The Fractal Link: Did a Lab Experiment Just Unify Turbulence and Quantum Mechanics?

How Laurent Nottale and Thierry Lehner solved the "Closure Problem" of fluid dynamics and revealed macroscopic quantum states in a simple air jet.

For a century, physics has been divided. On one side, we have Quantum Mechanics, describing the probabilistic behavior of the very small. On the other, Classical Fluid Dynamics, describing the chaotic flow of the very large. But a groundbreaking series of papers suggests these two worlds are connected by a single geometric concept: Fractality.

Solving the "Closure Problem"
Turbulence is often called the last unsolved problem of classical physics because of the "Closure Problem." When engineers try to average the Navier-Stokes equations to smooth out chaos, the math bites back: you end up with more unknowns than equations. You cannot solve the system without guessing or adding empirical "fudge factors."

Gemini representation: Please excuse the crudity of this model. I didn’t have time to build it to scale or paint it.

Nottale theory of Scale Relativity solves this. They propose that turbulent trajectories are fractal (non-differentiable). When you apply Newton’s laws to fractal paths, the equations spontaneously integrate into a macroscopic Schrödinger equation. This provides a direct way to calculate the probability of fluid motion, effectively "closing the loop" without guesswork.

The Macroscopic Planck Constant
How can a quantum equation describe a massive fluid jet? It comes down to dimensional equivalence.

• In quantum mechanics, ℏ (Planck’s constant) dictates the "fuzziness" of an electron's position.

• In this new theory, ℏ is replaced by a macroscopic diffusion constant, D. While the numbers are vastly different in size, they play the exact same mathematical role. D tells the fluid how "fuzzy" or fractal its trajectory is, allowing quantum-like behavior to emerge at the human scale.

Deriving the "Magic Numbers" of Physics
The true power of a theory lies in prediction. For decades, engineers have relied on empirical constants—numbers that we know exist but don't know why. Nottale and Lehner derived these from first principles:

• The von Karman Constant (κ): A dimensionless number dictating flow speed near walls. Long measured as ≈ 0.40, the theory predicts it exactly.

• The Velocity Correlation (ρ): A measure of how chaotic velocity in one direction affects another. The theory derives the universal value of ≈0.4 observed in shear flows.

• The Jet Opening Angle: Why do turbulent jets always spread at an angle of ≈11.5∘? The theory calculates this geometric constant (≈ 0.2 radians) based on the energy levels of a quantum harmonic oscillator.

The "Smoking Gun": A Hole in the Jet
Theory is cheap; proof is gold. In their "PROJET" experiment, the authors didn’t just do the math—they built a box.

A standard free jet behaves like a quantum particle in a "soft" potential—it settles into the Ground State, forming a bell-curve profile with maximum concentration in the center.

But the authors predicted that if they confined the jet inside a physical cone that matched the jet's natural opening angle perfectly, they could create an "infinite potential well." This should force the jet into a First Excited State. The math predicted that this state must have a node in the center.

They turned on the fog machine, and the prediction held: A stable, hollow jet. Instead of a central peak, there was a hole—zero concentration—running down the centerline. It was a macroscopic quantum state, visible to the naked eye.

Nobel-Worthy Physics?
The Nobel Prize is rarely awarded for pure theory; it rewards discovery that changes our understanding of the world and has practical application.

This work suggests that fractal geometry is the missing link between the classical and quantum worlds. It offers a new way to engineer flow—potentially revolutionizing how we design rocket nozzles, pipelines, and climate models. If these results are replicated, confirming that we can manipulate matter into quantized states simply by tuning geometry, we are looking at a discovery that deserves the highest recognition in science.

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Primary References:

• Macroscopic quantum-like behavior of a turbulent jet (Nottale & Lehner)

• The Turbulent Jet in the Scale-Relativity Framework (Nottale & Lehner)

• The Boundary Layer in the Scale-Relativity Theory of Turbulence (Nottale & Lehner)

Why Do Solar Systems Rotate on a Single Plane?

Why are we able to witness Planetary Alignments?

From the rings of Saturn to the grand spiral of the Andromeda galaxy, the universe seems to have a deep affinity for flatness. Our own solar system is the quintessential example: eight planets, countless asteroids, all orbiting the Sun in the same direction and on nearly the same plane—the ecliptic.

But why? At the birth of the solar system, in the primordial chaos of a giant molecular cloud, there was no "up" or "down," no preferred axis of rotation. How did this perfect chaos collapse into such a beautifully ordered, planar structure? The classical explanation is a story of physics at its most elegant. But does it tell the whole story? Or does the incredible stability of this structure hint at a deeper, quantum-like law governing the cosmos?

The five planets appeared in 2016 in a magic line from low in the East to high in the North.

The Classical Explanation: The Inevitable Emergence of Order

The standard model explains the formation of the ecliptic plane as the inevitable result of two fundamental laws acting on a collapsing cloud of gas and dust.

1. The "Imperfect" Chaos and the Birth of Spin:
The process begins with a vast, chaotic cloud. While the motions of its countless particles seem random, they never perfectly cancel out. Due to statistical probability within a finite system and the gentle gravitational twisting from nearby stars and the galaxy itself (tidal torques), the cloud always possesses a tiny, residual net spin—an overall angular momentum. This initial asymmetry, however infinitesimal, is the seed of all future order.

2. The Conservation of Angular Momentum: The Ice Skater Effect:
As gravity pulls the cloud inward, it begins to collapse. This is where the magic happens. Just as an ice skater spins faster by pulling her arms in, the collapsing cloud must spin faster and faster to conserve its initial angular momentum. The microscopic spin of the primordial cloud is amplified into a furious, macroscopic rotation.

3. The Battle of Forces: Gravity vs. Centrifugal Force:
This rapid rotation creates a powerful centrifugal force that pushes matter outward, but only in the plane perpendicular to the axis of spin. Gravity, on the other hand, pulls matter inward from all directions.

• Along the poles of the rotation axis, gravity wins unopposed, and material collapses onto the forming star.

• At the equator, the outward centrifugal force balances the inward pull of gravity, preventing material from falling in. It is forced to spread out into an orbiting disk.

The result, much like a spinning ball of pizza dough, is a flat, rapidly rotating protoplanetary disk. All the material within it inherits the same angular momentum, which is why the planets that form from this disk will all orbit on that same plane and in the same direction. This classical explanation is powerful and accounts for the basic structure we observe.

The Puzzles of Perfect Stability: Does the Classical Story Suffice?

The classical model is beautiful, but it leaves behind nagging questions, particularly concerning the extraordinary long-term stability of this arrangement.

Why One, and Only One, Plane?
The classical model explains that a single axis of rotation emerges as the system settles into its lowest energy state, dissipating any chaotic "wobble" through internal friction. But is this process sufficient to explain the near-perfect coplanarity we see over billions of years? What prevents external perturbations from re-introducing a wobble? What locks the system in with such rigidity?

The Problem of Intruders:
Our galaxy is a dynamic place. Stars, rogue planets, and interstellar clouds pass by. Why doesn't an external gravitational influence from a passing object disrupt this delicate arrangement? The classical model suggests stability, but it's a fragile stability. It feels as though something more fundamental is at play, actively maintaining this order.

The stability of the solar system feels less like a ball resting precariously at the bottom of a shallow bowl and more like an electron locked into its atomic orbital—a state it occupies by fundamental right, not by accident.

A Deeper Foundation: The Quantum Leap to Scale Relativity

This is where a more radical theory, Scale Relativity (SR), offers a compelling, albeit mind-bending, solution. As proposed by Laurent Nottale, this theory extends the principles of relativity to the scale of observation itself, postulating that spacetime is not smooth but fundamentally fractal.

In a seminal 1997 paper, "Scale relativity and quantization of the solar system" (Nottale, Schumacher, & Gay), the authors laid out the stunning consequences. In an SR framework, the equations governing planetary motion become Schrödinger-like. This doesn't mean planets are "fuzzy" like electrons, but that their possible trajectories are governed by probability densities.

1. Quantized Orbits as a Foundation for Stability:
SR predicts that planets cannot orbit at any arbitrary distance. Instead, they must occupy quantized orbits—specific, stable paths dictated by the structure of spacetime itself. The formula they provide, an = (GM/w₀²)n², where w₀ is a fundamental constant, remarkably matches the observed orbits in our solar system and has successfully predicted the existence of exoplanets in close-in orbits.

2. The Ecliptic as a "Fundamental State":
The theory also quantizes angular momentum, predicting that for a given orbital distance (principal quantum number n), the most probable state—the lowest energy state—is the one with the highest angular momentum, which corresponds to a nearly circular orbit. This directly explains why planets tend to be on circular, non-chaotic paths.

We can extrapolate this principle. If the positions of planets are quantized, perhaps their planes are as well. The ecliptic plane would not be just an accidental outcome of a collapsing cloud; it would be the system's "ground state"—the most probable, lowest-energy configuration for the entire system's angular momentum vector.

3. The Quantum "Immunity" to Perturbation:
This perspective fundamentally changes our view of stability. In the classical view, a perturbation could knock the system out of alignment. In a quantum-like view, the system is already in its most probable state. A small perturbation is not enough to make it "jump" to another, less probable state. The system would be inherently resilient, its stability actively enforced by the underlying laws of scale invariance. An intruder planet wouldn't just disrupt a fragile balance; it would have to fight against the very structure of spacetime.

Conclusion: An Improved, More Resilient Vision

The classical explanation for the ecliptic plane is not wrong. It beautifully describes the physical mechanism of formation. But Scale Relativity offers a deeper foundation, a reason why the outcome of that mechanism is so incredibly stable and ordered.

By adding a quantum-like layer of understanding, we move from a vision of accidental stability to one of fundamental, resilient order. The solar system isn't just a lucky survivor of a chaotic past; it is a macroscopic expression of the same quantization principles that govern the atom, locked into a stable configuration by the very fabric of the cosmos.

Hot Jupiters: Not So Jupiter After All: A Deeper Cosmic Order

The Impossible Planet and the Dawn of a New Physics: How Exoplanets Are Vindicating a Forgotten Theory

In 1995, astronomers found a monster: a gas giant orbiting its star in just four days. According to established theories, 51 Pegasi b should not exist. To explain this "impossible planet," a complex and chaotic story was born: planetary migration. Yet, a quieter, more elegant theory had already predicted such worlds—a theory that the latest astronomical evidence is now inadvertently proving right.

This isn't just a story about planets; it's about a potential revolution in our understanding of cosmic structure, moving from a paradigm of chaos to one of profound, predictable order.

The Standard Model: An Elaborate Story Facing Contradictory Evidence

The standard explanation for a "Hot Jupiter" is that it’s a cosmic nomad. Formed in the cold outer regions of its system, it was flung inward by gravitational chaos or dragged by the primordial gas disk, finally settling into a scorching orbit. This chaotic model became the default explanation for the bewildering diversity of exoplanetary systems.

But this narrative is cracking under the weight of new discoveries. Two key findings, in particular, refuse to fit the chaotic story:

1. The Survival of Fragile Neighbors: Astronomers are finding an increasing number of Hot Jupiters with small, adjacent planetary companions. As highlighted in a recent paper from the University of Wisconsin-Madison (Mathur & Becker, Publications of the Astronomical Society of the Pacific, 2025), these tiny worlds should have been obliterated by the passage of a migrating giant. Their mere existence suggests a much calmer formation history—a process known as in situ formation, where the planets assembled right where we see them.

2. The Mystery of Extreme "Metallicity": Hot Jupiters are bizarrely over-polluted with heavy elements—sometimes containing over 100 Earth-masses of rock and metal. A planet formed in the "clean" outer disk should be metal-poor, like our own Jupiter. How did they become so enriched?

These anomalies have forced the standard model into a corner, requiring ever more complex "patches" to an already convoluted story.

The Emergence of a Coherent Picture, Piece by Physical Piece

A groundbreaking paper by A. Morbidelli, K. Batygin, and E. Lega, leading figures in planetary science from the Observatoire de la Côte d'Azur and Caltech (Astronomy & Astrophysics, 2023), provides a stunning physical explanation for the metal-enrichment puzzle. They demonstrate that the physics of the inner protoplanetary disk is completely different from the outer regions.

Their work shows that when a giant planet is close to its star, it cannot block the inward flow of dust and pebbles. Instead, the gas flowing through the planet's orbit sweeps this material along with it. This process naturally turns the inner disk into a zone of incredibly high metallicity. Consequently, any planet forming in situ in this region will inevitably "breathe" this dust-rich gas, becoming massively enriched in heavy elements.

The conclusion is revolutionary: the extreme metallicity of Hot Jupiters is not an anomaly. It is the natural, expected signature of a planet born and raised in the inner system.

The Unifying Theory: Scale Relativity as the Deeper Cause

Here, the pieces of the puzzle snap together, but not in the way the authors of these papers might have intended. Both studies offer powerful support for in situ formation, but they still treat it as a special case within a chaotic framework. They describe what is happening, but not the deeper why.

The "why" may lie in Scale Relativity (SR), the theory developed by astrophysicist Laurent Nottale. SR postulates that spacetime is fundamentally fractal, leading to a profound consequence: planetary orbits are quantized. Like electrons around an atom, planets are not free to orbit anywhere; they are guided by a probabilistic law into specific, stable orbital paths dictated by the structure of spacetime itself.

From this perspective, the findings of these new papers are not just interesting phenomena; they are the physical manifestations of a deeper law:

1. SR predicts WHERE the stable zones are. The existence of a stable, quantized orbit very close to the star—predicted by SR before Hot Jupiters were ever discovered—is the reason in situ formation is possible. The planets are there because it's a "bright fringe" in the cosmic interference pattern.

2. SR explains the flow of materials. A probabilistic law has a physical consequence. If certain orbits are zones of high probability, then matter should not remain in the low-probability zones between them. Over time, there should be a natural flow of material away from the unstable "voids" and into the stable, quantized "potential wells." The mechanism described by Morbidelli et al.—the efficient inward funneling of dust—is not just a random gas-dynamic effect. It is the physical process that fulfills the structural mandate of Scale Relativity.

A New Paradigm: From Chaos to Quantized Order

We are at a turning point. We can continue to view the universe as a chaotic billiard table, patching our models with complex stories to account for every new observation.

Or we can embrace a simpler, more powerful vision. A vision where the universe is governed by an underlying order, where planets snap into place according to a fundamental law. In this paradigm:

• Hot Jupiters are not impossible migrants. They are natives, formed in situ in a location predicted by theory.

• Their extreme metallicity is not a mystery. It is the natural compositional signature of their birthplace.

• The flow of materials is not a coincidence. It is the physical mechanism by which the universe builds structure in its zones of highest probability.

The evidence is mounting. The once-radical idea of an ordered, quantized cosmos, as proposed by Scale Relativity, is no longer just a fringe theory. It is becoming the most elegant and predictive framework for understanding the worlds beyond our own.