The Logical Argument for the Declaration of Entropy as a Dynamic Field in the Theory of Entropicity (ToE) - The Bold Conceptual Leap from Entropy as a Microstate to Entropy as a Universal Field
How Entropy Can Propagate and Be a Field with Dynamics in ToE
A Technically Precise and Conceptually Rigorous Explanation
In the history of physics, the Theory of Entropicity (ToE), as first formulated and further developed by John Onimisi Obidi, is the first framework to make this universal, ontological, and fully dynamical declaration that entropy itself is the fundamental field of nature, with its own variational principle, spectral operator, field equations, coupling laws, speed limit, and geometric consequences.
To see how entropy can propagate and behave dynamically in the Theory of Entropicity (ToE), we must distinguish between two very different meanings of entropy:
1. Old View (Thermodynamics): Entropy as a Summary Statistic
In classical thermodynamics:
entropy is a macroscopic number,
it does not exist point-by-point,
and it cannot propagate.
This is the 19th-century view, where entropy is:
not local,
not a field,
and not dynamical.
This is not the entropy of ToE.
This is why people never imagined entropy could be a field.
2. Modern View (Quantum, Information, Geometry): Entropy is Local, Geometric, and Operator-Valued
Over the last 70 years, physics has quietly moved toward a new understanding:
Entropy is not a single number; it is encoded in local structures:
quantum states,
density matrices,
modular operators,
entanglement spectra,
Fisher–Rao metric,
Fubini–Study metric,
Amari–Čencov connections.
Every one of these is local, varies in space and time, and carries information geometry.
This is why ToE can treat entropy as a field.
The key transition is this:
Entropy became geometric in modern quantum theory.
Once geometry enters, fields enter.
Once fields enter, dynamics enter.
3. The Missing Step: ToE Identifies the Entropy Geometry with the Physical Ontology
Here is the breakthrough of the Theory of Entropicity (ToE):
ToE says:
The local geometric object that encodes entropy IS the fundamental physical field.
In other words:
Entropy is not a number.
It is a local geometric object.
That object has gradients.
Those gradients induce flux.
Flux produces curvature.
Curvature produces motion.
This what gives us a field.
And once we have a field, then we must have:
propagation,
equations of motion,
continuity relations,
dynamical evolution.
This is exactly what the Obidi Actions provide in the Theory of Entropicity (ToE).
4. Field Dynamics Arise from the Obidi Actions
Theory of Entropicity (ToE) hence defines entropy dynamics through two variational principles:
A. Local Obidi Action (LOA)
This produces a local field equation for entropy.
It has:
propagation terms,
potential terms,
source terms,
nonlinear feedback,
coupling with matter and geometry.
This is the Master Entropic Equation, the analogue of Einstein’s field equation but for entropy, not geometry.
So, LOA gives:
wave-like propagation of entropy disturbances
diffusion-like smoothing
nonlinear self-interaction
coupling to matter
This is standard field behavior.
B. Spectral Obidi Action (SOA)
This second Obidi Action provides the global constraint, ensuring that the entropy field:
organizes itself coherently,
conserves consistency between equilibrium and deformation,
maintains spectral balance.
The SOA is the entropic analogue of:
Spectral Action (Connes),
Zeta-regularized quantum actions,
Heat-kernel expansions.
But in ToE, it is not optional — it is fundamental.
Because SOA is defined through:
the modular operator Δ(S),
its eigenvalue spectrum,
its trace–log functional,
the entropy field has not just local dynamics but global spectral consistency, exactly like quantum fields.
5. Why Entropy Can Propagate
Propagation requires:
locality,
gradients,
continuity equations,
causal constraints.
The Theory of Entropicity (ToE) provides all four:
(1) Entropy is a local field S(x)
Defined at every spacetime point.
(2) Entropy has gradients ∂S
These gradients generate:
curvature,
force-like effects,
entropic flow.
(3) Entropy obeys a continuity equation
The entropic current Jᵐ is defined by:
Jᵐ = η ∂ᵐ S
and obeys:
∇ᵐ Jᵐ = 0
This is the same structure as:
charge conservation,
mass conservation,
probability conservation.
(4) Entropy obeys a wave-type equation
The Master Entropic Equation contains the analogue of a d’Alembertian, giving entropy finite-speed propagation.
Thus, entropy cannot change everywhere at once — it must propagate.
This produces:
entropic cones
entropic speed limit
causal structure
retarded entropic potentials
Exactly analogous to wave propagation in Maxwell, Klein–Gordon, and General Relativity (GR).
6. Why Entropy Must Have a Finite Speed (the Entropic Speed Limit)
Because the Master Entropic Equation contains:
second-order derivatives,
continuity constraints,
finite evolution operators.
The structure forces a finite propagation speed.
In fact, the speed of light emerges as:
the maximum rate of entropic re-computation of reality.
Hence:
light is fast because entropy is fast.
Matter moves slower because it consumes entropic capacity.
7. Why Entropy Produces Geometry in ToE
The Theory of Entropicity (ToE) shows:
Geometry = The way entropy arranges itself optimally.
Curvature = Variations in ∂²S.
Motion = Flow along ∇S.
In this formulation:
geometry is not primitive,
gravity is not a force,
spacetime is entropic structure,
mass is “frozen” entropy.
This is why entropy must propagate and as a field:
If geometry is produced by S(x), and geometry can change,
then S(x) must change.If S(x) changes, it must change with dynamics.
If it has dynamics, then it must propagate.
And if it propagates, it must therefore be a field.
8. Closure
Thus, we have seen and have been able to show from all of the above that Entropy behaves as a field with dynamics in ToE because:
(1) Entropy is not a thermodynamic number — it is a local geometric entity.
Quantum information theory already proved this.
(2) Entropy gradients generate physical effects.
(3) The Obidi Actions give entropy a variational principle.
(4) Field equations follow inevitably from that principle.
(5) These field equations enforce propagation, causality, and finite-speed evolution.
(6) Geometry, matter, time, and motion all emerge from the dynamics of ToE’s S(x).
Thus, entropy behaves exactly like:
a scalar field (Klein–Gordon),
a potential field (gravity),
a geometric field (metric),
a statistical field (information geometry),
and a modular field (operator algebra),
all unified into one object.
This is why the Theory of Entropicity (ToE) is fundamentally a field theory of entropy, not a thermodynamic reinterpretation.
Sources — help
References
Obidi, John Onimisi. (12th November, 2025). On the Theory of Entropicity (ToE) and Ginestra Bianconi’s Gravity from Entropy: A Rigorous Derivation of Bianconi’s Results from the Entropic Obidi Actions of the Theory of Entropicity (ToE). Cambridge University. https//doi.org/10.33774/coe-2025-g7ztq
John Onimisi Obidi. (6th November, 2025). Comparative analysis between john onimisi obidi’s theory of entropicity (toe) and waldemar marek feldt’s feldt–higgs universal bridge (f–hub) theory. International Journal of Current Science Research and Review, 8(11), pp. 5642–5657, 19th November 2025. URL: https://doi.org/10.47191/ijcsrr/V8-i11–21
Obidi, John Onimisi. 2025. On the Conceptual and Mathematical Foundations of the Theory of Entropicity (ToE): An Alternative Path toward Quantum Gravity and the Unification of Physics. Cambridge University. Published October 17, 2025. https://doi.org/10.33774/coe-2025-1dsrv
Obidi, John Onimisi (17th October 2025). On the Conceptual and Mathematical Foundations of the Theory of Entropicity (ToE): An Alternative Path toward Quantum Gravity and the Unification of Physics. Figshare. https://doi.org/10.6084/m9.figshare.30337396.v2
Obidi, John Onimisi. 2025. A Simple Explanation of the Unifying Mathematical Architecture of the Theory of Entropicity (ToE): Crucial Elements of ToE as a Field Theory. Cambridge University. Published October 20, 2025. https://doi.org/10.33774/coe-2025-bpvf3
Obidi, John Onimisi (15 November 2025). The Theory of Entropicity (ToE) Goes Beyond Holographic Pseudo-Entropy: From Boundary Diagnostics to a Universal Entropic Field Theory. Figshare. https://doi.org/10.6084/m9.figshare.30627200.v1
Obidi, John Onimisi. Unified Field Architecture of Theory of Entropicity (ToE). Encyclopedia. Available online: https://encyclopedia.pub/entry/59276 (accessed on 19 November 2025).
Obidi, John Onimisi. (4 November, 2025). The Theory of Entropicity (ToE) Derives Einstein’s Relativistic Speed of Light © as a Function of the Entropic Field: ToE Applies Logical Entropic Concepts and Principles to Derive Einstein’s Second Postulate. Cambridge University. https://doi.org/10.33774/coe-2025-f5qw8-v2
Obidi, John Onimisi. (28 October, 2025). The Theory of Entropicity (ToE) Derives and Explains Mass Increase, Time Dilation and Length Contraction in Einstein’s Theory of Relativity (ToR): ToE Applies Logical Entropic Concepts and Principles to Verify Einstein’s Relativity. Cambridge University. https://doi.org/10.33774/coe-2025-6wrkm
HandWiki contributors, “Physics:Theory of Entropicity (ToE) Derives Einstein’s Special Relativity,” HandWiki, https://handwiki.org/wiki/index.php?title=Physics:Theory_of_Entropicity_(ToE)_Derives_Einstein%27s_Special_Relativity&oldid=3845936
Further Resources on the Theory of Entropicity (ToE):
Website: Theory of Entropicity ToE — https://theoryofentropicity.blogspot.com
LinkedIn: Theory of Entropicity ToE — https://www.linkedin.com/company/theory-of-entropicity-toe/about/?viewAsMember=true
Substack: Theory of Entropicity (ToE) — John Onimisi Obidi | Substack
Medium: Theory of Entropicity (ToE) — John Onimisi Obidi — Medium
SciProfiles: Theory of Entropicity (ToE) — John Onimisi Obidi | Author
Encyclopedia.pub: Theory of Entropicity (ToE) — John Onimisi Obidi | Author
HandWiki contributors, “Biography: John Onimisi Obidi,” HandWiki, https://handwiki.org/wiki/index.php?title=Biography:John_Onimisi_Obidi&oldid=2743427 (accessed October 31, 2025).
HandWiki Contributions: Theory of Entropicity (ToE) — John Onimisi Obidi | HandWiki
HandWiki Home: Theory of Entropicity (ToE) — John Onimisi Obidi | HandWiki
HandWiki Homepage-User Page: Theory of Entropicity (ToE) — John Onimisi Obidi | HandWiki
Academia: Theory of Entropicity (ToE) — John Onimisi Obidi | Academia
ResearchGate: Theory of Entropicity (ToE) — John Onimisi Obidi | ResearchGate
Figshare: Theory of Entropicity (ToE) — John Onimisi Obidi | Figshare
Authoria: Theory of Entropicity (ToE) — John Onimisi Obidi | Authorea
Social Science Research Network (SSRN): Theory of Entropicity (ToE) — John Onimisi Obidi | SSRN
Wikidata contributors, Biography: John Onimisi Obidi “Q136673971,” Wikidata, https://www.wikidata.org/w/index.php?title=Q136673971&oldid=2423782576 (accessed November 13, 2025).
IJCSRR: International Journal of Current Science Research and Review - Theory of Entropicity (ToE) - John Onimisi Obidi | IJCSRR
Cambridge University Open Engage (CoE): Collected Papers on the Theory of Entropicity (ToE)
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