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 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.
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.
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.
Louis De Broglie is one of the principal architects of quantum theory. It is to him that we owe the idea of the particle as having some sort of wavelike functionality and the wave particle duality. Despite having invented this idea, de Broglie had his doubts and spent much of the rest of his life trying in vain to justify his assertion for which he had won the Nobel Prize. Essentially de Broglie’s notion of the particle as a wave is a restatement of the assumption adopted by Niels Bohr in his model for the hydrogen atom, notably that angular momentum is quantised. De Broglie does however provide us with two useful insights: that the energy levels of the atom are in some way associated with a harmonic sequence and that there is some sort of frequency multiplication process taking place within the atom.
Prologue A memorial conference for John Bell, open to the public, was held at Rutgers University around 20 years ago. I gave a short talk there on Bell's views about David Bohm's “hidden variables” formulation of quantum mechanics, a version of quantum mechanics often called the de Broglie–Bohm theory or Bohmian mechanics. This theory was in fact discovered by Louis de Broglie in 1927. In 1952, it was rediscovered and developed by Bohm, who was the first to appreciate its connection to the predictions of standard quantum mechanics. I did not publish the talk. I have decided that it would be appropriate to publish a very lightly edited version of it here, in this volume devoted to the work of Bell on the foundation of quantum mechanics. One reason for doing so is that the connection between Bell and Bohm continues to be somewhat underplayed, with the strength of his advocacy of Bohmian mechanics not properly appreciated. For example, about half of the papers in Bell's collected works on the foundations of quantum mechanics deal with Bohmian mechanics. But in his fine introduction to the revised edition of this great book [1], Alain Aspect mentions this theory in only a single sentence, and parenthetically at that. For several decades after Bell proved his nonlocality theorem, based in part on Bell's inequality, it was widely claimed that Bell had shown that hidden variables – and Bohmian mechanics in particular – were impossible, that they were incompatible with the predictions of quantum mechanics. For example, the great physicist Eugene Wigner, who, unlike most of his contemporaries, was profoundly concerned with the conceptual foundations of quantum mechanics and usually wrote on the subject with great clarity and insight, has written that This [hidden variables] is an interesting idea and even though few of us were ready to accept it, it must be admitted that the truly telling argument against it was produced as late as 1965, by J. S. Bell … This appears to give a convincing argument against the hidden variables theory.
Louis de Broglie put forward the idea of wave-particle duality in 1923-24, less than two years before the advent of Quantum Mechanics. This last theory assumes that duality, but in a more abstract and counter-intuitive manner than de Broglie did. In this article the wave behavior of matter is presented in a didactic way following de Broglie´s ideas. This kind of presentation may (hopefully) be helpful to better understand the quantum features in a more intuitive way. It is shown that this wave mechanics, in the framework of the theory of relativity, is compatible with Planck´s quantum and explains as well the particle´s dynamics, provided the uncertainty relations are properly incorporated. The main difficulty with this view lies in its interpreting the wave as a non-local entity
Louis de Broglie in the beginning 20th century voices his theory of a double solution, according to which a pilot wave accompanies a particle, simulated as a point singularity, along the most optimal path from its creation on a source up to the detection. The pilot wave is a real hidden wave, which is similar to the wave function resulting from the solution of the Schrödinger equation. This theory is in agreement with de Broglie’s postulate about the matter waves. In this article we are based on the Helmholtz decomposition theorem according to which any velocity may be represented as a sum of two velocities – irrotational and solenoidal ones. The first velocity stems from the gradient of the scalar field. The second occurs from a pseudo-vector field. We proclaim that the gradients of the scalar field define guiding paths of the pilot wave. While the pseudo-vector field defines a particle solenoidal filling. We give mathematical models of the irrotational and solenoidal flows simulating the position of a particle in a guiding wave. Modified Navier-Stokes equation in pair with the continuity equation resulting in the Schrödinger equation gives such solutions consisting of superposition of the irrotational and solenoidal flow. It is declared that the guiding wave forms from the irrotational flows. In turn, the solenoidal flows underlie forming particles that travel along optimal paths slave by the guiding wave. There are described mathematical solutions of the spherical particle washed by the superfluid representing the guiding wave along which it travels.
Wave-particle duality, the revolutionary concept that particles can exhibit both wave-like and particle-like properties, was first proposed by Louis de Broglie. His equations connecting a particle's energy and momentum to the properties of its corresponding wave laid the groundwork for quantum mechanics. This fundamental principle explains the surprising results of the double-slit experiment and underlies many of quantum mechanics' unique behaviors. In this work, relying solely on Einstein's two postulates of special relativity, and the idea that time is single valued, we will demonstrate that a complete theory of relativity predicts a wave-like quantizable behavior for particle trajectories. We will prove that the theory of relativity predicts a first-order, spin-independent transport equation as an equation of motion for particles, in addition to the classical relativistic Hamiltonian. As key results, we will demonstrate that our theory provides an integrable solution for a free scalar particle, offering insights into the wave-particle duality observed in nature. For spinor particles, we will show that our theory generalizes Weyl theory to accommodate massive particles, while also allowing for modified Dirac-like particles.