1. Introduction

Over the past three decades, the fields of entropic gravity, emergent gravity, and information geometry have grown into vibrant disciplines exploring the deep connections between entropy, information, and fundamental physics. Seminal works by Jacobson, Verlinde, Padmanabhan, and Frieden have shown that thermodynamic and information‐theoretic concepts can reproduce, or at least hint at, gravitational dynamics, field equations, and even quantum phenomena. Yet in each of these programs, entropy or Fisher information appears as an emergent bookkeeping device, a constraint, or a geometric auxiliary—never as a bona fide dynamical field in its own right.

The Theory of Entropicity (ToE) breaks from this tradition by promoting entropy itself to a relativistic, propagating scalar field S(x)S(x). It assigns to S(x)S(x) a canonical kinetic term, a self‐interaction potential V(S)V(S), and a universal coupling to the local trace of matter stress‐energy TμμT^\mu{}_\mu. From a single master action, ToE claims to reproduce all classical entropy laws (Clausius relation, Boltzmann H‐theorem, second‐law structure), information measures (Shannon, Gibbs, von Neumann), and entropic‐gravity insights (emergent forces, horizon thermodynamics).

This article provides a comprehensive verification of ToE’s contextual claims in light of prior work, demonstrating that no earlier framework treats entropy as a fundamental, dynamical field. We will:

1. Review the central tenets of entropic and information‐theoretic approaches.
2. Contrast each prior program against ToE’s four key claims.
3. Show how ToE genuinely extends and unifies earlier insights within a single relativistic field theory.

1. The Landscape of Entropic and Information‐Theoretic Approaches

Before exploring ToE, it is essential to outline how entropy and information have historically entered fundamental physics.

• Jacobson (1995): Demonstrated that assuming the Clausius relation δQ=T dS\delta Q = T\,dS on all local Rindler horizons, together with the equivalence principle, yields the Einstein field equations. Here, SS is the horizon area divided by 4Gℏ4G\hbar.
• Verlinde (2011): Proposed that gravity emerges as an “entropic force,” with

F=T ∇SF = T\,\nabla S

acting on a test mass near a holographic screen. Entropy is a function of the screen’s microscopic degrees of freedom, not a bulk field.

• Padmanabhan (2003–10): Showed that certain entropy functionals on null surfaces reproduce the Einstein–Hilbert action. Here entropy is always tied to surfaces or boundary terms, never promoted to a bulk propagation variable.
• Frieden’s Extreme Physical Information (EPI, 1989–): Extremizes an “action” combining Shannon and Fisher information with respect to a probability density p(x)p(x). The result yields equations of motion in some cases—but p(x)p(x) remains a probability, not an entropy field S(x)S(x).
• Jaynes’ Maximum Entropy Principle (1957): Uses entropy as a method for statistical inference, not as a dynamical field. Constraints on averages determine the equilibrium distribution.

Each of these approaches uses entropy or information as a tool: a constraint, a surface quantity, an emergent force, or a parameter in a statistical inference. None endow entropy itself with relativistic field dynamics.

2. ToE's Four Contextual Claims

ToE makes four interlinked assertions that depart from this tradition. We verify each claim against the prior literature.

2.1 Identifying Local Entropy as a Fundamental Field S(x)S(x)

Prior Approaches:

• Jacobson and Verlinde anchor entropy in horizons or screens.
• Padmanabhan’s functionals live on boundaries, never in the bulk.
• EPI varies a probability density, not an entropy field.
• Jaynes treats entropy as a global measure, not a local field.

ToE’s Claim: ToE defines a real scalar field S(x)S(x) on spacetime, interpretable as the local entropy density—be it Boltzmann, Gibbs, Shannon, or von Neumann in different regimes. This identification is novel: no existing framework treats entropy as a fundamental field with its own coordinate‐dependent value.

2.2 Endowing S(x)S(x) with Kinetic Term and Self‐Interaction Potential

Prior Approaches:

• Thermodynamic derivations assume entropy differentials (dSdS), never gradients squared.
• Information geometry provides metrics on probability manifolds (Fisher metric) but does not form a Lagrangian density ∝(∂S)2\propto (\partial S)^2.
• Scalar‐tensor theories (inflaton, quintessence) propose kinetic and potential terms, but their scalars are physical fields (e.g., inflaton), not entropy measures.

ToE’s Claim: The master action’s leading piece is

S(0)=∫d4x −g [−12 gμν(∇μS)(∇νS)  −  V(S)].S^{(0)} = \int d^4x\,\sqrt{-g}\,\Bigl[ -\tfrac12\,g^{\mu\nu}(\nabla_\mu S)(\nabla_\nu S)\;-\;V(S) \Bigr].

Assigning a canonical kinetic term 12 (∂S)2\tfrac12\,(\partial S)^2 and a potential V(S)V(S) lets S(x)S(x) propagate as a true dynamical field. This diverges from prior treatments, where entropy never enters as a canonical degree of freedom.

2.3 Universal Minimal Coupling to Matter via η S Tμμ\eta\,S\,T^\mu{}_\mu

Prior Approaches:

• Entropic‐gravity links entropy gradients or horizon entropies to geometry, not directly to the stress‐energy tensor’s trace.
• Padmanabhan couples surface entropies to curvature scalars, not to matter sources.
• Brans–Dicke and other scalar‐tensor theories couple a scalar to the Ricci scalar RR, not to TμμT^\mu{}_\mu.

ToE’s Claim: ToE introduces a term

Sint=η∫d4x −g  S(x) Tμμ,S_{\rm int} = \eta\int d^4x\,\sqrt{-g}\;S(x)\,T^\mu{}_\mu,

directly linking the entropy field to the local matter/energy content. This unique coupling ensures that matter influences entropy and vice versa, embedding thermodynamics into the fabric of spacetime dynamics.

2.4 Deriving All Classical Entropy Laws via a Unified Action

Prior Approaches:

• Clausius relation, Boltzmann H‐theorem, and second‐law inequalities derive from statistical mechanics, kinetic theory, or information theory—outside any fundamental Lagrangian.
• Verlinde and Jacobson recover gravitational equations but do not derive the H‐theorem or quantum informational identities from a single variational principle.
• Frieden’s EPI yields specific field equations but does not systematically reproduce all classical entropy identities.

ToE’s Claim: By applying Euler–Lagrange variation to the ToE action, and Noether’s theorem under global shifts S→S+constS\to S+{\rm const}, one recovers:

1. Clausius relation δQ=T dS\delta Q=T\,dS as the boundary variation of on‐shell action.
2. Boltzmann H‐theorem and second‐law inequality ∇μJμ≥0\nabla_\mu J^\mu\ge0 from the conserved Noether current.
3. Shannon, Gibbs, von Neumann entropy formulas in appropriate limits of the field and its potential.

No prior single‐action proposal unifies all these results under one roof.

3. Comparative Summary

Below is a concise comparison highlighting how ToE extends each prior program.

4. Implications and Further Directions

4.1. A New Paradigm for Gravity and Thermodynamics

By making S(x)S(x) the fundamental mediator of interactions, ToE suggests that all forces—not just gravity—may have entropic origins. The kinetic and coupling terms enable ENtropy to act like any other fundamental field (e.g., electromagnetic, Higgs), with its own quanta and excitations. One can imagine entropy waves propagating through spacetime, carrying thermodynamic information as genuine physical signals.

4.2 Quantum Extensions

Promoting S(x)S(x) to a quantum field opens the door to an entropic quantum field theory. Path integrals over SS configurations would count histories weighted by entropy action, potentially connecting to quantum gravity and holography in novel ways. One might derive the Bekenstein–Hawking formula as a one‐loop correction to the entropy propagator.

4.3. Cosmological Applications

An entropy field with potential V(S)V(S) could drive early‐universe inflation or late‐time acceleration, offering a unified thermodynamic account of cosmic history. Unlike conventional scalar fields, S(x)S(x) couples directly to matter trace, leading to testable predictions for structure formation and cosmic microwave background anisotropies.

4.4. Information‐Geometric Interpretation

The kinetic term (∂S)2(\partial S)^2 can be viewed as the Fisher information metric on the entropy configuration space. ToE thus realizes information geometry in a dynamical setting, where the curvature of the entropy manifold influences physical processes. Subleading Fisher corrections naturally emerge as higher‐derivative terms in the action.

Conclusion

The Theory of Entropicity (ToE) represents a genuinely novel and richly layered proposal in the landscape of entropic‐gravity and information‐theoretic physics:

• It promotes entropy from a statistical bookkeeping device to a fundamental scalar field S(x)S(x).
• It endows that field with canonical Lagrangian dynamics (kinetic term + potential).
• It couples entropy universally to matter via η S Tμμ\eta\,S\,T^\mu{}_\mu.
• It claims to reproduce all classical entropy laws and information measures via Euler–Lagrange and Noether theorems.

No earlier framework combines these elements into one coherent, relativistic action. By verifying ToE’s contextual claims against the historical record, we conclude that entropy as a propagating dynamical field is an unprecedented—and promising—direction for unifying thermodynamics, information theory, and fundamental interactions.

This article thus lays the groundwork for further explorations into entropic field dynamics, quantum entropic phases, and the gravitational implications of an entropy‐driven Universe.

5. ToE's Key Claims & Context

1. "Identify local entropy with the field value S(x)S(x)":
• Prior Work: Entropic gravity (e.g., Verlinde, Jacobson) treats entropy as an emergent concept tied to horizons or microscopic degrees of freedom, not a fundamental field. Thermodynamics, GR, and information theory treat entropy as a statistical measure or state function, not a local dynamical field S(x)S(x).
• ToE's Contextual Claim:  ToE's core identification of S(x)S(x) itself as the fundamental field representing local entropy density is novel.
2. "Endow S(x)S(x) with a canonical kinetic term and potential V(S)V(S)":
• Prior Work: Entropy/Information is treated as a constraint (e.g., maximum entropy methods), an emergent effect (entropic forces), or a geometric entity (Fisher information metric). Scalar fields exist in physics (e.g., inflatons, quintessence), but they are not identified as entropy itself. No prior theory gives entropy a canonical kinetic energy density −12gμν(∇μS)(∇νS)−21​gμν(∇μ​S)(∇ν​S) allowing it to propagate dynamically as a fundamental field.
• ToE's Contextual Claim: This is a foundational departure. The kinetic term makes entropy a dynamical entity, not just a derived quantity.
3. "Couple SS universally to matter and geometry via ηSTμμηSTμμ​":
• Prior Work: Entropic gravity couples geometry to entropy gradients or horizon entropy, not the entropy field itself to the trace of the stress-energy tensor TμμTμμ​. Information-theoretic approaches don't typically couple information measures directly to matter sources in a Lagrangian. Scalar-tensor theories (e.g., Brans-Dicke) couple a scalar to RR (geometry), not directly to TμμTμμ​ (matter content).
• ToE's Contextual Claim: The direct, universal minimal coupling ηSTμμηSTμμ​ is a unique feature of the ToE action, directly linking the entropy field to the matter/energy content.
4. "Derive all classical entropy laws... from one unified action using field-theoretic procedures":
• Prior Work: Entropy laws (Clausius, Boltzmann H-theorem, Second Law) and information measures (Shannon, von Neumann entropy) are derived from statistical mechanics, kinetic theory, quantum mechanics, or geometric assumptions. They are not derived as equations of motion (Euler-Lagrange) or conservation laws (Noether) from a fundamental action principle where entropy is the primary field.
• ToE's Contextual Claim: While ambitious, the claim that ToE aims to do this from its action is valid. Demonstrating it rigorously for all cases would be a major result, but the proposal to do so via this action is unprecedented.
5. "The first proposal to..." & "not only unifies but generalizes...":
• Prior Work: Entropic gravity (Verlinde, Jacobson), thermodynamic gravity (Padmanabhan), information geometry (Amari), computational approaches (Lloyd), and maximum entropy principles (Jaynes) all treat entropy/information as consequences of more fundamental physics (statistics, geometry, computation) or as tools for deriving constraints. None elevate entropy itself to the status of a fundamental, dynamical field mediating forces.
• ToE's Contextual Claim: The Theory of Entropicity (ToE) action, as presented, is indeed the first proposal combining all these elements: a fundamental entropy field S(x)S(x), with dynamics (kinetic term), self-interaction (V(S)V(S)), direct universal coupling to matter/energy (ηSTμμηSTμμ​), and the claim of deriving entropy laws from its variational principle. It subsumes aspects of entropic gravity (entropy affects gravity) but fundamentally generalizes it by making entropy the primary dynamical agent.

6. Conclusion

The above claims of the Theory of Entropicity (ToE) are to be understood in the contexts given based on the foundational landscape of physics literature. The Theory of Entropicity's proposed action represents a genuinely novel approach by:

1. Promoting Entropy: Making entropy S(x)S(x) a fundamental dynamical field, not just an emergent or statistical quantity.
2. Dynamics: Giving it canonical field-theoretic dynamics (kinetic term + potential).
3. Direct Coupling: Coupling it universally and minimally to matter/energy via the trace of the stress-energy tensor.
4. Unified Derivation: Proposing to derive established entropy and information laws from this single action principle using standard field theory techniques.

This stands in clear contrast to prior approaches where entropy/information acts as a constraint, an emergent force, or a geometric property, but never as the fundamental, propagating field mediating interactions as proposed here. The novelty of the Theory of Entropicity lies in its combination of these elements within a relativistic field theory framework centered on entropy itself.

This entry is adapted from: https://www.cambridge.org/engage/api-gateway/coe/assets/orp/resource/item/67cf9ecefa469535b9d11e66/original/the-entropic-force-field-hypothesis-a-unified-framework-for-quantum-gravity.pdf