Physics:Rényi Entropy Derived from Obidi's Theory of Entropicity(ToE)

Rényi Entropy Derived from Obidi's Theory of Entropicity (ToE)

The Theory of Entropicity(ToE), first formulated and developed by John Onimisi Obidi, stands within a lineage of profound attempts to link entropy, information, and geometry, yet it departs from all prior frameworks in a decisive way. Amari and Čencov pioneered the use of α‑connections to describe the geometry of statistical manifolds, but their work remained confined to the mathematics of inference. Verlinde proposed gravity as an emergent entropic force, but entropy in his framework is a derivative, holographic bookkeeping device. Jacobson derived Einstein’s equations from thermodynamic relations, but entropy there is still a statistical measure imposed on spacetime, not a field in its own right. Bianconi and others have recently advanced “gravity from entropy” approaches, but these remain couched in emergent or effective‑field perspectives.

By contrast, ToE makes the radical ontological move of declaring entropy itself to be the primary field of reality, with the Amari–Čencov α‑connections elevated from statistical tools to the universal transport law governing all flows — physical, informational, and cognitive, among others. Geometry, time, energy, and information are not independent primitives but emergent expressions of entropy dynamics. Where others treat entropy as secondary, the Theory of Entropicity (ToE) treats it as fundamental; where others restrict entropy to a small class in physics, ToE extends it to all domains of physics — and even beyond, including clinical practice and the cognitive sciences.

In this sense, ToE is not a rehash but a genuine leap: just as Einstein’s postulate of the constancy of (the speed of light) c redefined space and time, ToE’s postulate of entropy as the universal field redefines the foundations of physics and extends them into a unified science of existence and reality.

Hence, we can summarize as follows:

Where Amari and Čencov used (alpha) α‑connections to describe statistical inference, Jacobson derived Einstein’s equations from thermodynamics, Verlinde cast gravity as an emergent entropic force, and Bianconi framed spacetime as an entropic action, the Theory of Entropicity makes the radical leap of declaring entropy itself the fundamental field of reality, with (alpha) α‑connections elevated to the universal law of its flow across physics and cognitive science.

Thus:

- Amari/Čencov = statistical geometry.

- Jacobson = thermodynamic derivation.

- Verlinde = emergent entropic force.

- Bianconi = entropic action, but still emergent.

- ToE (Obidi) = entropy as the fundamental field, (alpha) α‑connections as universal law, cross‑domain universality.

References:

In this instalment, we wish to show how the Obidi Action of the Theory of Entropicity(ToE) helps us to derive the Rényi Entropy, like we have derived other expressions of entropy in thermodynamics and information theory. Thereafter, we demonstrate how we incorporate these notions into deriving the field equations of the Theory of Entropicity(ToE) as supersets of Einstein's field equations of his beautiful Theory of Relativity(ToE).

1. Definition

The Rényi entropy of order \( \alpha \) for a discrete probability distribution \( P = \{p_1, p_2, \dots, p_n\} \) is defined as:

[math]\displaystyle{ H_{\alpha}(P) = \frac{1}{1-\alpha} \, \log \left( \sum_{i=1}^n p_i^{\alpha} \right), \quad \alpha \gt 0, \, \alpha \neq 1. }[/math]

This generalizes Shannon entropy, which is recovered in the limit: [math]\displaystyle{ \lim_{\alpha \to 1} H_{\alpha}(P) = - \sum_{i=1}^n p_i \log p_i. }[/math]

2. Special Cases

• **Hartley entropy (max‑entropy):**

 [math]\displaystyle{  H_0(P) = \log n  }[/math] (all outcomes equally likely).  

• **Shannon entropy:**

 [math]\displaystyle{  H_1(P) = -\sum_i p_i \log p_i  }[/math].  

• **Collision entropy:**

 [math]\displaystyle{  H_2(P) = -\log \sum_i p_i^2  }[/math].  

• **Min‑entropy:**

 [math]\displaystyle{  H_{\infty}(P) = -\log \max_i p_i  }[/math].  

3. ToE Interpretation

In the *Theory of Entropicity*:

• The Rényi parameter \( \alpha \) is interpreted as a **curvature parameter** of the entropic manifold, governed by the Amari–Čencov α‑connections.
• By introducing \( \alpha \) into the **entropy action functional** (e.g., via a nonstandard kinetic term or weighted multiplier), ToE interpolates smoothly between different entropy regimes.
• This allows ToE to describe systems at different scales of **entropy sensitivity**:

 - \( \alpha \to 0 \): maximal ignorance (Hartley).  
 - \( \alpha = 1 \): balanced uncertainty (Shannon).  
 - \( \alpha = 2 \): pairwise collisions dominate (collision entropy).  
 - \( \alpha \to \infty \): extreme determinism (min‑entropy).  

4. Conceptual Significance

• In standard information theory, Rényi entropy is a mathematical generalization.
• In ToE, Rényi entropy is **physical**: the parameter \( \alpha \) encodes how entropy flows are transported across the entropic manifold.
• Different values of \( \alpha \) correspond to different **entropic geometries**, allowing ToE to unify statistical mechanics, quantum information, and cosmology under a single entropic law.

5. Summary

• Shannon entropy is not the end of the story — it is one point in a broader entropic spectrum.
• ToE elevates Rényi entropy from a mathematical generalization to a **physical principle**, with \( \alpha \) acting as the universal dial that tunes the curvature of entropy flows.

Overview

The Theory of Entropicity (ToE) is a proposed universal framework that elevates entropy from a secondary thermodynamic measure to the primary field of reality. In this view, geometry, time, energy, and information are emergent expressions of entropy flows governed by the Amari–Čencov α‑connections. The constancy of the speed of light, \(c\), is reinterpreted as the invariant propagation speed of the entropic field itself.

Positioning Statement

The Theory of Entropicity stands within a lineage of attempts to link entropy, information, and geometry, yet it departs from all prior frameworks in a decisive way. Amari and Čencov pioneered the use of α‑connections to describe the geometry of statistical manifolds, but their work remained confined to inference. Jacobson derived Einstein’s equations from thermodynamic relations, but entropy there is still a statistical measure imposed on spacetime. Verlinde cast gravity as an emergent entropic force, but entropy in his framework is a holographic bookkeeping device. Bianconi and others have advanced “gravity from entropy” approaches, but these remain emergent or effective‑field perspectives.

By contrast, ToE makes the radical ontological move of declaring entropy itself to be the fundamental field, with α‑connections elevated from statistical tools to the universal transport law governing all flows — physical, informational, cognitive, and civilizational. Geometry, time, energy, and information are not independent primitives but emergent expressions of entropy dynamics. Where others treat entropy as secondary, ToE treats it as fundamental; where others restrict entropy to physics, ToE extends it to life, mind, and society.

Einstein Analogy

• Einstein’s postulate: The speed of light \(c\) is constant and universal → relativity follows.
• ToE’s postulate: Entropy is the fundamental field → geometry, time, energy, and information follow.

In ToE, the constancy of \(c\) is not an independent axiom but the natural invariant of entropy transport. Thus, \(c\) is explained as the maximum rate at which entropy — and therefore information, energy, and causality — can propagate.

Rényi Entropy in ToE

The Rényi entropy of order \( \alpha \) for a distribution \(P = \{p_i\}\) is defined as: [math]\displaystyle{ H_{\alpha}(P) = \frac{1}{1-\alpha} \log \left( \sum_i p_i^{\alpha} \right), \quad \alpha \gt 0, \, \alpha \neq 1. }[/math]

Special cases:

• \( \alpha \to 1 \): Shannon entropy.
• \( \alpha = 0 \): Hartley entropy (max‑entropy).
• \( \alpha = 2 \): Collision entropy.
• \( \alpha \to \infty \): Min‑entropy.

In ToE:

• The Rényi parameter \( \alpha \) is interpreted as a curvature parameter of the entropic manifold.
• By introducing \( \alpha \) into the entropy action functional, ToE interpolates smoothly between different entropy regimes.
• Different values of \( \alpha \) correspond to different entropic geometries, allowing ToE to unify statistical mechanics, quantum information, and cosmology.

The Road to the Field Equations of the Theory of Entropicity(ToE)

Einstein’s Equations

[math]\displaystyle{ G_{\mu\nu} + \Lambda g_{\mu\nu} = \kappa T_{\mu\nu} }[/math]

ToE Master Entropic Equation (MEE)

The entropy‑field action generalized to Rényi entropy is the Obidi Action of the Theory of Entropicity(ToE), given as:

[math]\displaystyle{ \mathcal{S}_\alpha = \int d^4x \, \sqrt{-g} \left[ \frac{1}{2\kappa} F_\alpha(S) R + \frac{1}{2} A_\alpha(S) g^{\mu\nu} \nabla_\mu S \nabla_\nu S + U_\alpha(S) + \mathcal{L}_m \right]. }[/math]

Variation of the above Obidi Action [equation] yields:

[math]\displaystyle{ F_\alpha(S) G_{\mu\nu} = \kappa \left( T^{(m)}_{\mu\nu} + T^{(S)}_{\mu\nu} \right) + \nabla_\mu \nabla_\nu F_\alpha(S) - g_{\mu\nu} \Box F_\alpha(S), }[/math]

with

[math]\displaystyle{ T^{(S)}_{\mu\nu} = A_\alpha(S) \left( \nabla_\mu S \nabla_\nu S - \tfrac{1}{2} g_{\mu\nu} (\nabla S)^2 \right). }[/math]

Interpretation

• For \( \alpha = 1 \): reduces to Shannon entropy → standard ToE field equation.
• For \( \alpha \neq 1 \): introduces α‑dependent corrections, potentially explaining dark energy, dark matter, or inflation.
• The parameter \( \alpha \) acts as a universal dial tuning the curvature of entropy flows.

Conceptual Significance

• In relativity, \(c\) is postulated; in ToE, \(c\) is derived as the invariant of entropy transport.
• In thermodynamic gravity, entropy is emergent; in ToE, entropy is fundamental.
• Rényi entropy provides a spectrum of entropic geometries, embedding scale‑dependent corrections into the field equations.
• ToE extends beyond physics into life, mind, and civilization, treating them as entropic organisms at the edge of order and chaos.

Summary

The Theory of Entropicity reframes reality: not as matter in motion through spacetime, but as entropy in flow across the entropic manifold. From the smallest quantum fluctuation to the rise and fall of civilizations, all phenomena are expressions of the same entropic law. The constancy of the speed of light emerges as the signature of the entropic field itself, while Rényi entropy provides the universal dial tuning its geometry.