The Entropic Field Paradigm: A New Architecture for Gravity in the Theory of Entropicity (ToE) - Unifying Entropic Action, Entropic Geodesics, and Entropic Field Equations

The Entropic Field Paradigm: A New Architecture for Gravity in the Theory of Entropicity (ToE) 

Unifying Entropic Action, Entropic Geodesics, and Entropic Field Equations

🌀 The Entropic Field Paradigm: A New Architecture for Gravity in the Theory of Entropicity (ToE)

Unifying Entropic Action, Entropic Geodesics, and Entropic Field Equations

John Onimisi Obidi

Abstract

A variety of entropic and thermodynamic approaches to gravity have emerged over the past three decades, each illuminating a different facet of the deep relationship between information, entropy, and spacetime geometry. Yet none of these frameworks has produced a unified theory in which entropy itself is treated as a physical field with its own action, field equations, and geodesic principle. This paper introduces the Entropic Field Paradigm, a new theoretical architecture in which gravity arises from bodies moving through an entropic field and following paths that minimize entropic resistance. This approach incorporates an explicit action for entropy, from which field equations for the entropic field are derived. The resulting structure is distinct from and more comprehensive than previous entropic‑gravity proposals by Jacobson, Verlinde, Caticha, and Bianconi. This work positions the entropic field as a fundamental dynamical entity and establishes entropic geodesics as the mechanism underlying gravitational motion.

1. Introduction

The search for a deeper understanding of gravity has increasingly turned toward thermodynamic and information‑theoretic principles. Seminal contributions by Jacobson (1995), Verlinde (2010), Caticha (2000s), and more recently Bianconi (2025) have demonstrated that gravitational dynamics may emerge from entropy, information flow, or statistical inference.

However, these approaches share a common limitation: none treats entropy as a physical field with its own action and field equations, nor do they describe gravitational motion as the minimization of entropic resistance within such a field.

This paper presents a framework that fills this conceptual gap.

2. Background and Related Work

2.1 Jacobson (1995): Thermodynamic Derivation of Einstein Equations

Jacobson showed that Einstein’s field equations can be derived from the Clausius relation $\delta Q=TdS$ applied to local Rindler horizons. Limitation: No entropic field, no entropic action, no entropic geodesics.

2.2 Verlinde (2010): Gravity as an Entropic Force

Verlinde proposed that gravity arises as an entropic force associated with holographic screens. Limitation: No action principle; no field equations for entropy.

2.3 Caticha: Entropic Dynamics

Caticha developed a probabilistic framework in which dynamics emerge from entropic inference. Limitation: Not a gravitational theory; no entropic field or action.

2.4 Bianconi (2025): Entropic Action from Quantum Relative Entropy

Bianconi introduced an entropic action using quantum relative entropy and derived modified Einstein equations. Limitation: Does not propose an entropic field; does not describe motion as minimizing entropic resistance.

3. The Entropic Field Paradigm (Obidi)

3.1 Entropy as a Physical Field

In this framework, entropy is elevated from a thermodynamic descriptor to a dynamical field permeating spacetime. Let $S(x)$ denote the entropic field defined over a manifold $M$.

3.2 Entropic Resistance and Geodesics

Bodies move through the entropic field along paths that minimize entropic resistance, defined by a functional

$$\mathcal{R}[\gamma ]={\int }_{\gamma }f(S,\mathrm{\nabla }S)ds\mathrm{.}$$

The stationary paths of $\mathcal{R}$ are entropic geodesics, the analog of gravitational geodesics in General Relativity.

3.3 Action for Entropy

The decisive step is the formulation of an entropic action

$${\mathcal{A}}_S=\int \mathcal{L}(S,\mathrm{\nabla }S,g)d^{4}x,$$

where $\mathcal{L}$ couples the entropic field to geometry and matter.

3.4 Field Equations for Entropy

Variation of ${\mathcal{A}}_S$ with respect to $S$ yields entropic field equations

$$\frac{\delta {\mathcal{A}}_S}{\delta S}=0,$$

which govern the dynamics of the entropic field and, through it, the gravitational behavior of matter.

4. Distinction from Prior Entropic‑Gravity Theories

The Entropic Field Paradigm is the only framework that unifies:

Concept	Jacobson	Verlinde	Caticha	Bianconi	Obidi
Entropy as a physical field	❌	❌	❌	❌	✔️
Bodies move through entropic field	❌	❌	❌	❌	✔️
Motion minimizes entropic resistance	❌	❌	❌	❌	✔️
Explicit entropic action	❌	❌	❌	✔️	✔️
Field equations for entropy	❌	❌	❌	✔️	✔️
Entropic geodesics	❌	❌	❌	❌	✔️

This combination is unique to the present work.

5. Conclusion

The Entropic Field Paradigm introduces a new way of understanding gravity: not as curvature alone, nor as an emergent thermodynamic force, but as the dynamical consequence of motion through an entropic field governed by its own action and field equations. This framework synthesizes and extends prior entropic approaches while establishing a new foundation for gravitational theory.