De Broglie’s Hidden Thermodynamics and the Entropic Field: How a Forgotten Insight in Theoretical Physics Anticipates the Theory of Entropicity (ToE)
In the standard historical narrative of quantum theory, Louis de Broglie is primarily associated with the concept of wave–particle duality, which played a decisive role in the early development of quantum mechanics. Less widely appreciated is that, in the later phase of his career, de Broglie pursued a far more ambitious program: the construction of a unified framework in which mechanics, thermodynamics, and quantum theory would emerge as different aspects of a single underlying principle. He proposed that the motion of a particle is not merely a geometric trajectory in spacetime, nor merely a probabilistic wave evolution, but the visible manifestation of a deeper thermodynamic process, which he termed hidden thermodynamics.
Although this line of thought has largely receded from mainstream discussions, it contains a conceptual core that aligns closely with the modern Theory of Entropicity (ToE). In ToE, entropy is not treated as a secondary, statistical descriptor but as a fundamental physical field. When de Broglie’s late work is re‑examined through this entropic lens, it becomes evident that he was gesturing toward the very notion that ToE formalizes: the existence of an entropic substrate underlying all physical phenomena. De Broglie intuited this substrate but did not possess the mathematical machinery to express it as a field theory. The Theory of Entropicity provides precisely that missing structure.
Action and entropy as two forms of a single extremal principle
In his 1964 monograph Thermodynamics of the Isolated Particle, de Broglie advanced a radical proposal: the natural trajectory of a particle is determined by the simultaneous enforcement of two extremal principles. The first is the familiar principle of least action, which underlies classical mechanics and relativistic dynamics. The second is the principle of maximum entropy, which underlies thermodynamics and the statistical mechanics of macroscopic systems.
De Broglie argued that a particle’s path is the one that minimizes the action functional while simultaneously maximizing the entropy of a surrounding thermodynamic environment, which he described as a thermostat. In this picture, every particle is embedded in a thermodynamic medium that constrains and guides its motion. This was his attempt to synthesize the Maupertuis–Hamilton variational principle of mechanics with the Carnot–Boltzmann entropy principle of thermodynamics. He regarded dynamics as a special, simplified branch of thermodynamics, and he interpreted quantum behavior as the manifestation of a hidden entropic process.
However, de Broglie’s construction lacked a fully developed field‑theoretic ontology. He could not derive, from first principles, why the minimization of action and the maximization of entropy should be equivalent or mutually compatible. The duality he identified remained at the level of a profound but ungrounded insight: he had the conceptual intuition, but not the underlying field structure that would make it inevitable.
The Theory of Entropicity: entropy as the fundamental field of reality
The Theory of Entropicity (ToE) begins from a decisive inversion of the conventional hierarchy: entropy is not a derived, statistical quantity emerging from microscopic dynamics; instead, it is a fundamental field that determines which dynamics are possible. This field, denoted by \( S(x) \), is defined over an underlying manifold that, at macroscopic scales, is perceived as spacetime. The field \( S(x) \) possesses its own curvature, propagation law, and variational structure.
The dynamics of the entropic field are encoded in the Obidi Action, a variational functional whose extremization yields the Obidi Field Equations (OFE). These equations govern the evolution, redistribution, and constraint structure of the entropic field. In this framework, entropy is not a byproduct of physical processes; it is the entity that constrains and generates them. The familiar physical quantities and phenomena emerge as manifestations of the entropic field:
Time is interpreted as the irreversible flux of the entropic field, reflecting the monotonic evolution of \( S(x) \) along physically admissible trajectories. Gravity is understood as the curvature of the entropic field, with gravitational attraction arising from entropy gradients and entropic geodesics. Mass is identified with entropic resistance, i.e., the reluctance of the entropic field to reconfigure in the presence of localized structures. Motion is described as entropic reconfiguration, the rearrangement of the field’s configuration in response to constraints and gradients. Quantum probabilities are reinterpreted as measures of entropic accessibility of different configurations. The speed of light \( c \) becomes the maximum rate at which the entropic field can update its state, a universal entropic propagation limit.
Once entropy is treated as a field in this sense, the duality that de Broglie observed between least action and maximum entropy is no longer an unexplained coincidence. It becomes a structural necessity: action is the geometric encoding of entropic flow, while entropy is the thermodynamic encoding of the same underlying field. Minimizing action and maximizing entropy are two mathematically distinct but physically equivalent ways of describing the optimal evolution of the entropic field \( S(x) \).
From hidden thermodynamics to explicit entropic geometry
In de Broglie’s formulation, the hidden thermostat is a conceptual thermodynamic environment that influences particle motion but is not explicitly modeled as a field. In the Theory of Entropicity, this notion is made explicit and geometrically precise: the hidden thermostat is identified with the universal entropic field itself. What was previously a metaphor becomes a mathematically defined physical entity.
Within ToE, the wavefunction is reinterpreted as a representation of entropic accessibility across configuration space. The Born probabilities arise from entropic weighting of possible configurations, rather than from an axiomatic probability postulate. Wavefunction collapse is understood as an entropic synchronization event, in which the entropic field enforces a consistent configuration across interacting subsystems. Particle motion is the visible manifestation of reconfiguration of the entropic field. Mass corresponds to the resistance of the entropic field to reconfiguration in localized regions. Time is the irreversible flow of entropy, encoded in the monotonic evolution of \( S(x) \).
In this way, de Broglie’s hidden thermodynamics is no longer hidden. It is elevated to the status of an explicit entropic geometry that underlies both spacetime and quantum state space. The thermodynamic background that de Broglie postulated becomes the primary ontological layer in ToE.
Jaynes, Tsallis, and the expansion of entropy within the entropic field framework
De Broglie’s ideas emerged in a broader intellectual context in which the concept of entropy was being generalized beyond classical thermodynamics. Edwin T. Jaynes reformulated entropy as a universal principle of inference and information through the Maximum Entropy Principle, while Constantino Tsallis introduced a nonadditive entropy suitable for complex, non‑extensive systems. These developments demonstrated that entropy is not confined to heat engines or equilibrium states but is a general measure of uncertainty, information content, and configuration structure.
The Theory of Entropicity integrates these generalized entropic frameworks by embedding them into the dynamics of the entropic field \( S(x) \). Jaynes’ entropy appears as a special case of an entropic field configuration when the continuous entropic manifold is projected 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 and encodes long‑range correlations or complex interactions. Information theory itself becomes a projection of the entropic field onto discrete or coarse‑grained state spaces, rather than an independent foundational layer.
In this sense, ToE provides a field‑theoretic foundation that unifies classical thermodynamics, information theory, and generalized entropy formalisms. The entropic field serves as the common substrate from which these various entropy concepts arise as different approximations or projections.
The Theory of Entropicity as the completion of de Broglie’s program
De Broglie’s long‑standing objectives can be summarized as follows: to obtain a causal interpretation of quantum mechanics, to establish a thermodynamic foundation for dynamics, to unify action and entropy within a single principle, and to identify a deeper underlying structure beneath mechanical laws. The Theory of Entropicity addresses these aims by positing a field‑theoretic entropic substrate, formulating a universal variational principle in the form of the Obidi Action, and deriving Obidi Field Equations that govern the evolution of the entropic field and generate observable phenomena.
In ToE, motion, time, mass, gravity, and quantum behavior are all interpreted as emergent properties of the entropic field \( S(x) \). Where de Broglie perceived a duality between action and entropy, ToE identifies a single entropic field whose dynamics give rise to both. Where de Broglie invoked 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 entropic field as realization of a historical vision
The Theory of Entropicity (ToE) does not displace or negate de Broglie’s dual‑structure action principle. Instead, it provides the mathematical and ontological infrastructure that his intuition anticipated. De Broglie sensed that entropy and action were two expressions of a single 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 (OFE).
In this sense, the Theory of Entropicity is not merely a new theoretical proposal; it is the realization of a historical conceptual arc that began with de Broglie’s hidden thermodynamics and culminates in the recognition of the entropic field as the fundamental substrate of the universe. What was once an obscure and largely forgotten insight becomes, in ToE, a central organizing principle for a unified description of physical reality.