In the present work we present a didactic reconstruction of de Broglie’s demonstration of the equivalence between two fundamental physical concepts: action minimization and entropy maximization. The demonstration was first developed in de Broglie’s Thermodynamics of the Isolated Particle (an attempt to develop a causal interpretation of Quantum Theory in the 1960s). In this work, de Broglie argues from a particle in a hypothetical sub-quantum medium, which operates as a kind of hidden thermostat exchanging energy with the particle. In the discussion about the action minimization and entropy maximization, however, there is not any reference to the quantum nature of the system. So, it is a demonstration valid for any system in a thermal bath in some usual thermostat. Louis de Broglie concludes that the “natural” trajectory of a particle is not only the one that minimizes the action of the particle but also maximizes the entropy of the thermostat. Our objective is to reconstruct and restructure de Broglie’s original demonstration in order to rescue an historical debate, that has not been addressed in the literature, and that is of interest to contemporary fields such as non-equilibrium thermodynamics.
In the present work we present a didactic reconstruction of de Broglie’s demonstration of the equivalence between two fundamental physical concepts: action minimization and entropy maximization. The demonstration was first developed in de Broglie’s Thermodynamics of the Isolated Particle (an attempt to develop a causal interpretation of Quantum Theory in the 1960s). In this work, de Broglie argues from a particle in a hypothetical sub-quantum medium, which operates as a kind of hidden thermostat exchanging energy with the particle. In the discussion about the action minimization and entropy maximization, however, there is not any reference to the quantum nature of the system. So, it is a demonstration valid for any system in a thermal bath in some usual thermostat. Louis de Broglie concludes that the “natural” trajectory of a particle is not only the one that minimizes the action of the particle but also maximizes the entropy of the thermostat. Our objective is to reconstruct and restructure de Broglie’s original demonstration in order to rescue an historical debate, that has not been addressed in the literature, and that is of interest to contemporary fields such as non-equilibrium thermodynamics.
De Broglie’s Hidden Thermodynamics and the Entropic Field: How a Forgotten Insight Anticipates the Theory of Entropicity (ToE) Louis de Broglie 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.
From 1948 until around 1965, Louis de Broglie, awarded the Nobel Prize for Physics in 1929 for his fundamental contributions to quantum theory, pursued a systematic study of the formal analogies between wave mechanics and the thermomechanics of Boltzmann and Helmholtz. As part of this line of research, he produced several interesting observations, which were, however, published only in French, and, therefore, had a very limited diffusion. Here, we reconsider, in particular, a result of his relating to the analogy between the internal clock (de Broglie phase) of a free particle and a cyclic isothermal process in a thermomechanical system. We show that the fundamental equivalence obtained by him can be derived under more convenient hypotheses than the original ones, essentially tied to the quantization of the action exchanged by the particle with a suitable thermostat. In this emended formulation, the relations proposed by de Broglie describe the emergence of the particle proper time from a thermal background. They also suggest a specific physical meaning of the Wick rotation, often used in quantum mechanical calculations, and the thermal time that appears in it.
Abstract The problem of time is a notable obstacle towards the recognition of quantum theory as the ultimate fundamental description of nature. Quantum theory may not be complete if founded upon classical notions. Louis de Broglie, seeming to be more or less convinced about the ontology of his proposed matter waves, tried to develop a theory of sub-quantum degrees of freedom relying on statistical thermodynamics. He realized a quantum particle as a fluctuating dense corpuscle formed via non-linear effects from a sub-quantum medium. A wave on the medium guides the vibrating corpuscle. He argued that an intrinsic clock of a quantum particle is related to its Brownian motion at the sub-quantum level. This led him to conjecture a relation between the de Broglie clock frequency mc 2 / h and its implicit temperature, which equals that of the surrounding sub-quantum medium. About the same time, Mandelbrot was the first to derive in a classical setting a thermodynamic uncertainty relation between energy and temperature, that was, coincidentally or not, anticipated by Bohr and Heisenberg in the first years of development of quantum theory. We show here that, when the de Broglie temperature–time conjecture is assumed, the thermodynamic temperature–energy uncertainty relation leads to the quantum time–energy uncertainty relation.
How the Theory of Entropicity (ToE) Finalizes the Vision of de Broglie's Hidden Thermodynamics in His "Thermodynamics of the Isolated Particle", and Provides the Fundamental Origin for the "Arrow of Time" The Theory of Entropicity (ToE), first formulated and further developed by John Onimisi Obidi around 2025, functions as a modern and radical extension of Louis de Broglie’s hidden thermodynamics of the isolated particle. It proposes to finalize this vision by elevating entropy from a statistical, passive concept to a fundamental, active field that acts as the causal substrate for motion, gravity, spacetime, and quantum mechanics. In doing so, the Theory of Entropicity provides the structural and ontological framework that de Broglie lacked, transforming his intuitive thermodynamic insights into a unified, field‑theoretic foundation for physical law.
This paper provides an examination of the de Broglie relation, tracing its historical development from the quantum hypotheses proposed by Planck and Einstein to its covariant relativistic derivation. The discussion begins by situating de Broglie's seminal insight within the early framework of quantum theory. We then reconstruct his original heuristic derivation. The primary focus of this work, however, is the derivation of the de Broglie relation directly from the principles of special relativity, employing the four-momentum formalism. A comparative analysis between the heuristic and relativistic approaches underscores the universality and conceptual coherence of the latter. The paper concludes by highlighting the significance of relativistic mechanics in establishing a consistent foundation for wave-particle duality, further reinforcing this through a quantum field theoretical perspective.
The principle of least action, which has so successfully been applied to diverse fields of physics looks back at three centuries of philosophical and mathematical discussions and controversies. They could not explain why nature is applying the principle and why scalar energy quantities succeed in describing dynamic motion. When the least action integral is subdivided into infinitesimal small sections each one has to maintain the ability to minimize. This however has the mathematical consequence that the Lagrange function at a given point of the trajectory, the dynamic, available energy generating motion, must itself have a fundamental property to minimize. Since a scalar quantity, a pure number, cannot do that, energy must fundamentally be dynamic and time oriented for a consistent understanding. It must have vectorial properties in aiming at a decrease of free energy per state (which would also allow derivation of the second law of thermodynamics). Present physics is ignoring that and applying variation calculus as a formal mathematical tool to impose a minimization of scalar assumed energy quantities for obtaining dynamic motion. When, however, the dynamic property of energy is taken seriously it is fundamental and has also to be applied to quantum processes. A consequence is that particle and wave are not equivalent, but the wave (distributed energy) follows from the first (concentrated energy). Information, provided from the beginning, an information self-image of matter, is additionally needed to recreate the particle from the wave, shaping a “dynamic” particle-wave duality. It is shown that this new concept of a “dynamic” quantum state rationally explains quantization, the double slit experiment and quantum correlation, which has not been possible before. Some more general considerations on the link between quantum processes, gravitation and cosmological phenomena are also advanced.