De Broglie’s dual extremal principle: action and entropy as a unified structure

In his work Thermodynamics of the Isolated Particle (1964), de Broglie argued that the trajectory of a particle is determined by the simultaneous enforcement of two extremal principles: the least action principle (in the Hamilton–Maupertuis sense) and the maximum entropy principle (in the Carnot–Boltzmann sense). He proposed that a particle is effectively coupled to a hidden thermostat, a thermodynamic environment that guides its motion. This was his attempt to construct a causal interpretation of quantum mechanics in which the wave–particle duality and the pilot wave are manifestations of an underlying thermodynamic process.

However, de Broglie’s formulation lacked a fully developed field‑theoretic structure capable of encoding this dual extremal behavior in a single variational framework. He possessed the conceptual insight that mechanics and thermodynamics are deeply linked, but he did not have a universal field substrate that could carry both action and entropy as different aspects of the same dynamical entity.

The Theory of Entropicity as the missing entropic field substrate

The Theory of Entropicity asserts that entropy is not merely a statistical descriptor but a genuine field, denoted by \( S(x) \), defined over a manifold that underlies what is ordinarily interpreted as spacetime. This field possesses its own curvature, propagation law, and variational structure. These properties are encoded in the Obidi Action, a fundamental variational functional governing the dynamics of the entropic field, and in the associated Obidi Field Equations (OFE), which describe how the entropic field evolves, redistributes, and constrains physical configurations.

Within this framework, the duality that de Broglie identified between action minimization and entropy maximization is reinterpreted as a direct consequence of the underlying entropic dynamics. In ToE, action is understood as the geometric expression of entropic flow, while entropy is the thermodynamic expression of the same field. The two extremal principles are therefore not independent; they are different projections of a single variational structure defined on the entropic field. Minimizing action and maximizing entropy become equivalent statements about the optimal evolution of \( S(x) \).

From de Broglie’s hidden thermodynamics to explicit entropic geometry

De Broglie’s notion of hidden thermodynamics was an attempt to interpret quantum mechanics as the visible surface of a deeper thermodynamic process. He suspected that the wavefunction, the pilot wave, and the particle trajectory were all emergent manifestations of an underlying thermodynamic mechanism. In ToE, this intuition is made explicit and geometrically precise by identifying the universal entropic field as the substrate from which quantum behavior emerges.

In the entropic framework, the wavefunction is reinterpreted as a measure of entropic accessibility of configurations in the entropic manifold. Quantum probabilities arise from entropic weighting of possible configurations, rather than from an abstract postulate. Wavefunction collapse is understood as an entropic synchronization event, in which the entropic field enforces a consistent configuration across interacting subsystems. Motion is described as entropic reconfiguration of the field, mass as entropic resistance to reconfiguration, and time as entropic flux, i.e., the irreversible evolution of the entropic field.

What de Broglie conceived as a hidden thermostat becomes, in ToE, a fully explicit and universal entropic geometry. The thermodynamic background is no longer hidden; it is the primary ontological layer from which both quantum and classical phenomena arise.

Integration of Jaynes, Tsallis, and generalized entropic frameworks into ToE

The broader development of entropy-based theories in the twentieth and twenty‑first centuries—such as Edwin T. Jaynes’ Maximum Entropy Principle and Constantino Tsallis’ nonadditive entropy—has shown that entropy is not confined to classical thermodynamics or heat engines. Instead, entropy functions as a universal measure of information, uncertainty, and configuration space structure. These approaches generalize the concept of entropy to non‑equilibrium, non‑extensive, and information‑theoretic contexts.

In the Theory of Entropicity, these generalized entropic frameworks are naturally subsumed into the dynamics of the entropic field. Jaynes’ entropy appears as a special case of an entropic field configuration when one projects the continuous entropic manifold onto discrete probability distributions. Tsallis’ nonadditive entropy can be interpreted as a manifestation of nonlinear entropic curvature, where the geometry of the entropic field deviates from simple additive structures. Information theory itself becomes a projection of the entropic field onto discrete or coarse‑grained state spaces, rather than a separate foundational layer.

In this way, ToE provides a field‑theoretic foundation that unifies classical thermodynamics, information theory, and generalized entropy formalisms within a single entropic substrate, consistent with and extending de Broglie’s original thermodynamic intuition.

The key reconciliation: de Broglie’s duality as a consequence of the Obidi Action

De Broglie’s central discovery can be summarized as the claim that a particle’s natural path is simultaneously the path of least action and the path of maximum entropy. What he lacked was a mechanism explaining why these two extremal principles should coincide. The Theory of Entropicity provides this mechanism through the Obidi Action.

In ToE, the Obidi Action is the fundamental variational functional governing the evolution of the entropic field \( S(x) \). The action is constructed so that its extremization yields the Master Entropic Equation and the Obidi Field Equations, which encode the entropic curvature and flow of the field. Minimizing the Obidi Action corresponds to selecting trajectories and configurations 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.

Thus, the equivalence between least action and maximum entropy is no longer an unexplained duality but a direct reflection of the fact that both principles arise from the same entropic substrate. The Obidi Action unifies them as two aspects of a single variational principle defined on the entropic field.

Completion of de Broglie’s program within the Theory of Entropicity

De Broglie’s long‑term program aimed at several interconnected goals: a causal interpretation of quantum mechanics, a thermodynamic foundation for dynamics, a unification of action and entropy, and the identification of a deeper principle underlying mechanics. The Theory of Entropicity addresses these aims by providing a field‑theoretic entropic substrate, a universal variational principle in the form of the Obidi Action, and a set of governing equations—the Obidi Field Equations—from which motion, time, mass, and quantum behavior emerge as entropic phenomena.

Where de Broglie perceived a duality between action and entropy, ToE identifies a single entropic field whose dynamics generate both. Where de Broglie spoke of hidden thermodynamics, ToE introduces an explicit entropic geometry that underlies spacetime and quantum state space. Where de Broglie sought a synthesis of mechanics and thermodynamics, ToE offers a full unification in which relativity, quantum mechanics, and thermodynamics are different manifestations of the same entropic substrate.

Conclusion: the Theory of Entropicity as the fulfillment of de Broglie’s vision

The Theory of Entropicity does not replace or negate de Broglie’s dual‑structure action principle. Instead, it provides the mathematical and ontological infrastructure that his intuition anticipated but could not fully realize. De Broglie recognized that entropy and action are two expressions of a deeper underlying reality. ToE identifies that reality as the entropic field, formalizes it through the Obidi Action, and derives its dynamics via the Obidi Field Equations.

In this sense, the Theory of Entropicity is not a departure from de Broglie’s program but its natural continuation and completion. It transforms his qualitative insight into a rigorous field theory in which the unity of action and entropy is no longer conjectural but structurally encoded in the fundamental architecture of physical law.