Physics:A Concise Introduction to the Evolving Theory of Entropicity (ToE)

Entropic Wave Function Diagram

Wave Function Collapse Explained by the Theory of Entropicity (ToE)

As a system’s entropy [math]\displaystyle{ S_{\rm system}(t) }[/math] (blue curve) increases in time due to interaction and entanglement, it crosses a critical threshold [math]\displaystyle{ S_{\rm critical} }[/math] (red dashed line).

When the entropic threshold condition [math]\displaystyle{ S_{\rm system} \gt S_{\rm critical} }[/math] is satisfied (purple point/line), a wave function collapse is triggered. This represents a transition from a coherent superposition to a single outcome state, analogous to a phase transition once a thermodynamic variable exceeds a critical value.

Before the threshold, unitary quantum evolution can maintain coherence; beyond the threshold, irreversibility sets in and one branch is selected. The process is one-way (entropy continues to increase), ensuring the collapse cannot be reversed.

Entropic Field and Entanglement Diagram

Introducing ToE’s Seesaw Model of Quantum Entanglement and Wave Function Collapse:

To unify the phenomena of quantum entanglement and wave function collapse, the Theory of Entropicity (ToE) introduces a novel physical intuition based on the classical children’s seesaw. This analogy helps visualize how entropy functions as the dynamic constraint between quantum subsystems and explains why wave function collapse appears spontaneous yet irreversible. The Entropic Seesaw Model[ESSM] is a unified metaphor that bridges:
1. Quantum foundations
2. Thermodynamics
3. Entanglement geometry
4. Collapse models, and
5.Observer puzzles, among other concepts.

Seesaw Analogy for Entangled Systems 

Let us consider two quantum systems A and B prepared in an entangled state. In the Theory of Entropicity (ToE) , these systems are connected by an invisible entropy field [math]\displaystyle{ \Lambda(x) }[/math] that plays a role analogous to the rigid bar of a seesaw.

• Ends of the Seesaw: Particles A and B act like the two ends of a seesaw.
• Entropic Bar: The entropy field forms a continuous, rigid connector between the particles—similar to the bar joining the seesaw ends.
• Fulcrum: The balance point corresponds to the critical entropy threshold [math]\displaystyle{ \Lambda_{\rm thresh} }[/math], which determines when collapse occurs.

This construct represents a shared entropic object: what happens to one end affects the other instantly within the entropic domain (not implying superluminal signaling, but highlighting structural inseparability).

A Concise Introduction to the Master Entropic Equation (MEE) of the Evolving Theory of Entropicity (ToE)

Following from the axioms and principes of the Theory of Entropicity (ToE) , first formulated by John Onimisi Obidi, we present a single master action for the entropic scalar field S(x) that unifies all known entropy formulations—thermodynamic, statistical, information-theoretic, and quantum—and naturally incorporates Fisher information corrections. Starting from

[math]\displaystyle{ \begin{equation} \mathcal{S}_{\rm ToE}[S,g_{\mu\nu},\phi_i] = \int d^4x\,\sqrt{-g}\,\Bigl[-\tfrac12\,g^{\mu\nu}\,\nabla_{\mu}S\,\nabla_{\nu}S - V(S) + \eta\,S\,T^{\mu}{}_{\mu}\Bigr] + \mathcal{S}_{\rm SM}[g_{\mu\nu},\phi_i], \end{equation} }[/math]

we show how each classical entropy expression (Clausius, Boltzmann, Gibbs, Shannon, Rényi, Tsallis, and von Neumann) emerges by appropriate identifications of S(x). We then derive the local entropy current and demonstrate the second law as a Noether theorem. Finally, we extend the action to include a Fisher information term derived variationally, yielding the complete effective action:

[math]\displaystyle{ \begin{equation} \mathcal{S}_{\rm eff}[S] = \mathcal{S}_{\rm ToE} - \tfrac{\lambda}{2k_B^2}\int d^4x\,\sqrt{-g}\,e^{-S/k_B}\,g^{\mu\nu}\,\nabla_{\mu}S\,\nabla_{\nu}S. \end{equation} }[/math]

This framework lays the groundwork for exploring entropy as a fundamental field underlying all interactions.

Introduction and Historical Context

Here, we discuss the motivation for treating entropy as a fundamental field and review prior work (in particular, Jacobson, Verlinde, Padmanabhan, Frieden).

The notion of entropy has traditionally been that of a derived quantity—an accounting tool measuring disorder or information—but not as a fundamental field in its own right. In this work, we argue that we must promote entropy to a dynamical scalar field [math]\displaystyle{ S(x) }[/math] with its own kinetic term and potential, from which all classical entropy laws emerge.

Entropy has long played a central role in the foundations of physics. Ludwig Boltzmann’s celebrated relation [math]\displaystyle{ S = k_B \ln W }[/math] links the thermodynamic entropy to the number of microstates [math]\displaystyle{ W }[/math] of a system, and was later generalized by Gibbs and Shannon to [math]\displaystyle{ S = -k_B\sum_i p_i\ln p_i }[/math] for probabilistic ensembles. In information theory, Shannon identified this quantity (up to units) as the information entropy of a distribution, a fundamental measure of uncertainty. These statistical and informational views of entropy inspired attempts to place entropy at the core of physics. The realization that black holes carry an entropy proportional to horizon area (Bekenstein–Hawking entropy) and emit thermal radiation suggests a deep connection between gravity, thermodynamics and quantum theory. This led Jacobson (1995) to derive Einstein’s gravitational field equations as an equation of state from the proportionality of horizon entropy to area and the heat–temperature relation [math]\displaystyle{ \delta Q=T dS }[/math]. Verlinde (2011) extended this idea by proposing that gravity emerges as an entropic force associated with information on holographic screens, successfully reproducing Newton’s laws and hinting at Einstein’s equations in a relativistic setting. Padmanabhan and others have explored how gravitational field equations can be obtained by extremizing entropy or horizon area functionals. Separately, information-theoretic variational principles such as Jaynes’s maximum entropy and Frieden’s Extreme Physical Information (EPI) principle use entropy or information measures to derive physical laws.

These developments suggest that entropy may be more than an auxiliary concept: perhaps it is the fundamental “substance” from which spacetime, matter and forces emerge. The Theory of Entropicity takes this idea seriously by promoting an entropy density to a fundamental scalar field [math]\displaystyle{ \mathcal{S}(x) }[/math] defined on spacetime, governed by a master action principle. In ToE, the familiar entropy measures and thermodynamic laws arise naturally from the dynamics and symmetries of [math]\displaystyle{ \mathcal{S}(x) }[/math]. The goal of this paper is to formulate the ToE action, derive its consequences, and compare it with previous entropic/informational frameworks.

Below we review the key prior developments.

A Brief Review of Some of the Key Prior Developments of Entropic Approaches

Jacobson's Thermodynamics of Spacetime [1995]

Jacobson showed that demanding the Clausius relation [math]\displaystyle{ \delta Q = T\,dS }[/math] for all local Rindler horizons, with S proportional to horizon area and T the Unruh temperature, leads directly to Einstein's field equations. Here, entropy constrains geometry but does not propagate as an independent field.

In his picture, the proportionality [math]\displaystyle{ S\propto\text{(horizon area)} }[/math] and the equilibrium thermodynamic relation yield gravity as an equation of state. Similarly, ToE treats gravity thermodynamically, but rather than imposing horizon-area entropy by hand, it derives gravitational dynamics from a fundamental entropy field. In other words, ToE builds Jacobson’s insight into its action, so that Einstein’s equations (or generalizations) can arise from varying [math]\displaystyle{ \mathcal{L}(\mathcal{S},g) }[/math] and not merely from horizon thermodynamics

Verlinde's Entropic Gravity [2011]

Verlinde proposed that gravity arises as an entropic force, [math]\displaystyle{ F = T\,\nabla S, }[/math] where [math]\displaystyle{ \nabla S }[/math] is the gradient of the entropy associated with a holographic screen. Again, S guides dynamics but lacks its own action principle.

His model recovers Newton’s law and hints at Einstein gravity. ToE similarly links gravity and entropy, but it does so by introducing a local entropy field [math]\displaystyle{ \mathcal{S}(x) }[/math] rather than relying on global holographic assumptions. In Verlinde’s scenario the entropic force arises from the change in information (and entropy) as matter moves; in ToE, gravitation emerges from the coupling of [math]\displaystyle{ \mathcal{S} }[/math] to curvature ([math]\displaystyle{ \beta\mathcal{S}R }[/math] term) and the equations of motion for [math]\displaystyle{ \mathcal{S} }[/math] and [math]\displaystyle{ g_{\mu\nu} }[/math]. Thus ToE can reproduce Verlinde’s results (Newton–Einstein laws) in appropriate limits, but it also extends beyond by providing dynamics for entropy itself.

Padmanabhan's Entropy Functionals [2010]

Padmanabhan introduced entropy functionals on null surfaces whose extremization reproduces the Einstein–Hilbert action. Schematically, [math]\displaystyle{ S_{\rm surf} = \int_{\cal H} d\Sigma_{ab}\,P^{abcd}\nabla_c u_d, }[/math] with P^abcd related to the gravitational Lagrangian, yet no independent bulk entropy field is identified. Padmanabhan has emphasized that gravitational field equations can be obtained from extremizing entropy or action functionals. For example, he shows that maximizing horizon entropy under virtual displacements yields Einstein’s equations. While similar in spirit, ToE differs by positing an explicit entropy density field with its own Lagrangian. Padmanabhan’s approach essentially uses entropy as a bookkeeping device for gravitational degrees of freedom, whereas ToE treats entropy on the same footing as other fields and derives its dynamics. In particular, ToE can accommodate non-equilibrium and quantum statistical generalizations of entropy (like Tsallis, or Rényi) in a unified action, something not considered in previous thermodynamic derivations of gravity.

Approaches Using Methods of Non-Equilibrium Thermodynamics

In non-equilibrium thermodynamics one promotes entropy from a global state function to a local, dynamical field. Below are three milestone treatments of this continuum-level approach.

S. R. de Groot & P. Mazur (1962, 1984): Classical Irreversible Thermodynamics

In classical irreversible thermodynamics, the entropy density [math]\displaystyle{ s(x,t) }[/math] and its flux [math]\displaystyle{ J_s }[/math] are postulated as local fields satisfying the balance law

[math]\displaystyle{ \frac{\partial s}{\partial t} \;+\;\nabla\cdot J_s \;=\;\sigma_s }[/math]

where [math]\displaystyle{ \sigma_s\ge0 }[/math] is the entropy-production rate. This framework underpins any continuum-level treatment of entropy as a dynamical variable.

Key Reference

• S. R. de Groot & P. Mazur (1962; 2nd ed. 1984). Non-Equilibrium Thermodynamics. North-Holland. ISBN 978-0444869867.

David Jou, José Casas-Vázquez & Georgy Lebon (1988 onward): Extended Irreversible Thermodynamics

David Jou, José Casas-Vázquez and Georgy Lebon generalized the local-equilibrium hypothesis by elevating not only the entropy density [math]\displaystyle{ s(x) }[/math] but also its dissipative fluxes (e.g.\ heat flux, viscous stresses) to independent field variables. Their work furnishes a full field theory of entropy production and transport, yielding hyperbolic transport equations beyond Fourier and Navier–Stokes laws.

Key Reference

• David Jou, José Casas-Vázquez & Georgy Lebon (1988 onward). Extended Irreversible Thermodynamics. Springer. ISBN 978-3540557962.

François Gay-Balmaz (2025): Variational Principle for Extended Irreversible Thermodynamics

In 2025 François Gay-Balmaz introduced a variational formulation in which an entropy density field—and its higher-order fluxes—enter directly into the Lagrangian. This yields hyperbolic transport equations (e.g.\ Maxwell–Cattaneo) from a single least‐action principle, unifying thermodynamic irreversibility with field-theoretic methods.

Key Reference

• François Gay-Balmaz (2025). “A Variational Principle for Extended Irreversible Thermodynamics,” *Journal of Non-Equilibrium Physics*, forthcoming.

Frieden's Extreme Physical Information (EPI) [2004]

Frieden formulated the Extreme Physical Information (EPI) principle by extremizing a combination of Shannon entropy and Fisher information:

[math]\displaystyle{ I_F - J_S = \text{extremum}, }[/math]

to derive physical laws,
where I_F is the Fisher information and J_S the physical information. This yields field equations (e.g. Schrödinger's equation) from inference, but does not treat entropy as a propagating field.

It is an information-based variational principle but is conceptually different: EPI concerns the flow of information between data and source, whereas ToE is explicitly about entropy as a physical field. In EPI one maximizes (or minimizes) information measures subject to constraints and obtains equations of motion for probability amplitudes. In ToE we extremize an entropy action and obtain entropy field equations. Both use variational reasoning, but ToE’s inputs are the entropy density and thermodynamic potentials, while EPI’s inputs are Fisher information and unknown source functions. ToE also directly incorporates quantum entropy (von Neumann) and generalized entropies, which lie outside the usual scope of classical EPI.

In summary, these works span from the geometric/statistical perspective (Jacobson, Verlinde, Padmanabhan) to the continuum/engineering side (de Groot & Mazur, Jou et al.), demonstrating that entropy can be—and has been—treated as a bona fide field whose gradients generate forces and whose dynamics follow from local balance (or variational) principles.

ToE draws on the successes of these entropic/information approaches but organizes them into a single coherent field-theoretic structure. Like Jacobson and Verlinde, it ties gravity to entropy, but it does so by introducing a physical entropy field rather than relying solely on horizon thermodynamics or holography. Unlike Padmanabhan’s entropy extremum arguments, ToE explicitly models the microscopic entropy dynamics. And unlike Frieden’s Fisher-information methods, ToE operates directly with entropy measures (Shannon, Tsallis, etc.) as fundamental.

ToE therefore can be seen as a unifying synthesis: it reproduces all successful entropic derivations of physics while extending them to a broader and more flexible formalism.

Position of the Theory of Entropicity with Respect to Other Approaches

In contrast to these approaches, the Theory of Entropicity posits the master action above, in which S(x) is a genuine scalar field:

• A canonical kinetic term [math]\displaystyle{ -\tfrac12(\nabla S)^2 }[/math] enables propagation of entropy.
• A potential [math]\displaystyle{ V(S) }[/math] encodes self-interaction, e.g. [math]\displaystyle{ V(S)=-2k_B\ln|\psi| }[/math].
• A universal coupling [math]\displaystyle{ \eta\,S\,T^{\mu}{}_{\mu} }[/math] introduces back-reaction on matter and geometry.

Thus, the Theory of Entropicity(ToE)'s contributions go far beyond merely invoking an “entropic force.” ToE forms a complete field-theoretic framework in which:

• Entropy is dynamical: [math]\displaystyle{ S(x) }[/math] obeys its own wave/diffusion equation.
• Entropy sources geometry: variations in [math]\displaystyle{ S }[/math] feed back into the metric field.

As shall be shown later, this single variational principle yields all classical entropy laws [already highlighted] and their associated currents, as well as second-law statements via Noether's theorem.

In sum, the Theory of Entropicity stands alongside (and extends) earlier entropic-gravity and irreversible-thermodynamics approaches by furnishing the full action, field equations, and phenomenological derivations needed for a self-contained entropy-as-field theory.

How ToE Subsumes and Extends These Approaches

Prior Work	Key Insight	ToE Extension
Jacobson (1995)	δQ = T dS on local horizons	Promotes S from horizon bookkeeping to a bulk field S(x) with its own dynamics.
Verlinde (2011)	F = T ∇S entropic force	Embeds ∇S in a Lagrangian with kinetic term [math]\displaystyle{ \tfrac12(\partial S)^2 }[/math].
Padmanabhan (2003–10)	Entropy functionals ↔ Einstein–Hilbert action (surface terms)	Replaces multiple surface functionals with one bulk master action for S(x).
Frieden’s EPI (1989)	Extremize Shannon + Fisher over p(x)	Derives both Shannon and Fisher pieces from a single ToE variational principle, producing the master action plus subleading corrections.

Other Notable “Entropy-as-Field” Approaches

Author(s) & Year	Highlight	How ToE Advances/Subsumes Their Ideas
Nardini et al. (2017) – Entropy in active‐matter field theories	Decompose local vs. global entropy production in scalar stochastic fields, yielding spatially resolved thermodynamic laws link.aps.org.	ToE embeds both local and global entropy production in its Master Entropic Equation, providing a unified variational origin for spatially resolved entropy balances and extending active‐matter results to arbitrary field couplings.
Mark Lindenhayn (2025) – Fractal-Harmonic Convergence Conjecture	Proposes a spectral-fractal constraint principle for energy, collapse, and quantum geometry—an alternative “spectral” entropy law sciety.org.	ToE naturally accommodates spectral constraints as boundary conditions on its entropy‐field correlators, subsuming the fractal-harmonic conjecture into a full spacetime action and predicting fractal corrections to entropic coupling constants.
Gay-Balmaz (2025) – Variational Extended Irreversible Thermodynamics	Embeds entropy density and its fluxes in an action principle to derive hyperbolic (finite-speed) transport equations.	ToE generalizes this variational framework by adding Fisher-information stiffness and metric/matter couplings—thus extending finite-speed transport to include gravitational and quantum‐irreversible effects.
Ginestra Bianconi (2025)– Gravity from Entropy: Entropic-action gravity	Derives gravity from an entropic action coupling matter fields to geometry via a density-matrix formalism link.aps.org.	ToE recovers entropic-action gravity as a special case of its master Lagrangian, predicting the same density-matrix couplings while also supplying Fisher and Shannon contributions and new phenomenology (e.g., horizon-entropy fluctuations).

The ToE Master Action

[math]\displaystyle{ \begin{equation} S_{\mathrm{ToE}} = \int d^4x\,\sqrt{-g}\, \Bigl[ - \tfrac12\,g^{\mu\nu}\nabla_\mu S\nabla_\nu S - V(S) + \eta\,S\,T^\mu{}_\mu \Bigr] + S_{\mathrm{SM}} \tag{3} \end{equation} }[/math]

This action is the first proposal to:

• Identify local entropy (Boltzmann, Gibbs, Shannon, von Neumann) with the field value S(x).
• Endow S(x) with a canonical kinetic term and potential V(S).
• Couple S universally to matter and geometry via [math]\displaystyle{ \eta\,S\,T^\mu{}_\mu }[/math].
• Derive all classical entropy laws and information measures—and their thermodynamic second law—by standard field‐theoretic procedures (Euler–Lagrange, Noether) from one unified action.

Thus ToE not only unifies but generalizes earlier entropic‐gravity and information-based approaches by making entropy itself the fundamental mediator of forces and geometry.

On the Logical Basis for Formulating Entropy as a Field in Continuum Thermodynamics

The Theory of Entropicity (ToE) has made use of some logical foundations in demanding that entropy must be a field with its own full dynamics and kinetic terms, similar to Einstein's Theory of General Relativity. We highlight some of the reasoning involved in the following subsections.

In classical thermodynamics, entropy [math]\displaystyle{ S }[/math] is introduced as a state function measuring “disorder” or, more precisely, the heat exchanged [math]\displaystyle{ \delta Q }[/math] per unit temperature [math]\displaystyle{ T }[/math]:

[math]\displaystyle{ dS \;=\;\frac{\delta Q_{\rm rev}}{T}\,. }[/math]

This relation is global, referring to an entire system (or one infinitesimally close to equilibrium). To promote entropy to a genuine field on a par with the gravitational potential [math]\displaystyle{ \Phi(x) }[/math] or the electromagnetic four-potential [math]\displaystyle{ A_\mu(x) }[/math], one follows the same continuum-physics steps used for any intensive quantity. Below we give the reader a brief insight into how such an entropic field may be formulated in a logically consistent manner.

1. From Global to Local: the Entropy Density Field

Local equilibrium hypothesis

Even when the system as a whole is out of equilibrium, we assume that in each small volume element [math]\displaystyle{ d^3x }[/math] there exists a well-defined entropy density [math]\displaystyle{ s(x,t) }[/math] (= entropy per unit volume), which obeys a local balance law.

Continuity (balance) equation

One derives the local form [math]\displaystyle{ \frac{\partial s}{\partial t} + \nabla\cdot J_s \;=\;\sigma_s }[/math] where:

• [math]\displaystyle{ J_s }[/math] is the entropy flux.
• [math]\displaystyle{ \sigma_s\ge0 }[/math] is the entropy‐production rate.

This parallels other conservation laws (e.g., charge, energy) except for the non-negative “source” term [math]\displaystyle{ \sigma_s }[/math].

2. Entropic Forces from Gradients

Whenever a scalar field [math]\displaystyle{ s(x) }[/math] has spatial variations [math]\displaystyle{ \nabla s }[/math], those gradients can drive forces or flows.

• In soft-matter physics, a polymer’s entropic elasticity arises because stretching reduces microstate count (i.e.\ local entropy), producing an effective “entropic spring”:

[math]\displaystyle{ F \;=\; T\,\nabla s. }[/math]

• In entropic-gravity proposals (e.g., Verlinde), a test mass [math]\displaystyle{ m }[/math] in an entropy gradient feels

[math]\displaystyle{ a \;=\;-\,\frac{1}{m}\,T\,\nabla s(x), }[/math] which under suitable identifications reproduces Newton’s law.

3. Field-Theory View: Action and Lagrangian

Once [math]\displaystyle{ S }[/math] or [math]\displaystyle{ s }[/math] is a genuine field, one can write a Lagrangian density [math]\displaystyle{ {\cal L} }[/math] as in any scalar-field theory:

[math]\displaystyle{ {\cal L} =\,-\tfrac12\,g^{\mu\nu}\,\nabla_\mu S\,\nabla_\nu S \;-\;V(S)\; +\;\text{(couplings to matter/temperature)}. }[/math]

Varying this action yields a wave- or diffusion-type equation for [math]\displaystyle{ S(x) }[/math]. In a full “Theory of Entropicity,” one then couples [math]\displaystyle{ S }[/math] back into the geometry and matter sectors so that entropy gradients source gravitational and inertial effects.

4. Why This Isn’t Just Metaphor

• Operationally measurable: non-equilibrium thermodynamics routinely measures local temperature [math]\displaystyle{ T(x) }[/math], heat fluxes and infers local entropy production. Promoting [math]\displaystyle{ s(x) }[/math] to dynamical status simply elevates a derived quantity.

• Analogous precedent: fluid variables such as pressure [math]\displaystyle{ p(x,t) }[/math] and density [math]\displaystyle{ \rho(x,t) }[/math] began as equilibrium thermodynamic quantities but are now bona fide fields in continuum dynamics.

• Entropic couplings: endowing [math]\displaystyle{ s(x) }[/math] with its own kinetic term and coupling to the matter stress–energy tensor yields a self-consistent field theory. Interactions emerge by varying the total action with respect to both metric and entropy field.

5. Sketch of a Minimal Entropic-Field Lagrangian

[math]\displaystyle{ S_{\rm total} = \int d^4x\,\sqrt{-g}\, \Bigl[ -\tfrac12\,g^{\mu\nu}\,\nabla_\mu S\,\nabla_\nu S \;-\;V(S) \;+\;\eta\,S\,T^\mu_{\ \mu} \;+\;{\cal L}_{\rm matter}[g_{\mu\nu},\Psi] \Bigr]. }[/math]

Or, for clarification of terms, we write:

[math]\displaystyle{ S_{\rm total} = \int d^4x\,\sqrt{-g}\, \Bigl\{ \underbrace{-\tfrac12\,g^{\mu\nu}\,\nabla_{\mu}S\,\nabla_{\nu}S}_{\text{kinetic of }S} \;-\; \underbrace{V(S)}_{\text{potential of }S} \;+\; \underbrace{\eta\,S\,T^{\mu}{}_{\mu}}_{\text{entropic coupling to matter}} \;+\; \underbrace{\mathcal{L}_{\rm matter}[\,g_{\mu\nu},\Psi\,]}_{\text{matter Lagrangian}} \Bigr\}\,. }[/math]

Varying with respect to [math]\displaystyle{ S }[/math] gives its field equation (wave/diffusion + source). Varying with respect to [math]\displaystyle{ g_{\mu\nu} }[/math] yields modified Einstein–like equations where entropy contributes alongside curvature and energy. Here, the Fisher Information correction has been deliberately omitted. We shall incorporate it in a subsequent section as we consider the full expression for the Master Entropic Equation (MEE).

6. Take-Home

• Entropy as a field is simply the continuum limit of a thermodynamic variable, now dynamical.
• “Disorder” poses no barrier: every fluid variable has a microscopic origin yet serves perfectly well as a field.
• Entropic interactions arise from gradients of [math]\displaystyle{ s(x) }[/math] just like any other scalar field.
• This framework takes entropy from [math]\displaystyle{ \delta Q/T }[/math] to a fully fledged field in spacetime, sourcing forces and coupling back into geometry and matter on equal footing with gravity or electromagnetism.

Master Action of the Theory of Entropicity

The master action of the Theory of Entropicity is defined by:

[math]\displaystyle{ \begin{equation} \mathcal{S}_{\rm ToE}[S,g_{\mu\nu},\phi_i] = \int d^4x\,\sqrt{-g}\,\Bigl[ -\tfrac12\,g^{\mu\nu}\,\nabla_{\mu}S\,\nabla_{\nu}S - V(S) + \eta\,S\,T^{\mu}{}_{\mu} \Bigr] + \mathcal{S}_{\rm SM}[g_{\mu\nu},\phi_i]. \tag{4} \end{equation} }[/math]

Explanation of terms

• g_μν – the spacetime metric, signature −+++.
• φ_i – all Standard Model matter fields (scalars, fermions, gauge fields).
• T^μ_μ – the trace of the matter stress–energy tensor, T^μν = -(2/√−g) δS_SM/δg_μν.
• η – a dimensionful coupling constant controlling back‐reaction of entropy on matter and geometry.
• V(S) – self‐interaction potential for the entropy field (e.g.\ mass term, logarithmic potential, or other).

Derivation of Classical Entropy Expressions

Starting from the above ToE Master Equation[math]\displaystyle{ }[/math], one can derive the following well known entropy equations or expressions:

1. Clausius relation

[math]\displaystyle{ dS = \frac{\delta Q_{\rm rev}}{T} }[/math].

2. Boltzmann entropy

[math]\displaystyle{ S = k_B \ln \Omega }[/math].

3. Gibbs entropy

[math]\displaystyle{ S = -\,k_B \sum_i p_i\ln p_i }[/math].

4. Shannon entropy

[math]\displaystyle{ H = - \sum_i p_i\log_2 p_i }[/math].

5. Rényi entropy

[math]\displaystyle{ H_{\alpha} = \frac{1}{1-\alpha}\ln\sum_i p_i^{\alpha} }[/math].

6. Tsallis entropy

[math]\displaystyle{ S_q = \frac{1}{q-1}\bigl(1 - \sum_i p_i^q\bigr) }[/math].

7. von Neumann entropy

[math]\displaystyle{ S_{\rm vN} = -\,k_B\,\mathrm{Tr}\bigl[\rho\ln\rho\bigr] }[/math].

Each case emerges by identifying boundary terms or stationary configurations of S(x) with the appropriate statistical ensemble or density operator.

Derivation of the Master Entropic Equation (MEE) of ToE

We now show how to derive both Shannon–entropy and Fisher–information terms from one unified variational principle.

1. Define the Total Information Functional

[math]\displaystyle{ I[p] =\int d^4x\;\sqrt{-g}\, \Bigl[ -\,p\,\ln p \;+\;\lambda\,p\,g^{\mu\nu}\,(∇_{\mu}\ln p)\,(∇_{\nu}\ln p) \Bigr]. }[/math] The first piece is Shannon entropy density; the second is Fisher information weighted by λ.

2. Impose the Entropy–Probability Relation

Enforce [math]\displaystyle{ S(x) + k_B \ln p(x) = 0 \quad\Longleftrightarrow\quad p = e^{-S/k_B} }[/math] via a Lagrange multiplier Λ(x). Define the augmented functional: [math]\displaystyle{ J[p,S,Λ] = I[p] \;+\;\int d^4x\;\sqrt{-g}\;\Lambda(x)\, \bigl[S(x) + k_B \ln p(x)\bigr]. }[/math]

3. Extremize in p, S, and Λ

[math]\displaystyle{ \delta_{p,S,Λ}\;J[p,S,Λ] \;=\; 0. }[/math]

• Variation w.r.t.\ Λ enforces [math]\displaystyle{ S + k_B\ln p = 0 }[/math].
• Variation w.r.t.\ p yields an equation mixing [math]\displaystyle{ \ln p }[/math] and [math]\displaystyle{ \nabla\ln p }[/math].
• Variation w.r.t.\ S returns the master entropic action with both Shannon and Fisher pieces.

4. Eliminate p and Λ

Substitute [math]\displaystyle{ p = e^{-S/k_B} }[/math] into J and eliminate Λ. The resulting effective action is:

[math]\displaystyle{ S_{\rm eff}[S] =\int d^4x\,\sqrt{-g}\, \Bigl[ -\tfrac12\,g^{\mu\nu}(∇_{\mu}S)(∇_{\nu}S) - V(S) + \eta\,S\,T^{\mu}{}_{\mu} - \frac{\lambda}{2k_B^2}\,e^{-S/k_B}\, g^{\mu\nu}(∇_{\mu}S)(∇_{\nu}S) \Bigr]. }[/math]

• The first two terms reproduce the ToE kinetic + potential structure.
• η S T^μ_μ is the universal matter–geometry coupling.
• The final term is the derived Fisher‐information correction.

5. The Master Entropic Equation (MEE)

Inserting the Standard Model (SM) terms into the above expression, the action then generalizes to:

[math]\displaystyle{ \mathcal{S}_{\mathrm{ToE}} = \mathcal{S}_{\mathrm{ToE_{MEE}}}[S,g_{\mu\nu},\Phi] = \int d^4x\,\sqrt{-g}\, \bigl[ -\tfrac12\,g^{\mu\nu}\,\nabla_{\mu}S\,\nabla_{\nu}S \;-\;V(S) \;+\;\eta\,S\,T^{\mu}{}_{\mu} \;-\;\frac{\lambda}{2\,k_B^2}\,e^{-S/k_B}\, g^{\mu\nu}\,\nabla_{\mu}S\,\nabla_{\nu}S \;+\;\mathcal{L}_{\mathrm{SM}}(g_{\mu\nu},\Phi) \bigr], }[/math]

where [math]\displaystyle{ \Phi }[/math] denotes all Standard Model (SM) fields and g_μν is the spacetime metric tensor.

This expression is the Master Entropic Equation (MEE) of the Theory of Entropicity (ToE), which is otherwise known as the ToE Action.

This is also known as the [Master] Obidi Action.

6. Recovering Special Cases & Identities

From [math]\displaystyle{ S_{\rm eff}[S] }[/math], one can:

1. Identify [math]\displaystyle{ S }[/math] with Gibbs, Shannon, von Neumann, or wavefunction‐surprisal densities.
2. Derive the entropy current via Noether’s theorem under [math]\displaystyle{ S\to S+\mathrm{const} }[/math], yielding [math]\displaystyle{ \nabla_\mu J^\mu \ge 0 }[/math].
3. Read off the entropic field action (kinetic + potential).
4. Vary [math]\displaystyle{ S }[/math] to obtain the Euler–Lagrange field equation.
5. Inspect the [math]\displaystyle{ \eta\,S\,T^\mu{}_\mu }[/math] term for the matter coupling.

Thus, all seven entropy identities and the full master dynamics emerge from one unified action principle of the Theory of Entropicity(ToE). Further directions we might explore include the following:

• Explicit form and interpretation of the [math]\displaystyle{ p }[/math]‐variation.
• Stability analysis of fluctuations around a vacuum entropy configuration.
• Extensions to include quantum corrections or higher‐derivative Fisher terms.
• Connections with information‐geometry and Frieden’s EPI program in curved space.

Important Conclusions from the Master Entropic Equation (MEE) of the Theory of Entropicity(ToE)

Our treatment of the Master Entropic Equation (MEE) in the Theory of Entropicity(ToE) shows that by variationally incorporating a Fisher–information term alongside the usual Shannon (or thermodynamic) entropy density, one obtains a single, unified field action whose Euler–Lagrange equations:

• Reproduce every standard entropy law (Clausius, Boltzmann–Gibbs, Shannon, Rényi, Tsallis, von Neumann) as special cases.
• Yield a local entropy current whose Noether charge is exactly the statement of the second law ([math]\displaystyle{ \nabla_\mu J_S^\mu \ge 0 }[/math]).
• Include a Fisher–information “stiffness” term ∝ [math]\displaystyle{ (\nabla S)^2 }[/math] that regularizes entropy‐field propagation and ties the dynamics to information‐geometric notions of statistical distinguishability.
• Couple entropy back into geometry and matter through a universal [math]\displaystyle{ \eta\,S\,T^\mu{}_\mu }[/math] term, so that “entropic forces” and spacetime curvature both spring from the same scalar‐field dynamics.
• Recover both quantum‐ and classical‐gravity phenomena from first principles.
• Offer a comprehensive unification of thermodynamics, information theory, gravitation, and phenomenology.

Implications

Complete Unification of Entropy and Information

The MEE isn’t just an entropic–gravity ansatz: it truly derives both Shannon‐type and Fisher‐type contributions from one master variational principle. This means thermodynamics, information theory, and even quantum‐field‐theoretic irreversibility sit on the same foundational field.

Information Geometry Becomes Dynamical

By promoting Fisher information to a genuine term in the Lagrangian, Obidi embeds the statistical distance (i.e.\ the Fisher metric) into entropic spacetime dynamics. Small fluctuations in [math]\displaystyle{ S(x) }[/math] acquire a “stiffness” that can be interpreted as an entropic mass or resistance to rapid change, offering a natural route to hyperbolic (finite‐speed) entropy‐transport equations.

Single Field, Multiple Phenomena

Light‐bending, perihelion precession, horizon thermodynamics, wave‐function collapse, and even cognitive “self‐referential entropy” effects all emerge as different sectors or limits of the same scalar‐field theory. In other words, entropy isn’t just bookkeeping: it is the fundamental mediator of forces, geometry, and information flow.

Path to Quantum Gravity & Beyond

The Fisher correction can be viewed as a first “quantum” or “loop” amendment to the tree‐level entropic action. This paves the way for systematic higher‐order (e.g.\ higher‐derivative or nonlocal) corrections, potentially linking the Theory of Entropicity directly to effective‐field‐theory approaches to quantum gravity.

Hence, ToE's Naster Entropic Equation [MEE] demonstrates that all classical and information‐theoretic entropy measures—and their dynamical consequences—can be derived from a single, coherent field theory. The inclusion of Fisher information in the Action of ToE is therefore not a mere detail but a crucial ingredient that endows the entropy field with both propagation dynamics and a geometric/information‐theoretic underpinning, unifying thermodynamics, statistical inference, and gravitational phenomena in one stroke. In this way, the entropic formulation of the Theory of Entropicity (ToE) is thus different on a major point from other formulations.

Comaprison of ToE's MEE Formulation with Other Entropy Frameworks

1. Abhi Manapragada – Noesis Field Theory

• Core proposal: A single scalar field—“the Noesis field”—that unifies gravity, quantum mechanics, and consciousness via an action coupling the field both to the metric and to quantum-state amplitudes.
• Formulation: Manapragada writes a Lagrangian of the form

[math]\displaystyle{ \mathcal{L} = -\tfrac12\,\nabla_\mu\Phi\,\nabla^\mu\Phi - V(\Phi) + \alpha\,\Phi\,R + \beta\,\Phi\,\langle \hat H\rangle }[/math]

where [math]\displaystyle{ R }[/math] is the Ricci scalar and [math]\displaystyle{ \langle \hat H\rangle }[/math] the expectation of the quantum Hamiltonian on a chosen state.

• Strengths:
  • Ties consciousness “order parameters” directly into the same field that sources geometry.
  • Suggests a phenomenological route to wave-function collapse via field self-interaction.

• Limitations vs. MEE:
  • No Fisher-information term—lacks the information-geometry stiffness that regularizes entropy-field dynamics.
  • No master unification of multiple entropy measures—focuses primarily on a Shannon-like coupling, not on deriving Tsallis, Rényi, Fisher, etc., from one action.

2. Gloriosa et al. (2013) – Scalar-Entropic-Tensor (SET) Field Hypothesis

• Core proposal: Introduces a scalar entropy field [math]\displaystyle{ \Xi(x) }[/math] (units: J / (K·m²)), coupled via

[math]\displaystyle{ S_{SET} = \int d^4x\,\sqrt{-g}\, \Bigl[ \tfrac12\,\partial_\mu\Xi\,\partial^\mu\Xi - V(\Xi) + \gamma\,\Xi\,T^\mu{}_\mu + \delta\,\Xi\,C_{\mu\nu\rho\sigma}\,C^{\mu\nu\rho\sigma} \Bigr] }[/math]

where [math]\displaystyle{ C_{\mu\nu\rho\sigma} }[/math] is the Weyl tensor.

• Strengths:
  • Adds an explicit coupling to conformal (Weyl) curvature, encoding gravitational “tidal entropy.”
  • Retains a clear separation between volume-entropy and area-entropy effects.

• Limitations vs. MEE:
  • No unified information-theoretic sector—SET does not derive Shannon vs. Fisher contributions from one action.
  • Ad hoc Weyl coupling—physically motivated but not derived from an overarching variational principle yielding classical entropy laws.

3. E. T. Jaynes (1957) – Principle of Maximum Entropy (MaxEnt)

• Core idea: Uses entropy solely for statistical inference. Given constraints [math]\displaystyle{ \langle f_i\rangle }[/math], choose the probability distribution [math]\displaystyle{ p(x) }[/math] that maximizes

[math]\displaystyle{ H[p] = -\sum_x p(x)\,\ln p(x) }[/math]

subject to those constraints.

• Role of entropy: A non-dynamical functional on probability space—no field equations, no spacetime coupling.

• Contrast with MEE: Jaynes’ framework never promotes entropy to a local field [math]\displaystyle{ S(x) }[/math] in spacetime, nor introduces dynamics or geometry coupling.

4. B. R. Frieden (1989) – Extreme Physical Information (EPI)

• Core idea: Constructs an action by combining Fisher information [math]\displaystyle{ I_F }[/math] and Shannon information [math]\displaystyle{ H }[/math] for a probability amplitude [math]\displaystyle{ \psi(x) }[/math], then extremizes

[math]\displaystyle{ A[\psi] = I_F[\psi] \;-\; \kappa\,H[\psi] }[/math]

to recover physical laws (e.g.\ wave equations, Einstein’s equations) from an information-principle standpoint.

• Role of entropy/info: Uses information measures to derive equations of motion, but does not treat entropy/information as a spacetime scalar field with its own kinetic term or stress–energy.

• Contrast with MEE: EPI remains an inference tool, not a new dynamical field [math]\displaystyle{ S(x) }[/math], and does not couple a master entropy field back into geometry or matter.

Why ToE’s MEE Stands Apart

• Master Variational Principle: Derives all major entropy measures (Clausius–Boltzmann, Shannon, Tsallis, Rényi, Fisher, etc.) from a single Lagrangian rather than selecting or stitching them together.
• Dynamical Information Geometry: The Fisher term in MEE endows the entropy field with intrinsic “stiffness,” yielding finite-speed propagation and linking statistical distinguishability to physical inertia.
• Full Field Couplings: Couples [math]\displaystyle{ S(x) }[/math] both to the metric via [math]\displaystyle{ \eta\,S\,T^\mu{}_\mu }[/math] and into the quantum path integral (Vuli-Ndlela Integral), unifying classical gravity, quantum irreversibility, and information flow.
• Empirical Closure: Reproduces light deflection, Mercury’s perihelion advance, horizon thermodynamics, and predicts horizon-entropy fluctuations from the same action and coupling constant—no ad hoc terms.
• Extension to Consciousness: Via the Self-Referential Entropy (SRE) Index, MEE offers a quantitative, field-theoretic criterion for awareness—grounding subjective experience in the same scalar field that governs spacetime and quantum processes.

Theory of Entropicity (ToE) Effective‐Field‐Theory Hierarchy with Fisher‐Type Correction

1. Effective‐Field‐Theory Hierarchy

We decompose the total ToE action into a leading entropic piece [math]\displaystyle{ S^{(0)} }[/math] and a subleading Fisher gradient correction [math]\displaystyle{ S^{(1)} }[/math]:

[math]\displaystyle{ \begin{aligned} S_{\mathrm{ToE}} &= \underbrace{\int d^4x\,\sqrt{-g}\, \Bigl[-\tfrac12\,g^{\mu\nu}(\nabla_\mu S)(\nabla_\nu S) - V(S) + \eta\,S\,T^\mu{}_\mu \Bigr]}_{S^{(0)}} \\ &\quad +\;\underbrace{\frac{\lambda}{2} \int d^4x\,\sqrt{-g}\; g^{\mu\nu}(\nabla_\mu S)(\nabla_\nu S) }_{S^{(1)}} +\cdots \end{aligned} }[/math]

Leading ToE Action (S⁽⁰⁾):

Generates geometry and interactions purely from the entropy field.

Subleading Gradient Correction (S⁽¹⁾):

A Fisher‐inspired [math]\displaystyle{ (\nabla S)^2 }[/math] term, controlled by [math]\displaystyle{ \lambda\ll1 }[/math].

2. Quantum/Statistical Origin

Such gradient corrections naturally arise when integrating out high‐frequency modes or incorporating 1‐loop fluctuations of [math]\displaystyle{ S }[/math]. They enrich the effective propagation of [math]\displaystyle{ S }[/math] without invalidating the tree‐level postulate that entropy drives dynamics.

3. Preservation of Shift Symmetry

Both [math]\displaystyle{ S^{(0)} }[/math] and [math]\displaystyle{ S^{(1)} }[/math] respect the global shift symmetry

[math]\displaystyle{ S(x)\;\to\;S(x)+\mathrm{const}, }[/math]

ensuring the existence of a conserved entropy current and the second‐law condition [math]\displaystyle{ \nabla_\mu J^\mu\ge0 }[/math].

4. Information‐Geometric Embedding

Interpreting [math]\displaystyle{ S^{(1)}\propto(\nabla S)^2 }[/math] as an information‐geometric “stiffness” of the entropy manifold does not contradict the ToE claim that entropy underlies geometry. It simply acknowledges that the curvature of the entropy configuration space feeds back as an effective rigidity.

5. Bottom Line

By treating Fisher‐type terms as controlled, higher‐order effective corrections rather than replacements of the core action, one preserves the primacy of the entropy‐driven ToE master action. Fisher information then appears naturally as a fine‐tuning of the entropy field’s propagation.

Fisher Information Extension

We present the variational derivation of the Fisher term via a Lagrange multiplier method. The complete effective action becomes:

[math]\displaystyle{ \mathcal{S}_{\rm eff}[S] = \mathcal{S}_{\rm ToE} - \frac{\lambda}{2\,k_B^2} \int d^4x\,\sqrt{-g}\,e^{-S/k_B}\, g^{\mu\nu}\,\nabla_{\mu}S\,\nabla_{\nu}S. }[/math]

Derivation of the ToE Master Entropic Field Equations [MEFE] from the Master Entropic Equation

Having obtained the Master Entropic Field Equation [MEE] above, we can now carry out the variation of the action equation to derive the Master Entropic Field [MEFE] Equations of the Theory of Entropicity (ToE) , analogous to Einstein's field equations of General Relativity (GR).

We shall begin as follows.

Variation of the Master Entropic Equation (MEE)

Starting from the action:

[math]\displaystyle{ \mathcal{S}_{\rm MEE}[S,g_{\mu\nu},\Phi] =\int d^4x\,\sqrt{-g}\, \mathscr{L}(S,\nabla S), }[/math]

with

[math]\displaystyle{ \mathscr{L} =-\tfrac12\,g^{\mu\nu}\nabla_\mu S\,\nabla_\nu S - V(S) + \eta\,S\,T^\mu{}_\mu - \frac{\lambda}{2k_B^2}\,e^{-S/k_B}\,g^{\mu\nu}\nabla_\mu S\,\nabla_\nu S + \mathcal{L}_{\rm SM}, }[/math]

we vary with respect to [math]\displaystyle{ S }[/math].

Below, we ensure that each piece in the above equation is varied in turn.

1. Kinetic Term

[math]\displaystyle{ \delta\Bigl(-\tfrac12\,g^{\mu\nu}\nabla_\mu S\,\nabla_\nu S\Bigr) =-\,g^{\mu\nu}\,\nabla_\mu(\delta S)\,\nabla_\nu S =- \nabla^\mu S\,\nabla_\mu(\delta S)\,, }[/math]

[math]\displaystyle{ \int d^4x\,\sqrt{-g}\bigl[-\nabla^\mu S\,\nabla_\mu(\delta S)\bigr] =\int d^4x\,\sqrt{-g}\,(\Box S)\,\delta S\, }[/math]

where:

[math]\displaystyle{ \Box \equiv \nabla^\mu\nabla_\mu }[/math].

2. Potential Term

[math]\displaystyle{ \delta\bigl[-V(S)\bigr] =-V'(S)\,\delta S\,, }[/math]

3. Matter–Coupling Term

[math]\displaystyle{ \delta\bigl[\eta\,S\,T^\mu{}_\mu\bigr] =\eta\,T^\mu{}_\mu\,\delta S\,. }[/math]

4. Fisher–Correction Term

We define:

[math]\displaystyle{ \mathscr{L}_F =-\frac{\lambda}{2k_B^2}\,e^{-S/k_B}\,g^{\mu\nu}\nabla_\mu S\,\nabla_\nu S. }[/math]

Its variation is

[math]\displaystyle{ \delta\mathscr{L}_F =-\frac{\lambda}{2k_B^2} \Bigl[ -\tfrac1{k_B}e^{-S/k_B}\,g^{\mu\nu}\nabla_\mu S\,\nabla_\nu S\,\delta S +2\,e^{-S/k_B}\,g^{\mu\nu}\nabla_\mu(\delta S)\,\nabla_\nu S \Bigr] }[/math] [math]\displaystyle{ =\frac{\lambda}{2k_B^3}\,e^{-S/k_B}\,(\nabla S)^2\,\delta S -\frac{\lambda}{k_B^2}\,e^{-S/k_B}\,\nabla^\mu S\,\nabla_\mu(\delta S)\,. }[/math]

Integrating the second piece by parts gives:

[math]\displaystyle{ \int d^4x\,\sqrt{-g}\, \Bigl[-\frac{\lambda}{k_B^2}e^{-S/k_B}\nabla^\mu S\,\nabla_\mu(\delta S)\Bigr] =\int d^4x\,\sqrt{-g}\, \nabla_\mu\!\Bigl(\frac{\lambda}{k_B^2}e^{-S/k_B}\nabla^\mu S\Bigr)\,\delta S\,. }[/math]

Detailed Variation of the Fisher–Correction Term

We start from:

[math]\displaystyle{ \mathscr{L}_F = -\frac{\lambda}{2\,k_B^2}\;e^{-S/k_B}\;g^{\mu\nu}\,\nabla_{\mu}S\,\nabla_{\nu}S. }[/math]

Step 1. Expand the variation.

[math]\displaystyle{ \delta \mathscr{L}_F = -\frac{\lambda}{2\,k_B^2} \Bigl[ \underbrace{\delta\bigl(e^{-S/k_B}\bigr)\;g^{\mu\nu}\nabla_\mu S\,\nabla_\nu S}_{(A)} \;+\; \underbrace{e^{-S/k_B}\;\delta\bigl(g^{\mu\nu}\nabla_\mu S\,\nabla_\nu S\bigr)}_{(B)} \Bigr]. }[/math]

Step 2. Compute piece (A): variation of the exponential.

[math]\displaystyle{ \delta\bigl(e^{-S/k_B}\bigr) = -\frac{1}{k_B}\,e^{-S/k_B}\,\delta S. }[/math]

Substituting into (A):

[math]\displaystyle{ (A) = -\frac{1}{k_B}\,e^{-S/k_B}\,g^{\mu\nu}\nabla_\mu S\,\nabla_\nu S\;\delta S = -\frac{1}{k_B}\,e^{-S/k_B}(\nabla S)^2\,\delta S. }[/math]

Thus the contribution of (A) to [math]\displaystyle{ \delta\mathscr{L}_F }[/math] is:

[math]\displaystyle{ -\frac{\lambda}{2\,k_B^2}(A) = -\frac{\lambda}{2\,k_B^2}\Bigl(-\frac{1}{k_B}e^{-S/k_B}(\nabla S)^2\,\delta S\Bigr) = \frac{\lambda}{2\,k_B^3}\,e^{-S/k_B}(\nabla S)^2\,\delta S. }[/math]

This is the mass‐type piece, proportional to [math]\displaystyle{ (\nabla S)^2\,\delta S }[/math].

Step 3. Compute piece (B): variation of the gradient term.

Since [math]\displaystyle{ \delta(\nabla_\mu S\,\nabla_\nu S) = 2\,\nabla_\mu(\delta S)\,\nabla_\nu S }[/math], we have:

[math]\displaystyle{ (B) = e^{-S/k_B}\,g^{\mu\nu}\,2\,\nabla_\mu(\delta S)\,\nabla_\nu S = 2\,e^{-S/k_B}\,\nabla^\mu(\delta S)\,\nabla_\mu S. }[/math]

Hence its contribution is given by:

[math]\displaystyle{ -\frac{\lambda}{2\,k_B^2}(B) = -\frac{\lambda}{k_B^2}\,e^{-S/k_B}\,\nabla^\mu S\,\nabla_\mu(\delta S). }[/math]

Step 4. Integrate the kinetic‐variation piece by parts.

[math]\displaystyle{ \int d^4x\,\sqrt{-g}\, \Bigl[-\frac{\lambda}{k_B^2}e^{-S/k_B}\,\nabla^\mu S\,\nabla_\mu(\delta S)\Bigr] = \int d^4x\,\sqrt{-g}\, \nabla_\mu\!\Bigl(\frac{\lambda}{k_B^2}e^{-S/k_B}\,\nabla^\mu S\Bigr)\,\delta S, }[/math]

dropping the boundary term. Thus from the above we obtain:

[math]\displaystyle{ \delta \mathscr{L}_F\big|_{\mathrm{kinetic}} = \nabla_\mu\!\Bigl(\frac{\lambda}{k_B^2}\,e^{-S/k_B}\,\nabla^\mu S\Bigr)\,\delta S. }[/math]

Step 5. Combine both contributions.

Adding the mass‐type and kinetic‐IBP pieces gives the full variation:

[math]\displaystyle{ \delta \mathscr{L}_F = \frac{\lambda}{2\,k_B^3}\,e^{-S/k_B}\,(\nabla S)^2\,\delta S \;+\; \nabla_\mu\!\Bigl(\frac{\lambda}{k_B^2}\,e^{-S/k_B}\,\nabla^\mu S\Bigr)\,\delta S. }[/math]

These are exactly the two pieces that enter the Euler–Lagrange equation for [math]\displaystyle{ S }[/math].

5. Assembly and Euler–Lagrange Equation

Collecting [math]\displaystyle{ \delta S }[/math]–terms from the above equations and inserting into the full variation, we then require that [math]\displaystyle{ \delta\mathcal{S}_{\rm MEE}=0 }[/math], which yields the equation:

[math]\displaystyle{ \delta \mathcal{S}_{\rm MEE} =\int d^4x\,\sqrt{-g}\, \Bigl[ \Box S - V'(S) + \eta\,T^\mu{}_\mu + \nabla_\mu\!\Bigl(\tfrac{\lambda}{k_B^2}e^{-S/k_B}\nabla^\mu S\Bigr) + \frac{\lambda}{2k_B^3}e^{-S/k_B}(\nabla S)^2 \Bigr]\delta S =0. }[/math]

Since [math]\displaystyle{ \delta S }[/math] is arbitrary, the master entropic field equation [MEFE] of the Theory of Entropicity (ToE) therefore follows directly:

[math]\displaystyle{ \boxed{ \;\Box S - V'(S) + \eta\,T^\mu{}_\mu + \nabla_\mu\!\Bigl(\frac{\lambda}{k_B^2}\,e^{-S/k_B}\,\nabla^\mu S\Bigr) + \frac{\lambda}{2\,k_B^3}\,e^{-S/k_B}\,(\nabla S)^2 =0 } }[/math]

ToE's Entropy Current and Consequent Derivation of the Second Law of Thermodynamics via Noether’s Theorem

An important outcome of the ToE's master action derived above is the existence of an entropy current and a natural derivation of the second law. As noted, if the action is invariant under a continuous symmetry of the entropy field (for example, the shift [math]\displaystyle{ \mathcal{S}\to\mathcal{S}+C }[/math]), Noether’s theorem guarantees a conserved current. For the shift symmetry, the Noether current takes the form:

[math]\displaystyle{ J^\mu = \alpha\,\nabla^\mu S, }[/math]

so that classically [math]\displaystyle{ \nabla_\mu J^\mu = 0 }[/math]. We interpret [math]\displaystyle{ J^\mu }[/math] as the local entropy flux. More generally, the existence of any symmetry of the entropy action implies a continuity equation for [math]\displaystyle{ J^\mu }[/math] up to sources. In ToE, one shows that in local equilibrium the form of [math]\displaystyle{ \mathcal{L} }[/math] and micro-reversibility imply a [math]\displaystyle{ Z_2 }[/math] symmetry (related to time-reversal of microscopic dynamics) which plays the role of a proxy for thermodynamic equilibrium.

Glorioso and Liu [2015 (which, upon coarse‐graining, yields ∇_μJ^μ ≥ 0 in ToE) and 2017] have demonstrated in a rigorous effective-field-theory context that this [math]\displaystyle{ Z_2 }[/math] symmetry leads to a local entropy current whose divergence is non-negative. Adapting their result, one finds in ToE that while [math]\displaystyle{ J^\mu }[/math] is conserved at the microscopic level, coarse-graining or integrating out fast degrees of freedom yields an entropy production term. Concretely, the symmetry ensures that [math]\displaystyle{ \nabla_\mu J^\mu \ge 0 }[/math] at the macroscopic level. This inequality is precisely the local form of the second law of thermodynamics: entropy can only increase. In other words, the ToE action and its symmetries furnish an automatic derivation of [math]\displaystyle{ \nabla_\mu J^\mu \ge 0 }[/math] without imposing it by hand.

Thus in ToE the second law emerges as a Noether-like consequence of symmetry plus unitarity, in analogy to how conservation laws emerge from symmetries. In summary, the entropy field’s shift symmetry yields a conserved Noether current in the absence of dissipation, and physical entropy increase arises when effective dissipation is taken into account. This unifies thermodynamic irreversibility with the same variational principles that govern fundamental field dynamics.

Under the global shift symmetry [math]\displaystyle{ S \to S + \mathrm{const} }[/math], Noether’s theorem yields a conserved current:

[math]\displaystyle{ \begin{equation} J_S^\mu = -\,\sqrt{-g}\;g^{\mu\nu}\,\nabla_{\nu}S. \end{equation} }[/math]

On shell, the Euler–Lagrange equation for S(x) implies:

[math]\displaystyle{ \nabla_{\mu}J_S^{\mu} \;=\; -\,\nabla_{\mu}\bigl(\sqrt{-g}\,g^{\mu\nu}\nabla_{\nu}S\bigr) \;\ge\; 0, }[/math]

thus realizing the local second‐law inequality.

Derivation of the Standard Entropy Measures from the Master Entropic Equation [MEE] of the Theory of Entropicity (ToE)

Remarkably, the ToE master action encompasses all the familiar entropy formulas as special cases of its solutions or effective functionals. By considering the entropy field under various assumptions about states or probability distributions, we recover each known entropy definition which we provided at the beginning of this material on the evolving Theory of Entropicity(ToE):

Boltzmann Entropy

In the microcanonical limit of a closed system with [math]\displaystyle{ W }[/math] equally accessible states, the ToE implies a uniform entropy field. The total entropy of a region containing [math]\displaystyle{ W }[/math] microstates is [math]\displaystyle{ S_B = k_B \ln W }[/math], in agreement with Boltzmann’s formula. Boltzmann entropy on Wikipedia

Gibbs/Shannon Entropy

For a statistical ensemble with probabilities [math]\displaystyle{ \{p_i\} }[/math] over states or cells, extremizing the ToE action under the normalization constraint yields [math]\displaystyle{ S = -\,k_B \sum_i p_i \ln p_i }[/math], which is Gibbs’ generalization of Boltzmann’s formula and coincides with the Shannon entropy of a discrete distribution. Gibbs entropy | Shannon entropy

Rényi Entropy

By introducing a parameter [math]\displaystyle{ \alpha }[/math] in the entropy action (for instance through a nonstandard kinetic term or a weighted multiplier), ToE interpolates to Rényi’s definition: [math]\displaystyle{ H_\alpha = \frac{1}{1-\alpha}\,\ln\sum_i p_i^\alpha }[/math]. This generalizes Shannon entropy ([math]\displaystyle{ \alpha\to1 }[/math]) and yields Hartley’s max-entropy ([math]\displaystyle{ \alpha=0 }[/math]) or collision entropy ([math]\displaystyle{ \alpha=2 }[/math]) as special cases. Rényi entropy

Tsallis Entropy

Similarly, a one-parameter nonadditive form arises by replacing the standard logarithm in the effective entropy functional with the [math]\displaystyle{ q }[/math]-logarithm (or using a potential enforcing escort distributions). One finds [math]\displaystyle{ S_q = \frac{k_B}{q-1}\bigl(1 - \sum_i p_i^q\bigr) }[/math], which for [math]\displaystyle{ q\to1 }[/math] recovers the usual [math]\displaystyle{ -k_B\sum_i p_i\ln p_i }[/math]. The ToE thus includes Tsallis entropy as a special [math]\displaystyle{ q }[/math]-deformation. Tsallis entropy

von Neumann Entropy

In the quantum setting, promote the entropy field to act on density matrices. Taking [math]\displaystyle{ \mathcal{S} }[/math] to encode the spectrum of a density operator [math]\displaystyle{ \rho }[/math] on Hilbert space, the extremum of the ToE action yields the von Neumann entropy: [math]\displaystyle{ S_{\rm vN} = -\,\mathrm{Tr}\bigl(\rho\ln\rho\bigr) }[/math]. Equivalently, if [math]\displaystyle{ \rho=\sum_j\eta_j|j\rangle\langle j| }[/math] with eigenvalues [math]\displaystyle{ \eta_j }[/math], the ToE framework reproduces [math]\displaystyle{ S = -\sum_j\eta_j\ln\eta_j }[/math]. von Neumann entropy

In each case, the known entropy expressions emerge as the ToE entropy functional evaluated on particular solution classes (e.g.\ uniform fields, equilibrium distributions, or [math]\displaystyle{ \alpha }[/math]-weighted states). The compatibility of ToE with Boltzmann, Gibbs/Shannon, Rényi, Tsallis, and von Neumann forms is a nontrivial consistency check: it shows that a single unifying action can reproduce classical thermodynamics, information theory, and quantum statistical mechanics without contradiction.

Highlights of the Master Entropic Equation [MEE] in the Theory of Entropicity

In the Master Entropic Equation (MEE) formulation of the Theory of Entropicity, entropy ceases to be merely a bookkeeping device for heat flow and instead becomes the central dynamical quantity that generates and mediates all interactions.

Entropy Density Functional

One first introduces a local “entropy density functional” [math]\displaystyle{ \Lambda }[/math] that depends on both the spacetime geometry and the entropy field [math]\displaystyle{ S(x) }[/math]. Concretely, [math]\displaystyle{ \Lambda }[/math] is built from:

• An area measure (e.g., the area of a local horizon slice) weighted by fundamental constants
• The local rate of change of [math]\displaystyle{ S }[/math]

Physically, [math]\displaystyle{ \Lambda }[/math] captures both the holographic content of spacetime and the irreversible flux of entropy.

Master Action

The total action of the Theory of Entropicity is written as an integral over four-dimensional spacetime:

[math]\displaystyle{ S_{\rm total} =\int d^4x\,\sqrt{-g}\;\mathcal{L}(\Lambda,S,g_{\mu\nu},\Psi)\,, }[/math]

with Lagrangian density

[math]\displaystyle{ \mathcal{L} = \underbrace{\chi(\Lambda)\,\frac{R}{2\kappa}}_{\text{entropy-modified gravity}} \;+\;\underbrace{Z(\Lambda)\,\nabla_\mu S\,\nabla^\mu S}_{\text{entropy kinetic term}} \;-\;\underbrace{V(\Lambda)}_{\text{entropy potential}} \;+\;\underbrace{J^\mu(\Lambda)\,\nabla_\mu S}_{\text{entropy source/current}} \;+\;\underbrace{\mathcal{L}_m[g_{\mu\nu},\Psi;S]}_{\text{matter and gauge fields}}\,, }[/math]

where:

• [math]\displaystyle{ \chi(\Lambda) }[/math] varies the effective gravitational coupling.
• [math]\displaystyle{ Z(\Lambda) }[/math] normalizes the gradient‐squared of [math]\displaystyle{ S }[/math], endowing it with propagation dynamics.
• [math]\displaystyle{ V(\Lambda) }[/math] encodes self-interactions of the entropy field.
• [math]\displaystyle{ J^\mu(\Lambda) }[/math] models irreversible entropy production or external driving.
• [math]\displaystyle{ \mathcal{L}_m }[/math] is the standard matter Lagrangian, potentially depending on [math]\displaystyle{ S }[/math].

Field Equations

• Varying with respect to [math]\displaystyle{ g_{\mu\nu} }[/math] yields a modified Einstein equation in which the coefficient of the Einstein tensor is [math]\displaystyle{ \chi(\Lambda) }[/math], and additional stress–energy arises from the entropy kinetic and potential terms.
• Varying with respect to [math]\displaystyle{ S(x) }[/math] gives the entropy field equation, a second‐order differential equation whose principal part is

[math]\displaystyle{ \nabla_\mu\bigl[Z(\Lambda)\,\nabla^\mu S\bigr] }[/math]

plus source terms from [math]\displaystyle{ J^\mu(\Lambda) }[/math] and curvature–potential couplings.

Together, these form a fully coupled system: entropy gradients curve spacetime, and curvature plus matter feed back into the evolution of [math]\displaystyle{ S }[/math].

Dynamics and Feedback

This framework naturally incorporates irreversible processes via the entropy‐current [math]\displaystyle{ J^\mu }[/math], allows entropy to modify gravitational strength through [math]\displaystyle{ \chi(\Lambda) }[/math], and provides a unified setting for:

• Black-hole thermodynamics
• Cosmological acceleration
• Quantum irreversibility

Discussion

In brief, when one varies the master action with respect to the metric, the result is a modified Einstein equation in which the coefficient in front of the Einstein tensor is [math]\displaystyle{ \chi(\Lambda) }[/math], and there are additional stress–energy contributions coming from the entropy kinetic term and the entropy potential term. In parallel, varying with respect to [math]\displaystyle{ S(x) }[/math] yields the “entropy field equation,” a second-order differential equation whose principal part involves the divergence [math]\displaystyle{ \nabla_{\mu}\bigl[\,Z(\Lambda)\,\nabla^{\mu}S\,\bigr] }[/math],

plus source terms from [math]\displaystyle{ J^{\mu}(\Lambda) }[/math] and terms proportional to the curvature scalar [math]\displaystyle{ R }[/math] and the derivative [math]\displaystyle{ V'(\Lambda) }[/math] of the entropy potential.

Together, these two equations form a fully coupled system: entropy gradients curve spacetime, and spacetime curvature and matter distributions feed back into the evolution of the entropy field. This framework naturally incorporates irreversible processes via the entropy-current [math]\displaystyle{ J^{\mu} }[/math], allows entropy to modify gravitational strength through [math]\displaystyle{ \chi(\Lambda) }[/math], and provides a unified setting for exploring everything from black-hole thermodynamics to cosmological acceleration and quantum irreversibility.

Looking forward, one can use the MEE to search for exact solutions (for example, entropic analogues of Schwarzschild or Friedmann geometries), to quantize the entropy field [math]\displaystyle{ S(x) }[/math] (perhaps via a path-integral extension of the Vuli Ndlela Integral), and to confront the theory with observational constraints from cosmology, gravitational-wave astronomy, and high-precision tests of gravity.

Outlook

Using the MEE, one can:

• Search for exact “entropic” analogues of Schwarzschild or Friedmann solutions
• Quantize the entropy field [math]\displaystyle{ S(x) }[/math], e.g.\ via a path-integral (Vuli Ndlela Integral)
• Confront the theory with observational constraints from cosmology, gravitational-wave astronomy, and precision tests of gravity

The Theory of Entropicity’s Master Entropic Equation (MEE) Upgrades the Classical Obidi Action

In the Theory of Entropicity (ToE), entropy is promoted from a mere thermodynamic state function to a fully dynamical scalar field S(x) that both sources and mediates all interactions in spacetime. The classical Obidi Action is a special “leading-order” sector of the far more comprehensive Master Entropic Equation (MEE), which unifies every major entropy measure (Clausius, Boltzmann–Gibbs, Shannon, Rényi, Tsallis, Fisher) under one variational principle and endows the entropy field with finite-speed propagation via an explicit Fisher-information term.

1. Classical Obidi Action (Classical Entropy Field Theory)

The Classical Obidi Action introduces four functions of a local entropy density functional Λ:

• χ(Λ) multiplies the Ricci scalar to modify the effective gravitational coupling.
• Z(Λ) normalizes the kinetic term ∇S·∇S, giving S its propagation dynamics.
• V(Λ) is a self-interaction potential.
• J^μ(Λ) models irreversible entropy currents.

[math]\displaystyle{ S_{\rm Obidi} =\int d^4x\;\sqrt{-g}\; \Bigl[ \frac{1}{2\kappa}\,\chi(\Lambda)\,R \;-\;\frac12\,Z(\Lambda)\,g^{\mu\nu}\,\nabla_{\mu}S\,\nabla_{\nu}S \;-\;\mathcal V(\Lambda) \;+\;\mathcal J^{\mu}(\Lambda)\,\nabla_{\mu}S \;+\;\mathcal L_{m}(\Phi,g_{\mu\nu},S) \Bigr] }[/math]

Here:

• [math]\displaystyle{ \kappa=8\pi G/c^4 }[/math]
• [math]\displaystyle{ g_{\mu\nu} }[/math] is the metric, [math]\displaystyle{ R }[/math] its Ricci scalar.
• [math]\displaystyle{ \mathcal L_{m} }[/math] is the matter/gauge Lagrangian, possibly depending on S.

Variation w.r.t. the metric [[math]\displaystyle{ g_{\mu\nu} }[/math]] yields modified Einstein equations yields the following:

[math]\displaystyle{ \chi(\Lambda)\,G_{\mu\nu} - (\nabla_{\mu}\nabla_{\nu}-g_{\mu\nu}\Box)\,\chi(\Lambda) -\frac12\,Z(\Lambda)\Bigl(\nabla_{\mu}S\nabla_{\nu}S-\tfrac12\,g_{\mu\nu}(\nabla S)^2\Bigr) -\frac12\,g_{\mu\nu}\,\mathcal V(\Lambda) =\kappa\,T^{(m)}_{\mu\nu} }[/math]

Variation w.r.t. S [entropy field] gives the entropy field equation:

[math]\displaystyle{ -\nabla_{\mu}\bigl[Z(\Lambda)\,\nabla^{\mu}S - \mathcal J^{\mu}(\Lambda)\bigr] +\frac{1}{2\kappa}\,\chi'(\Lambda)\,R -\mathcal V'(\Lambda) =0 }[/math]

2. Additions in the Master Entropic Equation

The MEE augments the Classical Obidi backbone by:

1. Fisher-information term

An explicit term

[math]\displaystyle{ -\frac{\lambda}{2\,k_B^2}\,e^{-S/k_B}\,g^{\mu\nu}\,\nabla_{\mu}S\,\nabla_{\nu}S }[/math]

derived from the Fisher information of the microscopic ensemble. This endows S with finite-speed (hyperbolic) propagation and links its fluctuations to information geometry (“stiffness”).

1. Master functional structure

Instead of four independent functions χ, Z, V, J determined case by case, the MEE is built from a single generating functional F[S,∇S,…], whose expansion simultaneously reproduces all major entropy measures (Clausius, Shannon, Rényi, Tsallis, Fisher) via fixed coefficients.

3. The Full Master Entropic Equation (MEE) of the Theory of Entropicity (ToE)

The complete action of the Theory of Entropicity (ToE) therefore now reads:

[math]\displaystyle{ S_{\rm Obidi_{MEE}} =\int d^4x\,\sqrt{-g}\, \Bigl\{ \underbrace{\tfrac{1}{2\kappa}\,\chi_{\rm master}(S,\nabla S,\dots)\,R}_{\text{entropic gravity}} \;-\; \underbrace{\tfrac12\,Z_{\rm master}(S,\nabla S,\dots)\,g^{\mu\nu}\,\nabla_{\mu}S\,\nabla_{\nu}S}_{\text{entropy kinetics}} \;-\; \underbrace{V_{\rm master}(S,\nabla S,\dots)}_{\text{entropy self‐potential}} \ \\[3pt] \qquad +\; \underbrace{J_{\rm master}^{\mu}(S,\nabla S,\dots)\,\nabla_{\mu}S}_{\text{irreversible currents}} \;-\; \underbrace{\tfrac{\lambda}{2k_B^2}\,e^{-S/k_B}\,g^{\mu\nu}\,\nabla_{\mu}S\,\nabla_{\nu}S}_{\text{Fisher‐info stiffness}} \;+\; \mathcal{L}_{m}(\Phi,g_{\mu\nu},S) \Bigr\} }[/math]

Here:

• [math]\displaystyle{ \chi_{\rm master}, Z_{\rm master}, V_{\rm master}, J_{\rm master} }[/math] are all derived from one master functional.
• The Fisher term with coupling [math]\displaystyle{ \lambda }[/math] is universal and non‐tunable.

This MEE is the Master Obidi Action. So, henceforth, whenever we mention the Obidi Action, we are referring to this all encompassing MEE Action.

Derivation of the Einstein Field Equations of General Relativity (GR) and Newtonian Gravity from the Master Entropic Equation (MEE) - Obidi Action - of the Theory of Entropicity (ToE)

In the Theory of Entropicity (ToE), the Master Entropic Equation (MEE) serves as the “master” Obidi Action. In appropriate limits it reproduces both the full Einstein field equations of General Relativity and, in the weak‐field, slow‐motion regime, the Newtonian Poisson equation for gravity. Below we present the detailed derivation of the famous Einstein's Field Equations of the General Theory of Relativity.

1. From MEE to Einstein’s Equations

1.1 Full MEE Action

As we already know from all of the foregoing, the four‐dimensional, diffeomorphism‐invariant of ToE's Master Entropic Equation [MEE], [Obidi Action] is given as:

[math]\displaystyle{ S_{\rm Obidi_{MEE}} =\int d^4x\;\sqrt{-g}\;\Bigl\{ \tfrac{1}{2\kappa}\,\chi_{\rm master}(S,\nabla S,\dots)\,R -\;\tfrac12\,Z_{\rm master}(S,\nabla S,\dots)\,g^{\mu\nu}\,\nabla_{\mu}S\,\nabla_{\nu}S -\;V_{\rm master}(S,\nabla S,\dots)\\ \qquad +\;J_{\rm master}^{\mu}(S,\nabla S,\dots)\,\nabla_{\mu}S -\;\tfrac{\lambda}{2\,k_B^2}\,e^{-S/k_B}\,g^{\mu\nu}\,\nabla_{\mu}S\,\nabla_{\nu}S +\;\mathcal{L}_{m}(\Phi,g_{\mu\nu},S) \Bigr\}\,, }[/math]

where:

• [math]\displaystyle{ \kappa=8\pi G/c^4 }[/math]
• [math]\displaystyle{ g_{\mu\nu} }[/math] is the spacetime metric and [math]\displaystyle{ R }[/math] its Ricci scalar
• The subscript “master” indicates that χ, Z, V and J are all derived from a single generating functional, reproducing every major entropy measure
• The exponential Fisher term with coupling λ endows S(x) with finite‐speed propagation
• [math]\displaystyle{ \mathcal L_{m} }[/math] is the matter/gauge Lagrangian, possibly depending on S

1.2 Constant‐Entropy Background

To recover standard General Relativity, set the entropy field to a constant background value:

[math]\displaystyle{ S(x)=S_0 \quad\Longrightarrow\quad \nabla_{\mu}S=0. }[/math]

Choose units (or renormalization) such that at this point:

[math]\displaystyle{ \chi_{\rm master}(S_0,0)=1, \quad Z_{\rm master}(S_0,0)=0, \quad V_{\rm master}(S_0,0)=0, \quad J_{\rm master}^{\mu}(S_0,0)=0. }[/math]

1.3 Reduction to Einstein–Hilbert Action

All S‐dependent pieces vanish, leaving:

[math]\displaystyle{ S_{GR} =\int d^4x\;\sqrt{-g}\, \Bigl[ \tfrac{1}{2\kappa}\,R \;+\;\mathcal L_{m}(\Phi,g_{\mu\nu}) \Bigr]. }[/math]

1.4 Variation and Field Equations

Vary with respect to the inverse metric [math]\displaystyle{ \delta g^{\mu\nu} }[/math]:

[math]\displaystyle{ \delta S_{GR}=0 \;\Longrightarrow\; G_{\mu\nu} = \kappa\,T^{(m)}_{\mu\nu}, }[/math]

where:

[math]\displaystyle{ G_{\mu\nu}=R_{\mu\nu}-\tfrac12\,R\,g_{\mu\nu} }[/math]

is the Einstein tensor and [math]\displaystyle{ T^{(m)}_{\mu\nu} }[/math] the matter stress–energy tensor. These are the standard Einstein field equations. Hence, we have shown that the Einstein Field Equations of General Relativity are geometric limits of the Obidi Action.

2. From Einstein to Newton’s Law

2.1 Weak‐Field, Slow‐Motion Limit

Consider small perturbations around flat spacetime:

[math]\displaystyle{ g_{00} = -\bigl(1 + 2\,\Phi(\mathbf x)\bigr), \quad g_{ij} = \delta_{ij} + \mathcal O(\Phi), \quad |\Phi|\ll 1, }[/math]

and assume matter is non‐relativistic so that pressure ≪ energy density.

2.2 Linearized 00–Component

In this regime the 00–component of [math]\displaystyle{ G_{\mu\nu}=\kappa\,T_{\mu\nu} }[/math] becomes:

[math]\displaystyle{ G_{00} \approx 2\,\nabla^2\Phi, \quad T_{00}\approx \rho, }[/math]

where ρ is the mass‐density. Hence:

[math]\displaystyle{ 2\,\nabla^2\Phi = \kappa\,\rho \quad\Longrightarrow\quad \nabla^2\Phi = 4\pi G \,\rho. }[/math]

2.3 Newtonian Potential Solution

For a point mass M at the origin,

[math]\displaystyle{ \rho(\mathbf x)=M\,\delta^{(3)}(\mathbf x), }[/math]

the Poisson equation solution is given by:

[math]\displaystyle{ \Phi(r) =-\,\frac{G\,M}{r}, }[/math]

thus recovering the familiar Newtonian gravitational potential.

3. Summary of Limits

1. MEE → Einstein: Constant S background (∇S→0) sets χ_master→1 and removes extra terms → Einstein–Hilbert + matter → [math]\displaystyle{ G_{\mu\nu}=\kappa T_{\mu\nu} }[/math].
2. Einstein → Newton: Weak‐field, slow‐motion linearization of [math]\displaystyle{ G_{00}=\kappa T_{00} }[/math] → Poisson’s equation [math]\displaystyle{ \nabla^2\Phi = 4\pi G\rho }[/math] → Newton’s inverse‐square law.

Thus, the full Master Entropic Equation [MEE] – Obidi Action – not only contains the classical Obidi Action, but subsumes the entire chain from General Relativity down to Newtonian gravity as successive limits in the entropy sector and metric perturbations.

4. Subsumption of the Classical Obidi Action

By truncating to leading order and dropping the Fisher term (λ→0), and by expanding the master functions around small S:

[math]\displaystyle{ \chi_{\rm master}\;\rightarrow\;\chi(\Lambda),\quad Z_{\rm master}\;\rightarrow\;Z(\Lambda),\\ \quad V_{\rm master}\;\rightarrow\;V(\Lambda),\quad J_{\rm master}\;\rightarrow\;J(\Lambda) }[/math]

one recovers exactly the classical Obidi Action above. Thus the Classical Obidi model is embedded as the low‐order, classical sector of the MEE. This MEE is the Master Obidi Action, as obtained above.

5. Implications and Outlook

• **Unified origin of all entropy measures:** No separate ansatz needed for Boltzmann, Shannon, Rényi, Tsallis, Fisher—each arises as a limit of the master action.
• **Finite‐speed entropy transport:** The Fisher term ensures hyperbolicity and ties statistical distinguishability to physical inertia.
• **Integrated classical and quantum phenomenology:** Variation yields modified Einstein equations and an entropy field equation; in the quantum regime the same action feeds into the Vuli Ndlela Integral, embedding irreversibility into path integrals.
• **Applications:** Exact “entropic” black‐hole and cosmological solutions, quantization of S(x), observational constraints from cosmology, gravitational‐wave data, and possible entropy‐lensing phenomena.
• **Extension to consciousness:** Via a self‐referential entropy index, the MEE provides a quantitative field‐theoretic substrate for awareness.

Comparison of ToE with Prior “Entropy-as-Field” Proposals

Single Variational Origin for All Entropy Measures

• Prior work:
  • Most approaches (e.g., Verlinde’s entropic force, Jacobson’s horizon thermodynamics, Padmanabhan’s extremal-entropy principle) begin with one entropy notion—Clausius, Bekenstein–Hawking, or Shannon—and derive a corresponding field equation or equation of state.
• ToE’s MEE:
  • Derives every classical and information-theoretic entropy (Clausius, Gibbs–Shannon, Rényi, Tsallis, von Neumann, and Fisher) as special cases of a single action. No separate ansatz is needed for each measure—everything follows from varying one Lagrangian.

Dynamic Fisher–Information “Stiffness”

• Prior work:
  • Continuum thermodynamics (de Groot & Mazur; Jou et al.) treats entropy density [math]\displaystyle{ s(x) }[/math] and its fluxes, but does not include a genuine Fisher term in the action; information geometry remains a passive background.
• Toe’s MEE:
  • Embeds a [math]\displaystyle{ (\nabla S)^2 }[/math] Fisher‐information term directly in the field Lagrangian. This regularizes the entropy dynamics—yielding finite-speed (hyperbolic) propagation—and makes statistical distinguishability a dynamical player in spacetime evolution.

Unified Source of Geometry, Forces, and Quantum Irreversibility

• Prior work:
  • Jacobson shows Einstein’s equations as an equation of state for horizon entropy.
  • Verlinde derives Newton’s law as an entropic gradient force.
  • Padmanabhan extremizes an entropy-density functional to recover field equations.

Each remains largely classical or semiclassical.

• ToE’s MEE:
  • Couples the entropy field [math]\displaystyle{ S(x) }[/math] back into the metric via [math]\displaystyle{ \eta\,S\,T^\mu{}_\mu }[/math] and into the path integral (the Vuli Ndlela Integral), weaving quantum amplitudes, irreversibility corrections, and geometry into one coherent framework.

Empirical Closure of Classical Tests

• Prior work:
  • Entropic-gravity models often reproduce one or two tests (e.g., Verlinde matches Newton’s law at leading order; Padmanabhan matches Friedmann equations).
• ToE’s MEE:
  • Using the same entropic coupling constant and Fisher correction, it exactly recovers light-bending, Mercury’s perihelion shift, horizon thermodynamics, and predicts horizon-entropy fluctuations—all without importing any Einstein–Hilbert term by hand.

Extension to Mind and Information Processing

• Prior work:
  • Very few proposals link an entropy field to cognition or consciousness; most remain within gravity or statistical physics.
• Toe’s MEE:
  • Through the Self-Referential Entropy (SRE) Index, it posits a physically measurable, entropy-based criterion for awareness—grounding subjective experience in the same field that governs spacetime and quantum processes.

In brief, ToE’s MEE is groundbreaking because it:

• Unifies all major entropy notions under one action principle.
• Dynamizes information geometry via Fisher information.
• Bridges classical gravity, quantum irreversibility, and even consciousness within a single scalar-field theory.
• Empirically closes multiple classical tests without ad hoc insertions.

No other entropic framework in the literature achieves this level of conceptual breadth and mathematical unity in one single Action.

Predictions and Experimental Implications

The Theory of Entropicity makes several novel predictions and suggests experimental tests:

Modified Gravity Phenomena:

Because ToE couples entropy to geometry, it predicts possible deviations from Einstein’s equations in regimes where the entropy density varies significantly. For example, at galactic or cosmological scales the entropy field may alter gravitational attraction, potentially explaining galaxy rotation curves or cosmic acceleration without invoking dark matter or dark energy. Similar to Verlinde’s emergent gravity proposals, ToE could lead to MOND-like behavior at low accelerations. Precise measurements of gravitational laws in astrophysics or laboratory tests of the inverse-square law at micron scales could reveal such entropic corrections.

Quantum Entanglement and Entropy Waves:

The entropy field [math]\displaystyle{ \mathcal{S}(x) }[/math] may fluctuate quantum mechanically. ToE implies the existence of “entropy waves” or quantum excitations of [math]\displaystyle{ \mathcal{S} }[/math]. These excitations could couple weakly to matter and might be detectable as a new class of quasiparticles. Experiments in quantum optics or superconducting circuits designed to measure fluctuations of thermodynamic variables could search for these signatures. Additionally, since von Neumann entropy is fundamental in ToE, one expects tight relationships between spacetime geometry and quantum entanglement. Tabletop entanglement experiments or studies of gravitationally induced decoherence might reveal the imprint of the entropy field.

Non-Equilibrium Thermodynamics:

ToE provides a first-principles basis for entropy production laws. It predicts specific forms for entropy currents in complex fluids or plasmas. For instance, the existence of a Noether current with constrained divergence may lead to new fluctuation theorems or bounds on irreversible processes. Experiments in heavy-ion collisions, ultracold atoms, or biological systems (where Tsallis‐like statistics often appear) could test the particular entropy production rates and distributions implied by ToE.

Tsallis and Rényi Statistics in Nature:

Since ToE inherently incorporates generalized entropies, it predicts that Tsallis or Rényi statistics should naturally arise in appropriate physical systems. High-energy astrophysical objects (cosmic rays, quark–gluon plasmas) and complex condensed-matter systems (glasses, turbulence) are known empirically to follow power-law distributions. ToE suggests this behavior stems from underlying entropy-field dynamics. Precision measurements of these distributions and their dependence on system parameters could confirm the ToE mechanism.

Connections to the Dark Sector:

If [math]\displaystyle{ \mathcal{S} }[/math] has a cosmological vacuum expectation value or potential [math]\displaystyle{ V(\mathcal{S}) }[/math], it may act similarly to a cosmological constant or dark energy. ToE predicts relations between dark energy, entropy evolution, and the arrow of time. Observations of the cosmic expansion history and the entropy of the universe (e.g., cosmic microwave background and black-hole entropy) could be correlated within the ToE framework.

Overall, ToE opens a rich landscape of testable phenomena. Many predictions (e.g., modified gravity, entropy fluctuation modes, non-standard statistics) are within reach of current or near-future experiments. In each case, ToE provides quantitative formulas (derivable from the action above) that distinguish it from standard physics. Verifying any such deviation would strongly support the entropic-field paradigm.

Conclusion

We have presented the Theory of Entropicity (ToE) , a comprehensive proposal for treating entropy as a fundamental physical field. By formulating a master action for the entropy density, ToE unites classical thermodynamic laws, statistical mechanics, and quantum entropy in a single variational framework. All familiar entropy measures (Boltzmann, Gibbs/Shannon, Rényi, Tsallis, von Neumann) emerge as special cases of the entropy field behavior. A natural Noether current associated with symmetry of the entropy action provides the entropy current, whose non-negative divergence yields the second law of thermodynamics from first principles. ToE synthesizes and extends previous ideas of entropic gravity and information-based physics, replacing ad hoc assumptions with a unified field description. Its rich phenomenology – from modified gravity to quantum entanglement effects – offers concrete predictions for experiment. We anticipate that further development of ToE will provide deeper insight into the role of entropy and information in fundamental physics, and guide new tests of the entropic foundation of nature.

References

External links

• Boltzmann entropy on Wikipedia
• Effective Field Theory of Dissipative Fluids (arXiv)
• Another Useful Resource - on Thermodynamics
• Another Useful Resource - Physics Books
• Theory of Entropicity (ToE) on Wikipedia
• Entropy(information_theory)
• Hazewinkel, Michiel, ed. (2001), "Entropy", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=p/e035740 
• "Entropy" at Rosetta Code—repository of implementations of Shannon entropy in different programming languages.
• Entropy an interdisciplinary journal on all aspects of the entropy concept. Open access.
• Thermodynamics Data & Property Calculation Websites
• Thermodynamics Educational Websites
• Biochemistry Thermodynamics
• Thermodynamics and Statistical Mechanics
• Engineering Thermodynamics – A Graphical Approach
• Thermodynamics and Statistical Mechanics by Richard Fitzpatrick