient using a $70\%/15\%/15\%$ train/validation/test split and $100$ bootstrap runs [2507.02683]. 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-$10$ logarithms [2507.02683].

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
$$
H^\alpha(f)=\int_0^1 f(\hat\nu)\,[-\log f(\hat\nu)]^\alpha d\hat\nu
$$
subject to normalization and a mean-velocity constraint, with $\hat\nu=\nu/\nu_{\max}\in[0,1]$. Setting $\alpha=1/2$ yields an explicit pdf and, after series approximation and cdf matching with $F(\hat\nu)=(y/M)^k$, the velocity profile
$$
\hat\nu(y)=\frac{1}{b}\left[-a-\sqrt{a^2-2\sqrt{2}\,b\,e^{1/2}(y/M)^k}\right],
$$
where
$$
a=\sqrt{2}e^{1/2}(6\hat\nu_m-4),\qquad
b=-2\sqrt{2}e^{1/2}(6\hat\nu_m-3).
$$
Regression coefficients $R^2$ range from $0.97233$ to $0.99996$, and the model is compared against the Chiu, SL, KG, and KT entropy-based formulations [2507.02323].

A closely related construction is used for the vertical distribution of suspended sediment concentration. The normalized concentration $\hat{\mathcal C}=\mathcal C/c_r$ is treated as a random variable on $[0,1]$, with zero surface concentration and a type I monotone profile. Maximization of
$$
H^\alpha(f)=\int_0^1 f(\hat c)(-\log f(\hat c))^\alpha d\hat c
$$
under normalization and mean-concentration constraints, again with $\alpha=1/2$, produces an explicit pdf and the normalized concentration profile
$$
\hat c(y)=\frac{1}{\lambda_1}\left[
-\lambda_0+\sqrt{\lambda_0^2+\lambda_1\,(1-\hat Y^a)e^{-\hat Y^a}\,2\sqrt2\,e^{1/2}}
\right],
\qquad
\hat Y=\frac{y-y_r}{h-y_r},
$$
with
$$
\lambda_0=(4-6\hat c_m)\sqrt2\,e^{1/2},\qquad
\lambda_1=6(2\hat c_m-1)\sqrt2\,e^{1/2}.
$$
Across experimental and field datasets, reported $R^2$ values lie between $0.9603$ and $0.9958$, and comparisons are made with Tsallis-, Shannon-, Rouse-, Rényi-, and Fractional Wang-based models [2507.05986].

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.

Source: https://www.emergentmind.com/topics/fractional-differential-entropy-fde