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

Unifying Entropic Action, Entropic Geodesics, and Entropic Field Equations

Author: John Onimisi Obidi

Abstract

Over the past several decades, a variety of entropic and thermodynamic approaches to gravity have been proposed, each revealing a different aspect of the deep relationship between information, entropy, and spacetime geometry. Notable contributions by Jacobson, Verlinde, Caticha, and Bianconi have demonstrated that gravitational dynamics can be related to thermodynamic identities, entropic forces, or principles of entropic inference. However, none of these frameworks has produced a unified theory in which entropy itself is treated as a physical field endowed with its own action functional, field equations, and geodesic principle.

This work introduces the Entropic Field Paradigm, a new theoretical architecture within the Theory of Entropicity (ToE), in which gravity arises from bodies moving through an underlying entropic field and following paths that minimize entropic resistance. An explicit entropic action is formulated, from which entropic field equations are derived. The resulting structure is conceptually and formally distinct from previous entropic‑gravity proposals and is more comprehensive in that it unifies entropic action, entropic geodesics, and entropic field dynamics within a single coherent framework. The entropic field is thereby positioned as a fundamental dynamical entity, and entropic geodesics are established as the mechanism underlying gravitational motion.

1. Introduction

The modern search for a deeper understanding of gravity has increasingly turned toward thermodynamic and information‑theoretic principles. The discovery of black hole thermodynamics, the holographic principle, and the intimate connection between entropy and horizon area have suggested that gravitational phenomena may be emergent from more fundamental entropic or informational processes. Within this context, several influential approaches have been developed that interpret aspects of gravity in thermodynamic or entropic terms.

Jacobson demonstrated that the Einstein field equations can be derived from the Clausius relation applied to local Rindler horizons, thereby suggesting that spacetime dynamics can be viewed as an equation of state. Verlinde proposed that gravity can be understood as an entropic force associated with holographic screens. Caticha developed a framework of entropic dynamics in which dynamical laws emerge from principles of entropic inference. More recently, Bianconi introduced an entropic action based on quantum relative entropy and derived modified gravitational field equations.

Despite their conceptual richness, these approaches share a common limitation. None of them treats entropy as a fully fledged physical field defined over spacetime, equipped with its own action functional, field equations, and geodesic principle governing the motion of matter. In particular, they do not describe gravitational motion as the minimization of entropic resistance within such a field. The Entropic Field Paradigm introduced here is designed to fill this conceptual and structural gap by elevating entropy to the status of a dynamical field and by formulating a unified architecture in which gravitational phenomena arise from the interaction between matter and the entropic field.

2. Background and Related Work

2.1 Jacobson (1995): Thermodynamic Derivation of Einstein Equations

In a seminal contribution, Jacobson showed that the Einstein field equations can be derived from the Clausius relation

\( \delta Q = T\, dS \)

applied to local Rindler horizons, where \( \delta Q \) is the energy flux across the horizon, \( T \) is the Unruh temperature associated with the horizon, and \( dS \) is the change in horizon entropy. By imposing this thermodynamic relation for all local causal horizons and assuming proportionality between entropy and horizon area, Jacobson recovered the Einstein equations as an equation of state for spacetime. This result provides a profound thermodynamic interpretation of gravitational dynamics.

However, in Jacobson’s formulation, entropy is not treated as a dynamical field defined over spacetime. There is no explicit entropic action functional, no entropic field equations obtained by varying such an action, and no notion of entropic geodesics along which matter moves by minimizing an entropic functional. The thermodynamic relation constrains spacetime geometry but does not introduce an independent entropic field with its own dynamics.

2.2 Verlinde (2010): Gravity as an Entropic Force

Verlinde proposed that gravity can be understood as an entropic force arising from changes in the entropy associated with holographic screens. In this picture, when a test mass approaches a holographic screen, the entropy of the screen changes, and an effective force emerges that can be identified with gravity. By combining this entropic force concept with the holographic principle and equipartition of energy, Verlinde recovered Newton’s law of gravitation and aspects of Einstein’s equations.

While this approach provides an elegant entropic interpretation of gravitational attraction, it does not introduce an explicit action principle for entropy, nor does it yield field equations for an entropic field. The entropy is associated with holographic screens rather than being represented as a continuous field \( S(x) \) defined on spacetime. As a result, there is no formulation of entropic geodesics or of motion as the minimization of entropic resistance in a field‑theoretic sense.

2.3 Caticha: Entropic Dynamics

Caticha developed the framework of entropic dynamics, in which dynamical laws are derived from principles of entropic inference. In this approach, the evolution of a system is obtained by maximizing entropy subject to appropriate constraints, leading to equations that resemble familiar dynamical laws. Entropic dynamics provides a powerful conceptual bridge between probability theory, information theory, and dynamics.

However, entropic dynamics, in its original formulation, is not a theory of gravity. It does not introduce a gravitational entropic field, nor does it provide an entropic action for such a field. The framework is probabilistic and inferential rather than field‑theoretic, and it does not describe gravitational motion as the minimization of entropic resistance in a spacetime field.

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

Bianconi introduced an entropic action constructed from quantum relative entropy and used it to derive modified gravitational field equations. In this formulation, the action functional incorporates information‑theoretic quantities, and the resulting equations can be viewed as a generalization of Einstein’s equations with entropic corrections. This work represents an important step toward integrating entropy and information into the action principle of gravity.

Nevertheless, even in this approach, entropy is not explicitly promoted to a standalone physical field \( S(x) \) with its own field equations. The entropic action modifies the gravitational sector but does not describe matter as moving through an entropic field or as following trajectories that minimize entropic resistance. The notion of entropic geodesics is not developed, and the entropic structure remains embedded in the action rather than being represented as an independent dynamical field.

3. The Entropic Field Paradigm

3.1 Entropy as a Physical Field

The Entropic Field Paradigm proposed within the Theory of Entropicity (ToE) elevates entropy from a thermodynamic descriptor to a dynamical field permeating spacetime. Let \( M \) denote a spacetime manifold equipped with a metric \( g_{\mu\nu} \). The entropic field is represented by a scalar function \( S(x) \), where \( x \in M \). This field \( S(x) \) assigns to each spacetime point a value of entropic potential, encoding the local structure of entropic accessibility.

By treating \( S(x) \) as a genuine field, the entropic field paradigm allows one to formulate an action principle for entropy, derive field equations for \( S(x) \), and define entropic geodesics as the paths that extremize appropriate entropic functionals. This represents a conceptual shift from viewing entropy as a derived quantity to regarding it as a fundamental component of the dynamical architecture of spacetime.

3.2 Entropic Resistance and Entropic Geodesics

In this framework, bodies moving through spacetime are simultaneously moving through the entropic field \( S(x) \). Their motion is governed not only by geometric considerations but also by the structure of the entropic field. The central concept is that of entropic resistance, which quantifies the entropic “cost” associated with a given trajectory.

Consider a curve \( \gamma \) in spacetime, parameterized by an appropriate parameter, and let \( ds \) denote the line element along this curve. The entropic resistance associated with \( \gamma \) is defined by a functional

\( R[\gamma] = \int_{\gamma} f\big(S, \nabla S\big)\, ds, \)

where \( f(S, \nabla S) \) is a scalar function that depends on the entropic field \( S \) and its gradient \( \nabla S \). The precise form of \( f \) encodes how variations in the entropic field influence the resistance experienced by a moving body. The physically realized trajectories are those for which the functional \( R[\gamma] \) is stationary under variations of the path \( \gamma \). These stationary paths are termed entropic geodesics.

Entropic geodesics are the entropic analogue of gravitational geodesics in General Relativity. Whereas geodesics in GR extremize the spacetime interval determined by the metric \( g_{\mu\nu} \), entropic geodesics extremize the entropic resistance functional determined by the entropic field \( S(x) \). In this way, gravitational motion is reinterpreted as motion that minimizes entropic resistance rather than purely geometric distance.

3.3 Action for the Entropic Field

The decisive structural step in the entropic field paradigm is the formulation of an explicit entropic action for the field \( S(x) \). Let \( L(S, \nabla S, g) \) denote a Lagrangian density that depends on the entropic field, its gradient, and the spacetime metric. The entropic action is defined as

\( A_{S} = \int L\big(S, \nabla S, g\big)\, d^{4}x, \)

where the integration is taken over the spacetime manifold \( M \) with the appropriate volume element. The Lagrangian density \( L \) encodes the self‑dynamics of the entropic field as well as its coupling to geometry and matter. For example, \( L \) may contain kinetic terms involving \( \nabla S \), potential terms depending on \( S \), and interaction terms coupling \( S \) to curvature invariants constructed from \( g_{\mu\nu} \) or to matter fields.

By introducing this action, the entropic field paradigm places entropy on the same conceptual footing as other dynamical fields in field theory. The entropic field is no longer a passive descriptor but an active participant in the dynamics of spacetime and matter.

3.4 Field Equations for the Entropic Field

The dynamics of the entropic field \( S(x) \) are obtained by varying the entropic action \( A_{S} \) with respect to \( S \). The condition for stationarity of the action is

\( \frac{\delta A_{S}}{\delta S} = 0. \)

This variational principle yields entropic field equations that govern the evolution and configuration of the entropic field. The explicit form of these equations depends on the choice of Lagrangian density \( L(S, \nabla S, g) \), but in general they will be second‑order partial differential equations relating \( S \), its derivatives, and possibly curvature and matter terms.

Through these field equations, the entropic field influences the motion of matter by shaping the entropic resistance landscape in spacetime. Matter, in turn, may act as a source for the entropic field, leading to a coupled system in which geometry, matter, and entropy are dynamically interrelated. Gravitational behavior is thus mediated by the entropic field, and entropic geodesics emerge as the natural trajectories in this entropic‑geometric environment.

4. Distinction from Prior Entropic‑Gravity Theories

The Entropic Field Paradigm is distinguished from earlier entropic and thermodynamic approaches to gravity by the fact that it simultaneously incorporates several structural features that are absent, in combination, from the prior literature. Specifically, it treats entropy as a physical field, describes bodies as moving through this entropic field, formulates motion as the minimization of entropic resistance, introduces an explicit entropic action, derives field equations for entropy, and defines entropic geodesics as the fundamental mechanism underlying gravitational motion.

4.1 Comparative Structural Features

This combination of features is unique to the entropic field paradigm. While individual elements, such as an entropic action or thermodynamic derivations of gravitational equations, appear in earlier works, none of the prior frameworks unifies entropy as a field, entropic action, entropic field equations, and entropic geodesics into a single coherent architecture. The entropic field paradigm therefore represents a new level of structural completeness in entropic approaches to gravity.

5. Conclusion

The Entropic Field Paradigm introduces a new way of understanding gravity within the Theory of Entropicity (ToE). Rather than viewing gravity solely as curvature of spacetime or as an emergent entropic force associated with horizons or holographic screens, this paradigm treats gravity as the dynamical consequence of motion through an entropic field governed by its own action and field equations. Bodies move along entropic geodesics, which are trajectories that minimize entropic resistance in the entropic field.

By synthesizing and extending prior entropic approaches, the entropic field paradigm establishes a new foundation for gravitational theory in which entropy is a fundamental dynamical entity. This architecture provides a natural setting for unifying thermodynamic, informational, and geometric aspects of gravity and opens the way for further developments in which the entropic field may play a central role in the unification of fundamental interactions and the deeper structure of spacetime.

Appendix: Entropic Field Paradigm – Extended Introduction and Background

1. Introduction

The quest to understand the fundamental nature of gravity has undergone a profound conceptual shift over the past several decades. While General Relativity (GR) remains one of the most successful physical theories ever formulated, its geometric description of gravitation as curvature of spacetime has increasingly been supplemented—and in some cases challenged—by approaches grounded in thermodynamics, information theory, and statistical mechanics. This shift is motivated by a growing recognition that gravitational phenomena exhibit deep structural parallels with entropic processes, and that spacetime itself may encode information in ways that transcend classical geometric intuition.

The modern entropic turn in gravitational theory can be traced to several landmark contributions. Jacobson’s 1995 derivation of the Einstein field equations from the Clausius relation marked the first major step toward interpreting gravity as an emergent thermodynamic phenomenon. Verlinde’s 2010 proposal of gravity as an entropic force further advanced the idea that gravitational attraction may arise from changes in information associated with the positions of material bodies. Caticha’s development of entropic dynamics introduced a probabilistic framework in which physical laws emerge from principles of inference rather than from fundamental geometric postulates. More recently, Bianconi’s 2024–2025 work has demonstrated that an entropic action constructed from quantum relative entropy can reproduce modified Einstein equations, suggesting that gravitational dynamics may be encoded in information‑theoretic functionals.

Despite their conceptual diversity, these approaches share a common philosophical orientation: they treat gravity not as a primitive interaction but as a phenomenon emerging from deeper informational or thermodynamic structures. Yet they also share a common limitation. None of these frameworks treats entropy as a physical field in its own right—one that permeates spacetime, possesses its own action functional, and obeys field equations analogous to those governing curvature or matter. Likewise, none describes gravitational motion as the minimization of entropic resistance, a variational principle that would define geodesics not in terms of metric length but in terms of entropic cost.

The entropic field framework introduced in the main text is designed to fill this conceptual gap. It proposes that entropy should be understood as a dynamical field defined over spacetime, with its own action functional and associated field equations. Within this framework, bodies move through the entropic field along paths that minimize entropic resistance, giving rise to what may be termed entropic geodesics. These entropic geodesics serve as the fundamental mechanism underlying gravitational motion, replacing or supplementing the metric geodesics of General Relativity.

The significance of this approach lies not merely in its novelty but in its potential to unify disparate strands of entropic and information‑theoretic research into a coherent dynamical theory. By treating entropy as a field with its own variational structure, the framework provides a natural way to integrate thermodynamic, statistical, and geometric insights into a single theoretical architecture. It also offers a new perspective on longstanding problems in gravitational physics, including the nature of inertia, the origin of gravitational attraction, and the relationship between information and spacetime structure.

Within this broader context, the entropic field paradigm can be viewed as a natural continuation of the entropic turn in gravitational theory. It extends earlier insights by promoting entropy from a derived quantity to a fundamental field‑theoretic entity, thereby enabling a unified treatment of entropic action, entropic field equations, and entropic geodesics. The remainder of this appendix situates this framework within the landscape of entropic‑gravity research by providing a structured review of key contributions and their limitations.

2. Background and Related Work

The entropic and information‑theoretic approaches to gravity form a diverse and rapidly evolving research landscape. Although these approaches differ in methodology and emphasis, they share a common ambition: to reinterpret gravitational phenomena in terms of entropy, information flow, or statistical inference. This section reviews four major contributions that have shaped the field and provides a critical assessment of their conceptual scope.

2.1 Jacobson (1995): Thermodynamic Derivation of Einstein Equations

Ted Jacobson’s 1995 paper, “Thermodynamics of Spacetime: The Einstein Equation of State”, is widely regarded as the foundational work in the thermodynamic interpretation of gravity. Jacobson demonstrated that the Einstein field equations can be derived from the Clausius relation

\( \delta Q = T\, dS, \)

applied to local Rindler horizons. In this formulation, the heat flux \( \delta Q \) across a horizon is related to the change in entropy \( dS \), with the Unruh temperature \( T \) providing the thermodynamic link between acceleration and temperature. Jacobson’s insight suggested that the Einstein equations are not fundamental dynamical laws but rather equations of state, analogous to those governing thermodynamic systems. Spacetime, in this view, behaves like a medium whose macroscopic properties emerge from microscopic degrees of freedom that encode entropy.

Despite its conceptual power, Jacobson’s framework does not introduce an entropic field, nor does it define an action for entropy or derive field equations for such a field. The thermodynamic relations he employs are applied to horizons rather than to spacetime as a whole, and gravitational motion is not described in terms of entropic resistance or entropic geodesics. Entropy appears as a horizon property rather than as a dynamical field permeating spacetime.

2.2 Verlinde (2010): Gravity as an Entropic Force

Erik Verlinde’s 2010 proposal, “On the Origin of Gravity and the Laws of Newton”, advanced the idea that gravity is an entropic force arising from changes in information associated with the positions of material bodies. Drawing on holographic principles and the thermodynamics of horizons, Verlinde argued that gravitational attraction can be understood as a statistical tendency toward configurations of higher entropy. In this framework, the force experienced by a test mass near a holographic screen is proportional to the change in entropy with respect to displacement, leading to Newton’s law of gravitation and, in certain limits, to aspects of General Relativity.

Verlinde’s approach provides an elegant entropic interpretation of gravitational attraction, but it does not include an action principle for entropy, nor does it introduce field equations governing an entropic field. The entropic force arises from boundary information associated with holographic screens rather than from a dynamical field permeating spacetime. Motion is not described as minimizing entropic resistance, and no entropic geodesic structure is defined. As a result, the entropic content remains tied to surfaces rather than being encoded in a spacetime field \( S(x) \).

2.3 Caticha: Entropic Dynamics

Ariel Caticha’s entropic dynamics program offers a different perspective, grounded in the idea that physical laws emerge from principles of entropic inference. In this framework, the evolution of a system is determined by maximizing entropy subject to constraints, leading to equations of motion that resemble those of classical and quantum mechanics. Caticha’s work is notable for its methodological clarity and its emphasis on inference as the foundation of dynamics, providing a powerful conceptual bridge between statistical reasoning and physical law.

However, entropic dynamics is not a gravitational theory. It does not introduce an entropic field, does not define an action for entropy, and does not derive field equations analogous to those of General Relativity. Its relevance to gravity is primarily conceptual rather than structural. There is no description of bodies moving through an entropic field, nor any formulation of entropic geodesics or entropic resistance as governing principles of gravitational motion.

2.4 Bianconi (2024–2025): Entropic Action from Quantum Relative Entropy

Ginestra Bianconi’s recent work represents one of the most mathematically explicit attempts to construct an entropic action capable of reproducing gravitational dynamics. By employing quantum relative entropy as the core functional, Bianconi derives modified Einstein equations and demonstrates that gravitational behavior can emerge from information‑theoretic principles. Her framework introduces a coupling between matter fields and geometry mediated by an entropic action, leading to a “dressed” Einstein–Hilbert structure and an emergent cosmological constant.

Despite its sophistication, Bianconi’s approach does not treat entropy as a physical field defined over spacetime. It does not describe bodies as moving through an entropic field, nor does it introduce the concept of entropic resistance or entropic geodesics. The entropic action is informational rather than field‑theoretic in nature. Entropy enters as a functional of quantum states and their relative information content, but it is not represented as a scalar field \( S(x) \) with its own local dynamics and variational principle.

3. Conclusion: The Necessity and Role of the Theory of Entropicity (ToE)

The preceding analysis of thermodynamic and information‑theoretic approaches to gravity reveals a research landscape that is rich in conceptual innovation yet marked by a persistent structural gap. The contributions of Jacobson, Verlinde, Caticha, and Bianconi have each reshaped the modern understanding of gravitational phenomena, but none has succeeded in constructing a unified dynamical theory in which entropy itself functions as a physical field endowed with its own action, field equations, and geodesic principle. This absence is not a minor omission; it represents a fundamental limitation in the current entropic‑gravity paradigm. The Theory of Entropicity (ToE), as first formulated and further developed by John Onimisi Obidi, emerges precisely to address this limitation and to provide the missing theoretical architecture required to elevate entropic gravity from a collection of partial analogies to a coherent field theory [of gravity].

To appreciate the necessity of ToE, it is essential to recognize the structural asymmetry in existing approaches. Jacobson’s thermodynamic derivation of the Einstein equations demonstrates that gravitational dynamics can be interpreted as emergent from horizon thermodynamics, but it does not endow entropy with dynamical degrees of freedom. Verlinde’s entropic‑force proposal reframes gravity as a statistical tendency toward higher entropy, yet it relies on holographic screens rather than a spacetime‑filling entropic field. Caticha’s entropic dynamics provides a powerful inferential framework, but it does not attempt to model gravity or spacetime structure. Bianconi’s entropic action represents the closest analogue to a field‑theoretic formulation, but even this approach treats entropy as an informational functional rather than as a physical field capable of guiding motion.

What is missing from all these frameworks is a field‑theoretic ontology of entropy—a recognition that entropy may not merely describe the statistical state of matter or information but may itself constitute a field woven into the fabric of spacetime. Without such an ontology, entropic gravity remains conceptually incomplete. It lacks a variational principle for entropy, a set of field equations governing its evolution, and a mechanism by which bodies respond dynamically to entropic gradients. The Theory of Entropicity fills this void by proposing that entropy is a fundamental field

\( S(x) \)

defined over spacetime, possessing its own action functional and obeying field equations derived from that action.

The introduction of an entropic action is the decisive step that transforms entropy from a descriptive quantity into a dynamical entity. By constructing an action

\( A_{S} \)

that depends on the entropic field and its derivatives, ToE places entropy on the same conceptual footing as the metric in General Relativity or scalar fields in scalar‑tensor theories. This action serves as the foundation from which entropic field equations are derived, providing a systematic and mathematically rigorous description of how entropy evolves and interacts with matter and geometry. In doing so, ToE establishes entropy as a participant in the dynamics of spacetime rather than as a passive bookkeeping device.

Equally significant is the introduction of entropic geodesics, the paths that bodies follow as they move through the entropic field. In ToE, gravitational motion is not defined by the extremization of metric length but by the minimization of entropic resistance, a functional that quantifies the entropic cost of a trajectory. This principle provides a natural and intuitive explanation for gravitational attraction: bodies move along paths that minimize resistance within the entropic field, just as objects in classical mechanics follow paths that minimize action. Entropic geodesics thus serve as the dynamical mechanism that links the entropic field to observable gravitational behavior.

The necessity of ToE becomes even more apparent when considering the broader implications of treating entropy as a field. Such a framework offers a new perspective on the relationship between information and spacetime, suggesting that the structure of spacetime may emerge from or be shaped by the distribution and dynamics of entropy. It provides a natural setting for exploring the origin of inertia, the nature of gravitational mass, and the deep connections between thermodynamics, quantum information, and geometry. Moreover, by introducing a field‑theoretic description of entropy, ToE opens the door to new approaches to longstanding problems such as dark energy, dark matter, and the unification of gravity with quantum mechanics.

In this sense, the Theory of Entropicity is not merely an extension of existing entropic‑gravity models; it is a new theoretical architecture that redefines the role of entropy in fundamental physics. It synthesizes the insights of Jacobson, Verlinde, Caticha, and Bianconi while addressing their limitations and integrating their partial contributions into a coherent dynamical framework. By doing so, ToE provides the conceptual and mathematical tools necessary to advance entropic gravity from a collection of heuristic analogies to a fully developed field theory.

The role of ToE is therefore twofold. First, it serves as the completion of the entropic‑gravity program, supplying the missing elements required to construct a self‑contained theory in which entropy is a dynamical field. Second, it functions as a foundational framework for future research, offering a platform upon which new theoretical developments can be built. Whether in the context of cosmology, quantum gravity, or the physics of information, the Theory of Entropicity provides a unifying perspective that has the potential to reshape our understanding of gravity and spacetime.

In conclusion, the Theory of Entropicity is necessary because it fills a structural gap left by all previous entropic and information‑theoretic approaches to gravity. It introduces the missing dynamical elements—an entropic field, an entropic action, entropic field equations, and entropic geodesics—that are required to transform entropy from a descriptive quantity into a fundamental component of physical law. Its role is to unify, extend, and complete the entropic‑gravity paradigm, establishing a new foundation for the study of gravitational phenomena and the interplay between entropy, information, and spacetime. As such, ToE represents not merely a new theory but a new conceptual lens through which the nature of gravity [and the foundation of reality in the universe] may be understood.

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