De Broglie’s Dual‑Structure Action Principle and the Theory of Entropicity (ToE): From Hidden Thermodynamics to the Entropic Field — From De Broglie’s Profound Thermodynamic Insight to Obidi’s Entropic Architecture of ToE

De Broglie’s "Hidden Thermodynamics" and the Entropic Field: How a Forgotten Insight Anticipates the Theory of Entropicity (ToE)

Louis de Broglie^[1] is remembered in every physics textbook for one idea: wave–particle duality. Yet the most profound insight of his career came decades later, when he attempted something far more ambitious — a unification of mechanics, thermodynamics, and quantum theory under a single principle.

In his late work, de Broglie argued that the motion of a particle is not simply a geometric path in spacetime nor a probabilistic wave evolution. Instead, he believed it was the visible expression of a deeper thermodynamic process, which he called hidden thermodynamics.

This idea faded from mainstream physics, but it contains a conceptual seed that aligns strikingly with the modern Theory of Entropicity (ToE)^{[2] [3][4][5][6][7][8][9][10][11][12]}— a framework that was first formulated and further developed by John Onimisi Obidi,^[13] which elelevates entropy from a statistical descriptor to a fundamental physical field. When we revisit de Broglie’s late writings through the lens of ToE, a remarkable picture emerges: he was pointing toward the very entropic substrate that ToE formalizes with mathematical precision.

The Dual‑Structure Action Principle: De Broglie’s Attempt at Unification

In his momentous work, Thermodynamics of the Isolated Particle (1964), de Broglie proposed that a particle’s natural trajectory is determined by two simultaneous extremal principles:

1. the principle of least action, the foundation of classical and relativistic mechanics
2. the principle of maximum entropy, the foundation of thermodynamics

He argued that every particle is embedded in a thermodynamic environment — a conceptual “thermostat” — that guides its motion. In this view, dynamics is a special case of thermodynamics, and quantum behavior reflects a hidden entropic process.

But de Broglie's momentous framework lacked the mathematical substrate to support this idea. He could not explain why minimizing action and maximizing entropy should be equivalent. He had the intuition, but not the field‑theoretic machinery.

The Theory of Entropicity (ToE) provides exactly what he was missing.

Entropy as a Field: The Core of the Theory of Entropicity (ToE)

The Theory of Entropicity (ToE) begins with a conceptual inversion: entropy is not derived — it is fundamental. It is represented as a field S(x) defined over a manifold that underlies what we perceive as spacetime. This field has:

1. curvature
2. propagation limits
3. a variational structure
4. governing field equations

These properties are encoded in the Obidi Action, whose extremization yields the Obidi Field Equations (OFE). In this framework, entropy is not something that results from physical processes — it is the entity that determines which processes are possible.

This reinterpretation transforms the foundations of physics:

1. Time becomes the irreversible flux of the entropic field.
2. Gravity becomes the curvature of that field.
3. Mass becomes entropic resistance to reconfiguration.
4. Motion becomes entropic reconfiguration.
5. Quantum probabilities become entropic accessibility.
6. The speed of light (c) becomes the maximum rate at which the entropic field can update its state.

Once entropy is treated as a field, the duality de Broglie observed becomes a structural necessity: action is the geometric encoding of entropic flow, and entropy is the thermodynamic encoding of the same underlying field.

From Hidden Thermodynamics to Explicit Entropic Geometry

De Broglie’s “hidden thermostat” becomes, in ToE, the universal entropic field. What he treated as a conceptual metaphor becomes a mathematically defined physical entity.

In the entropic framework:

1. the wavefunction corresponds to entropic accessibility
2. Born probabilities arise from entropic weighting
3. collapse is an entropic synchronization event
4. motion is the reconfiguration of the entropic field
5. mass is the resistance of the field to reconfiguration
6. time is the irreversible evolution of entropy

De Broglie’s hidden thermodynamics is no longer hidden — it becomes explicit entropic geometry.

Jaynes, Tsallis, and the Expansion of Entropy: A Natural Fit Within ToE

The twentieth century saw major generalizations of entropy:

1. Jaynes reframed entropy as a universal principle of inference
2. Tsallis introduced a nonadditive entropy for complex systems

These developments broadened entropy beyond heat engines and equilibrium physics.

The Theory of Entropicity (ToE) incorporates these frameworks seamlessly:

1. Jaynes’ entropy becomes a special case of entropic field configuration
2. Tsallis’ entropy becomes a special case of nonlinear entropic curvature
3. information theory becomes a projection of the entropic field onto discrete states

The Theory of Entropicity (ToE) thus provides the field‑theoretic foundation that unifies classical thermodynamics, information theory, and generalized entropy formalisms.

The Obidi Action: Why Least Action Equals Maximum Entropy

De Broglie discovered that a particle’s natural path is both the path of least action and the path of maximum entropy. What his monumental framework lacked was a mechanism explaining why these two principles coincide.

The Obidi Action provides this mechanism.

Its extremization yields the Master Entropic Equation (MEE) and the Obidi Field Equations (OFE), which encode the curvature and flow of the entropic field (EF). Minimizing the Obidi Action corresponds to selecting trajectories (via the ToE potent Vuli-Ndlela Integral) that optimize the efficiency of entropic flow.

Because entropy production and entropic flux are built into the structure of the action, the path of least action is simultaneously the path that maximizes the appropriate entropic functional.

The duality is no longer mysterious — it is a direct consequence of the entropic substrate.

The Theory of Entropicity (ToE) as the Completion of De Broglie’s Vision

De Broglie sought:

1. a causal interpretation of quantum mechanics
2. a thermodynamic foundation for dynamics
3. a unification of action and entropy
4. a deeper principle underlying mechanics

The Theory of Entropicity (ToE) provides all of these.

It [ToE] offers:

1. a field‑theoretic entropic substrate
2. a universal variational principle in the Obidi Action
3. governing equations (the Obidi Field Equations—OFE) from which motion, time, mass, gravity, and quantum behavior emerge

• Where de Broglie saw a duality, ToE sees a single field.
• Where de Broglie saw hidden thermodynamics, ToE sees explicit entropic geometry.
• Where de Broglie sought a synthesis, ToE provides a full unification.

The Theory of Entropicity (ToE) does not replace de Broglie’s dual‑structure action principle — it fulfills it. It provides the mathematical and ontological foundation that his intuition required.

In this sense, ToE is not merely a new theoretical framework. It is the realization of a historical vision — the completion of a conceptual arc that began with de Broglie’s hidden thermodynamics and culminates in the entropic field as the fundamental substrate of the universe.

Reference(s) — 1

1. On the Conceptual Foundations of the Theory of Entropicity (ToE): ToE-Google: ToE-Google Resources on the Theory of Entropicity (ToE) — Placeholder — Theory of Entropicity: https://entropicity.github.io/Theory-of-Entropicity-ToE/concepts/index1.html

References — 2

1. Grokipedia — Theory of Entropicity (ToE): https://grokipedia.com/page/Theory_of_Entropicity
2. Grokipedia — John Onimisi Obidi: https://grokipedia.com/page/John_Onimisi_Obidi
3. Google Blogger — Live Website on the Theory of Entropicity (ToE): https://theoryofentropicity.blogspot.com
4. GitHub Wiki on the Theory of Entropicity (ToE): https://github.com/Entropicity/Theory-of-Entropicity-ToE/wiki
5. Canonical Archive of the Theory of Entropicity (ToE): https://entropicity.github.io/Theory-of-Entropicity-ToE/
6. LinkedIn — Theory of Entropicity (ToE): https://www.linkedin.com/company/theory-of-entropicity-toe/about/?viewAsMember=true
7. Medium — Theory of Entropicity (ToE): https://medium.com/@jonimisiobidi
8. Substack — Theory of Entropicity (ToE): https://johnobidi.substack.com/
9. Figshare — Theory of Entropicity (ToE):https://figshare.com/authors/John_Onimisi_Obidi/20850605
10. Encyclopedia — SciProfiles — Theory of Entropicity (ToE): https://sciprofiles.com/profile/4143819
11. HandWiki — Theory of Entropicity (ToE): https://handwiki.org/wiki/User:PHJOB7
12. John Onimisi Obidi. Theory of Entropicity (ToE): Path to Unification of Physics and the Laws of Nature: https://encyclopedia.pub/entry/59188