q-Homogeneous Fractal Structure

• q-Homogeneous fractal structure is a hierarchical system where the entropic index q governs self-similar scaling and q-exponential distributions.
• It bridges microscopic dynamics with observable phenomena in thermofractals, Yang–Mills theory, scattering experiments, and cosmic structure analyses.
• The framework quantitatively links scaling laws, entropy, and fractal dimensions, enabling transitions from multifractality to monofractality in complex systems.

A q-homogeneous fractal structure is a hierarchical system wherein the scaling properties of observables or measures are controlled by the entropic index $q$, with self-similarity or self-affinity under level iteration and the emergence of q-exponential laws. Such systems appear in statistical mechanics (thermofractals), quantum field theory (Yang–Mills fractals), spatial correlation analysis (cosmic structure), optical scattering, and in the algebraic framework of deterministic geometric fractals. The parameter $q$ prescribes not just the probability law at every hierarchical stage, but also fixes the Hausdorff (fractal) dimension and determines whether the structure is strictly monofractal or multi-fractal at different scales. The concept generalizes traditional notions of homogeneity and allows for rigorous quantitative connection between scaling, entropy, and measure concentration in highly structured systems.

1. Definition and Core Principles

A q-homogeneous fractal structure is exemplified by a system—such as a thermofractal—wherein:

• The total quantity of interest (e.g., energy, structure factor, probability) recursively decomposes into kinetic and internal components, with the internal component distributed among $N'$ or $N$ subsystems.
• Each subsystem recursively inherits the same statistical (thermodynamic, geometric, or algebraic) law as the whole, i.e., self-similarity.
• The probability distribution for the key variable at every hierarchical level is q-invariant up to scale, adopting the q-exponential (Tsallis–Pareto) form:

$$P(\varepsilon) = A \left[1 + (q-1)\frac{\varepsilon}{kT}\right]^{-1/(q-1)}$$

where $q$ is the non-extensive entropic index, $T$ an effective temperature, and $A$ a normalization constant.

• The structure is called q-homogeneous because dilation scales the variable's distribution by a degree $-1/(q-1)$:

$$P(\lambda \varepsilon) = \lambda^{-1/(q-1)} P(\varepsilon), \qquad \forall \lambda > 0$$

which defines the fractal's homogeneity class (Deppman, 2016, Deppman et al., 2020).

2. Mathematical Framework: Fractal Dimensions and Scaling Laws

The scaling character of a q-homogeneous fractal structure is encapsulated by its Hausdorff dimension $D_H(q)$. For a thermofractal with $N'$ subsystems, the covering and scaling relations give (Deppman, 2016, Deppman et al., 2020):

$$D_H(q) = 1 + \frac{\ln\left[\frac{(q-1)N'}{3-2q}\right]}{\ln N'}$$

A generalization to physical observables leads to the following paradigm:

• For mass or structure factor scaling (e.g., small-angle scattering):

$S(q) \propto q^{-d_f}$

over the genuine fractal regime, where $d_f$ is the actual fractal (mass) dimension, and $q$ here denotes the scattering vector, not the entropic parameter (Katyal et al., 2016).

• For generalized (Rényi or Minkowski–Bouligand) dimension spectrum $D_q$ measured from data distributions or field configurations:

$$D_q(r) = \frac{1}{q-1} \frac{d\, \log C_q(r)}{d \log r}$$

with $C_q(r)$ the q-th moment of neighbor-counts within a ball of radius $r$. When $D_q$ is independent of $q$, the structure is monofractal; otherwise, multifractal (Goyal et al., 2024).

In the multifractal formalism, the Lipschitz–Hölder exponent $\alpha(q)$ satisfies

$\alpha(q) = D_H(q)$

Thus, in a truly q-homogeneous fractal, the singularity spectrum is degenerate, implying all local exponents collapse to a single value (monofractality) (Deppman, 2016).

3. Physical Realizations: Statistical Mechanics and Field Theory

Thermofractals and Non-Extensivity

In statistical mechanics, q-homogeneous fractal structure is realized by “thermofractals” whose subsystem energy distributions follow Tsallis (q-exponential) statistics at every hierarchy level. The recursive thermodynamic structure mandates that at every depth, the observable distributions are not strictly Boltzmann–Gibbs but take a power-law (q-exponential) form, with the index $q$ determined by the system parameters:

$q-1 = \frac{1-v}{3N'}$

The physical manifestation is that scaling of subsystem energies, and hence all macroscopic observables, preserves the distributional form under level iteration, reflecting in experimentally measured observables such as hadron spectra (Deppman, 2016).

q-Homogeneous Fractals in Yang–Mills Theory

In Yang–Mills gauge theories, the emergence of q-homogeneous fractal structure arises when the requirement of self-similarity in effective partonic configurations is imposed on statistical ensembles. The field-theoretic analysis demonstrates that the q-parameter is directly computed from the one-loop $\beta$-function:

$$q = 1 + \frac{1}{\left(\frac{11}{3}N_c - \frac{2}{3}N_f\right)}$$

For QCD with $N_c = 3, N_f = 3$, this gives $q = 1.1428...$, in agreement with phenomenology. The multiplicative homogeneity property,

$P(q \varepsilon) = q^{-1/(q-1)} P(\varepsilon)$

demonstrates that the energy flow among partons respects q-scaling, and this scaling fully determines the fractal dimension and the pattern of non-extensivity (Deppman et al., 2020).

4. Experimental and Observational Signatures

Small-Angle Scattering and Electromagnetic Structure

In SAXS/SANS and related experiments, the genuine fractal regime is characterized by a power-law scaling of the structure factor $S(q) \propto q^{-d_f}$ over a well-defined $q$-window. The multiplicative q-homogeneity is robust under weak multiple scattering as established by the mean-field criterion:

$(N-1)|\chi(x)A(kR_g)| \ll 1$

If this is not met, anomalous scaling regimes emerge (with exponents $\delta \neq d_f$) that must not be confused with intrinsic fractality. The existence of an unambiguous, $q$-homogeneous scaling window is thus essential for model-free extraction of the true fractal dimension of aggregates (Katyal et al., 2016).

Cosmic Structure: Multi- and Mono-Fractality

Cosmological surveys reveal a scale-dependent transition between multifractal (q-dependent $D_q$) and monofractal ($D_q$ independent of $q$) regimes in the spatial distribution of cosmic objects such as quasars. At scales $r \lesssim 80h^{-1}$ Mpc, the spectrum $D_q(r)$ is strongly q-dependent, indicative of hierarchical clustering (multi-fractality). For $r \gtrsim 110h^{-1}$ Mpc, $D_q(r) \simeq$ const $\simeq 2.85-2.90$ for $-5 \leq q \leq +5$, signaling statistical homogeneity and q-homogeneous monofractality (Goyal et al., 2024). This transition scale is fundamental for confronting the cosmological principle and constructing large-scale structure models.

5. Algebraic and Geometric Realizations: q-Deformation, Squeezing, and Quantum Geometry

The algebraic underpinning of q-homogeneous fractal structure is formalized via q-deformed algebras and coherent states:

• The operator $q^N$ acts as a scaling generator in Bargmann–Fock space, producing fractal copies $u_{n,q}(\alpha) = (q\alpha)^n$ invariant under $n \to n+1$, $\alpha \to q\alpha$.
• The q-derivative (Jackson derivative), $$D_q f(\alpha) = \frac{f(q\alpha) - f(\alpha)}{(q-1)\alpha}$$, encodes fractal self-similarity; the self-similarity dimension $d$ is given by $d = -(\ln q)/(\ln b)$.
• In coordinate representation, q-deformation generates noncommutative geometry, with commutator $[x_1,x_2] = i q^2$, linking fractality to quantized phase space and the emergence of interference phases analogous to the Aharonov–Bohm effect (Vitiello, 2012).
• The squeezing operator $q^N = e^{\zeta N}$ (with $\zeta = \ln q$) provides a dynamical realization of geometric fractals (e.g., Koch curve, logarithmic spiral) as squeezed coherent condensates, unifying fractal morphogenesis with quantum statistical structure.

6. Implications for Interactions, Dynamics, and Measure Theory

The reconciliation of q-homogeneous fractal structure with underlying microscopic dynamics is achieved by linking the macroscopic scaling of measures (energy, correlation, structure factor) to the form of microscopic interactions or combinatorial rules:

• In thermofractals, the equivalence between the thermodynamic potential and Dashen–Ma–Bernstein formalism constraints the S-matrix, requiring the multiparticle scattering phase to organize into a form producing Tsallis q-exponential statistics. Deviations from ordinary Boltzmann behavior trace directly to the presence and scale of the underlying fractal hierarchy (Deppman, 2016).
• In measure-theoretic terms, a q-homogeneous structure is defined by measures $\mu$ that satisfy

$\mu(\lambda X) = \lambda^{-\frac{1}{q-1}}\mu(X)$

over the dilation group, which ensures invariance of local concentrations and a power-law scaling of singularities precisely at the order set by $q$.

7. Summary Table: Defining Properties Across Contexts

Context	Homogeneity Law / Dimension	Scaling Regime/Observable
Thermofractals (Deppman, 2016)	$$P(\lambda \varepsilon) = \lambda^{-1/(q-1)} P(\varepsilon)$$	Energy distributions, S-matrix
Yang–Mills (Deppman et al., 2020)	$q = 1 + 1/(\frac{11}{3}N_c - \frac{2}{3}N_f)$	Effective vertices/partons
SAXS/SANS fractals (Katyal et al., 2016)	$S(q) \propto q^{-d_f}$, $d_f$ robust if mean-field holds	Scattering structure factor
Cosmic structure (Goyal et al., 2024)	$D_q(r) \rightarrow \text{const}$ for $r \gtrsim 110h^{-1}$ Mpc	Counts-in-spheres, correlation dim.
Algebraic construction (Vitiello, 2012)	$q^N f(\alpha) = f(q\alpha)$, $d = -(\ln q)/(\ln b)$	Operator iterates, coherent states

This synthesis highlights that a q-homogeneous fractal structure encodes a precise algebraic, analytic, and physical recipe for invariant scaling across hierarchical levels, with the entropic index $q$ central to its geometry, observable laws, and dynamical or combinatorial origins.