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Entropic Wave Function Diagram

 Entropic collapse diagram

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

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

When the entropic threshold condition $S_{\rm {system}}>S_{\rm {critical}}$ 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

 ToE’s Seesaw Model of Quantum Entanglement

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 $\Lambda (x)$ 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 $\Lambda _{\rm {thresh}}$, 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)

We introduce the new Theory of Entropicity (ToE) PHJOB7 (talk) 08:09, 7 July 2025 (UTC)

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

Failed to parse (unknown function "\begin{equation}"): {\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} }

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:

Failed to parse (unknown function "\begin{equation}"): {\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} }

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 $S(x)$ 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 $S=k_{B}\ln W$ links the thermodynamic entropy to the number of microstates $W$ of a system, and was later generalized by Gibbs and Shannon to $S=-k_{B}\sum _{i}p_{i}\ln p_{i}$ 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 $\delta Q=TdS$. 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 ${\mathcal {S}}(x)$ 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 ${\mathcal {S}}(x)$. 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 $\delta Q=T\,dS$ 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 $S\propto {\text{(horizon area)}}$ 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 ${\mathcal {L}}({\mathcal {S}},g)$ and not merely from horizon thermodynamics

Verlinde's Entropic Gravity [2011]

Verlinde proposed that gravity arises as an entropic force, $F=T\,\nabla S,$ where $\nabla S$ 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 ${\mathcal {S}}(x)$ 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 ${\mathcal {S}}$ to curvature ($\beta {\mathcal {S}}R$ term) and the equations of motion for ${\mathcal {S}}$ and $g_{\mu \nu }$. 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, $S_{\rm {surf}}=\int _{\cal {H}}d\Sigma _{ab}\,P^{abcd}\nabla _{c}u_{d},$ 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.

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: $I_{F}-J_{S}={\text{extremum}}toderivephysicallaws,$ 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, 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 $-{\tfrac {1}{2}}(\nabla S)^{2}$ enables propagation of entropy.
• A potential $V(S)$ encodes self-interaction, e.g. $V(S)=-2k_{B}\ln |\psi |$.
• A universal coupling $\eta \,S\,T^{\mu }{}_{\mu }$ introduces back-reaction on matter and geometry.

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.

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 ${\tfrac {1}{2}}(\partial S)^{2}$.
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.

The ToE Master Action

Failed to parse (unknown function "\begin{equation}"): {\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} }

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 $\eta \,S\,T^{\mu }{}_{\mu }$.
• 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.

Master Action of the Theory of Entropicity

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

Failed to parse (unknown function "\begin{equation}"): {\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} }

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${\displaystyle }$, one can derive the following well known entropy equations or expressions:

1. Clausius relation

$dS={\frac {\delta Q_{\rm {rev}}}{T}}$.

2. Boltzmann entropy

$S=k_{B}\ln \Omega$.

3. Gibbs entropy

$S=-\,k_{B}\sum _{i}p_{i}\ln p_{i}$.

4. Shannon entropy

$H=-\sum _{i}p_{i}\log _{2}p_{i}$.

5. Rényi entropy

$H_{\alpha }={\frac {1}{1-\alpha }}\ln \sum _{i}p_{i}^{\alpha }$.

6. Tsallis entropy

$S_{q}={\frac {1}{q-1}}{\bigl (}1-\sum _{i}p_{i}^{q}{\bigr )}$.

7. von Neumann entropy

$S_{\rm {vN}}=-\,k_{B}\,\mathrm {Tr} {\bigl [}\rho \ln \rho {\bigr ]}$.

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

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

2. Impose the Entropy–Probability Relation

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

3. Extremize in p, S, and Λ

Failed to parse (syntax error): {\displaystyle \delta_{p,S,Λ}\;J[p,S,Λ] \;=\; 0. }

• Variation w.r.t.\ Λ enforces $S+k_{B}\ln p=0$.
• Variation w.r.t.\ p yields an equation mixing $\ln p$ and $\nabla \ln p$.
• Variation w.r.t.\ S returns the master entropic action with both Shannon and Fisher pieces.

4. Eliminate p and Λ

Substitute $p=e^{-S/k_{B}}$ into J and eliminate Λ. The resulting effective action is:

Failed to parse (syntax error): {\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]. }

• 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:

${\mathcal {S}}_{\mathrm {ToE} }={\mathcal {S}}_{\mathrm {ToE_{MEE}} }[S,g_{\mu \nu },\Phi ]=\int d^{4}x\,{\sqrt {-g}}\,{\bigl [}-{\tfrac {1}{2}}\,g^{\mu \nu }\,\nabla _{\mu }S\,\nabla _{\nu }S\;-\;V(S)\;+\;\eta \,S\,T^{\mu }{}_{\mu }\;-\;{\frac {\lambda }{2\,k_{B}^{2}}}\,e^{-S/k_{B}}\,g^{\mu \nu }\,\nabla _{\mu }S\,\nabla _{\nu }S\;+\;{\mathcal {L}}_{\mathrm {SM} }(g_{\mu \nu },\Phi ){\bigr ]},$

where $\Phi$ 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.

6. Recovering Special Cases & Identities

From $S_{\rm {eff}}[S]$, one can:

1. Identify $S$ with Gibbs, Shannon, von Neumann, or wavefunction‐surprisal densities.
2. Derive the entropy current via Noether’s theorem under $S\to S+\mathrm {const}$, yielding $\nabla _{\mu }J^{\mu }\geq 0$.
3. Read off the entropic field action (kinetic + potential).
4. Vary $S$ to obtain the Euler–Lagrange field equation.
5. Inspect the $\eta \,S\,T^{\mu }{}_{\mu }$ 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 you might explore:

• Explicit form and interpretation of the $p$‐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.

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 $S^{(0)}$ and a subleading Fisher gradient correction $S^{(1)}$:

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

Leading ToE Action (S⁽⁰⁾):

Generates geometry and interactions purely from the entropy field.

Subleading Gradient Correction (S⁽¹⁾):

A Fisher‐inspired $(\nabla S)^{2}$ term, controlled by $\lambda \ll 1$.

2. Quantum/Statistical Origin

Such gradient corrections naturally arise when integrating out high‐frequency modes or incorporating 1‐loop fluctuations of $S$. They enrich the effective propagation of $S$ without invalidating the tree‐level postulate that entropy drives dynamics.

3. Preservation of Shift Symmetry

Both $S^{(0)}$ and $S^{(1)}$ respect the global shift symmetry

$S(x)\;\to \;S(x)+\mathrm {const} ,$

ensuring the existence of a conserved entropy current and the second‐law condition $\nabla _{\mu }J^{\mu }\geq 0$.

4. Information‐Geometric Embedding

Interpreting $S^{(1)}\propto (\nabla S)^{2}$ 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:

${\mathcal {S}}_{\rm {eff}}[S]={\mathcal {S}}_{\rm {ToE}}-{\frac {\lambda }{2\,k_{B}^{2}}}\int d^{4}x\,{\sqrt {-g}}\,e^{-S/k_{B}}\,g^{\mu \nu }\,\nabla _{\mu }S\,\nabla _{\nu }S.$

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:

Failed to parse (unknown function "\mathscr"): {\displaystyle \mathcal{S}_{\rm MEE}[S,g_{\mu\nu},\Phi] =\int d^4x\,\sqrt{-g}\, \mathscr{L}(S,\nabla S), }

with

Failed to parse (unknown function "\mathscr"): {\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}, }

we vary with respect to $S$.

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

1. Kinetic Term

$\delta {\Bigl (}-{\tfrac {1}{2}}\,g^{\mu \nu }\nabla _{\mu }S\,\nabla _{\nu }S{\Bigr )}=-\,g^{\mu \nu }\,\nabla _{\mu }(\delta S)\,\nabla _{\nu }S=-\nabla ^{\mu }S\,\nabla _{\mu }(\delta S)\,,$

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

where:

$\Box \equiv \nabla ^{\mu }\nabla _{\mu }$.

2. Potential Term

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

3. Matter–Coupling Term

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

4. Fisher–Correction Term

We define:

Failed to parse (unknown function "\mathscr"): {\displaystyle \mathscr{L}_F =-\frac{\lambda}{2k_B^2}\,e^{-S/k_B}\,g^{\mu\nu}\nabla_\mu S\,\nabla_\nu S. }

Its variation is

Failed to parse (unknown function "\mathscr"): {\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] } $={\frac {\lambda }{2k_{B}^{3}}}\,e^{-S/k_{B}}\,(\nabla S)^{2}\,\delta S-{\frac {\lambda }{k_{B}^{2}}}\,e^{-S/k_{B}}\,\nabla ^{\mu }S\,\nabla _{\mu }(\delta S)\,.$

Integrating the second piece by parts gives:

$\int d^{4}x\,{\sqrt {-g}}\,{\Bigl [}-{\frac {\lambda }{k_{B}^{2}}}e^{-S/k_{B}}\nabla ^{\mu }S\,\nabla _{\mu }(\delta S){\Bigr ]}=\int d^{4}x\,{\sqrt {-g}}\,\nabla _{\mu }\!{\Bigl (}{\frac {\lambda }{k_{B}^{2}}}e^{-S/k_{B}}\nabla ^{\mu }S{\Bigr )}\,\delta S\,.$

Detailed Variation of the Fisher–Correction Term

We start from:

Failed to parse (unknown function "\mathscr"): {\displaystyle \mathscr{L}_F = -\frac{\lambda}{2\,k_B^2}\;e^{-S/k_B}\;g^{\mu\nu}\,\nabla_{\mu}S\,\nabla_{\nu}S. }

Step 1. Expand the variation.

Failed to parse (unknown function "\mathscr"): {\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]. }

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

$\delta {\bigl (}e^{-S/k_{B}}{\bigr )}=-{\frac {1}{k_{B}}}\,e^{-S/k_{B}}\,\delta S.$

Substituting into (A):

$(A)=-{\frac {1}{k_{B}}}\,e^{-S/k_{B}}\,g^{\mu \nu }\nabla _{\mu }S\,\nabla _{\nu }S\;\delta S=-{\frac {1}{k_{B}}}\,e^{-S/k_{B}}(\nabla S)^{2}\,\delta S.$

Thus the contribution of (A) to Failed to parse (unknown function "\mathscr"): {\displaystyle \delta\mathscr{L}_F} is:

$-{\frac {\lambda }{2\,k_{B}^{2}}}(A)=-{\frac {\lambda }{2\,k_{B}^{2}}}{\Bigl (}-{\frac {1}{k_{B}}}e^{-S/k_{B}}(\nabla S)^{2}\,\delta S{\Bigr )}={\frac {\lambda }{2\,k_{B}^{3}}}\,e^{-S/k_{B}}(\nabla S)^{2}\,\delta S.$

This is the mass‐type piece, proportional to $(\nabla S)^{2}\,\delta S$.

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

Since $\delta (\nabla _{\mu }S\,\nabla _{\nu }S)=2\,\nabla _{\mu }(\delta S)\,\nabla _{\nu }S$, we have:

$(B)=e^{-S/k_{B}}\,g^{\mu \nu }\,2\,\nabla _{\mu }(\delta S)\,\nabla _{\nu }S=2\,e^{-S/k_{B}}\,\nabla ^{\mu }(\delta S)\,\nabla _{\mu }S.$

Hence its contribution is given by:

$-{\frac {\lambda }{2\,k_{B}^{2}}}(B)=-{\frac {\lambda }{k_{B}^{2}}}\,e^{-S/k_{B}}\,\nabla ^{\mu }S\,\nabla _{\mu }(\delta S).$

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

$\int d^{4}x\,{\sqrt {-g}}\,{\Bigl [}-{\frac {\lambda }{k_{B}^{2}}}e^{-S/k_{B}}\,\nabla ^{\mu }S\,\nabla _{\mu }(\delta S){\Bigr ]}=\int d^{4}x\,{\sqrt {-g}}\,\nabla _{\mu }\!{\Bigl (}{\frac {\lambda }{k_{B}^{2}}}e^{-S/k_{B}}\,\nabla ^{\mu }S{\Bigr )}\,\delta S,$

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\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. }

Step 5. Combine both contributions.

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

Failed to parse (unknown function "\mathscr"): {\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. }

These are exactly the two pieces that enter the Euler–Lagrange equation for $S$.

5. Assembly and Euler–Lagrange Equation

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

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

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

Failed to parse (unknown function "\boxed"): {\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 } }

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 ${\mathcal {S}}\to {\mathcal {S}}+C$), Noether’s theorem guarantees a conserved current. For the shift symmetry, the Noether current takes the form:

$J^{\mu }=\alpha \,\nabla ^{\mu }S,$

so that classically $\nabla _{\mu }J^{\mu }=0$. We interpret $J^{\mu }$ as the local entropy flux. More generally, the existence of any symmetry of the entropy action implies a continuity equation for $J^{\mu }$ up to sources. In ToE, one shows that in local equilibrium the form of ${\mathcal {L}}$ and micro-reversibility imply a $Z_{2}$ 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 $Z_{2}$ symmetry leads to a local entropy current whose divergence is non-negative. Adapting their result, one finds in ToE that while $J^{\mu }$ 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 $\nabla _{\mu }J^{\mu }\geq 0$ 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 $\nabla _{\mu }J^{\mu }\geq 0$ 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 $S\to S+\mathrm {const}$, Noether’s theorem yields a conserved current:

Failed to parse (unknown function "\begin{equation}"): {\displaystyle \begin{equation} J_S^\mu = -\,\sqrt{-g}\;g^{\mu\nu}\,\nabla_{\nu}S. \end{equation} }

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

$\nabla _{\mu }J_{S}^{\mu }\;=\;-\,\nabla _{\mu }{\bigl (}{\sqrt {-g}}\,g^{\mu \nu }\nabla _{\nu }S{\bigr )}\;\geq \;0,$

thus realizing the local second‐law inequality.

Derivation of the Standard Entropy Measures

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 $W$ equally accessible states, the ToE implies a uniform entropy field. The total entropy of a region containing $W$ microstates is $S_{B}=k_{B}\ln W$, in agreement with Boltzmann’s formula. Boltzmann entropy on Wikipedia

Gibbs/Shannon Entropy

For a statistical ensemble with probabilities $\{p_{i}\}$ over states or cells, extremizing the ToE action under the normalization constraint yields $S=-\,k_{B}\sum _{i}p_{i}\ln p_{i}$, 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 $\alpha$ in the entropy action (for instance through a nonstandard kinetic term or a weighted multiplier), ToE interpolates to Rényi’s definition: $H_{\alpha }={\frac {1}{1-\alpha }}\,\ln \sum _{i}p_{i}^{\alpha }$. This generalizes Shannon entropy ($\alpha \to 1$) and yields Hartley’s max-entropy ($\alpha =0$) or collision entropy ($\alpha =2$) 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 $q$-logarithm (or using a potential enforcing escort distributions). One finds $S_{q}={\frac {k_{B}}{q-1}}{\bigl (}1-\sum _{i}p_{i}^{q}{\bigr )}$, which for $q\to 1$ recovers the usual $-k_{B}\sum _{i}p_{i}\ln p_{i}$. The ToE thus includes Tsallis entropy as a special $q$-deformation. Tsallis entropy

von Neumann Entropy

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

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 ${\mathcal {S}}(x)$ may fluctuate quantum mechanically. ToE implies the existence of “entropy waves” or quantum excitations of ${\mathcal {S}}$. 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 ${\mathcal {S}}$ has a cosmological vacuum expectation value or potential $V({\mathcal {S}})$, 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

This article incorporates text from this source, which is in the public domain: A Concise Introduction to the Evolving Theory of Entropicity (ToE)