Investigations into K-Statistical Mechanics and K-Entropy Through the Principles of the Theory of Entropicity (ToE)

The concept that entropy is a field from which relativistic kinematics can be derived, or more specifically, that relativistic kinematic effects emerge from an entropic field (S), is postulated in the Theory of Entropicity (ToE), primarily developed by John Onimisi Obidi. 

Key aspects of this theory include: 

• The Entropic Field as a Regulatory Agent: ToE redefines entropy not just as a statistical measure of disorder, but as a dynamic, universal field (
  

  $S\left(x\right)$
  or
  

  $A\left(x\right)$
  ) that governs the structure of spacetime, time, and interaction.
• Derivation of the Speed of Light (
  

  $c$
  ): The constant speed of light is derived as the maximum propagation rate of disturbances within this entropic field, rather than being a standalone, unexplained postulate.
• Relativistic Effects as Entropic Resistance: Time dilation and length contraction are interpreted as manifestations of local entropic field distortions, where motion near
  

  $c$
  increases "entropic drag".
• The No-Rush Theorem: This principle states that no physical process can occur faster than the entropic field permits, effectively enforcing the relativistic speed limit through thermodynamics. 

Additionally, related research by G. Kaniadakis on

$\kappa$

-entropy has demonstrated that the self-duality and scaling axioms of

$\kappa$

-statistical mechanics can be derived directly from the first principles of special relativity, showing that

$\kappa$

-entropy acts as a relativistic generalization of classical entropy. However, the "Theory of Entropicity" by Obidi is the specific, recent framework explicitly framing entropy as a field that generates relativistic motion constraints. 

Generalized entropies such as Tsallis, Rényi, and Kaniadakis entropies have expanded the toolkit of modern statistical mechanics. They allow researchers to model systems with long‑range interactions, heavy‑tailed distributions, relativistic constraints, and non‑Gaussian behavior. Among these, the Kaniadakis framework—built around the so‑called “kappa‑entropy”—is especially notable for its compatibility with relativistic symmetries and its elegant deformation of the exponential and logarithmic functions.

This paper examines K‑Statistical Mechanics and K‑Entropy through the conceptual lens of the Theory of Entropicity (ToE), clarifying how these frameworks relate, where they differ, and why they do not threaten the foundations of ToE.

1. What K‑Entropy and K‑Statistical Mechanics Actually Are

Kaniadakis entropy is defined using a deformed logarithm. In plain text form:

- K‑entropy:  

  Sk = – Σ pi * lnk(pi)  

- K‑logarithm:  

  ln_k(x) = (x^k – x^(–k)) / (2k)

When k → 0, this reduces to the classical Boltzmann–Gibbs entropy.

K‑statistics is therefore a generalized statistical mechanics, not a geometric or ontological theory. It is designed to model systems where classical statistics fail, especially in relativistic or heavy‑tailed regimes.

2. Entropy in ToE: A Completely Different Concept

In the Theory of Entropicity, entropy is not a statistical measure. It is a fundamental field, written conceptually as S(x), defined over spacetime. This field has:

- gradients  

- curvature  

- geometric structure  

- dynamical evolution governed by PoLER (the Principle of Least Entropic Resistance)

Entropy in ToE is therefore:

- ontological  

- geometric  

- continuous  

- field‑theoretic  

It is not a probability‑based functional.

This distinction is essential: ToE does not depend on any particular statistical entropy formula. It does not rely on Boltzmann, Shannon, Tsallis, or Kaniadakis entropy.

3. Why K‑Entropy Is Fully Compatible With ToE

Because ToE defines entropy as a geometric field, generalized statistical entropies cannot contradict or undermine it. They operate at different conceptual levels:

- K‑entropy describes statistical behavior of ensembles.  

- ToE entropy describes the geometric substrate of physical reality.

K‑entropy modifies the algebraic form of statistical entropy.  

ToE defines entropy as the underlying field from which geometry and dynamics emerge.

These two frameworks do not collide.

4. K‑Entropy as a Statistical Shadow of Entropic Geometry

From the viewpoint of ToE, statistical entropies arise when the entropic field is coarse‑grained over ensembles or macroscopic observables. Different coarse‑grainings produce different effective entropy functionals.

In this sense:

K‑entropy corresponds to a particular statistical regime of the entropic field.

Heavy‑tailed distributions, relativistic constraints, and non‑Gaussian behavior can be interpreted as manifestations of specific entropic curvature structures. K‑statistics therefore fits naturally within ToE as a derived approximation, not a competing ontology.

5. Clarifying the Role of Hilbert‑Space Representations

Some generalized entropies—including K‑entropy—use operator‑based or Hilbert‑space representations. This does not imply:

- that entropy is fundamentally quantum  

- that ToE requires a Hilbert‑space ontology  

- that ToE reduces to spectral geometry  

In ToE, such representations are simply mathematical tools for encoding aspects of entropic geometry. They do not define the ontology of the theory.