Not Just Transport: After Parisi, a Second Family of Classical Laws Emerges in Quantum Systems

An equation born to describe surfaces growing in cigarette smoke now describes quantum materials. A research group at the University of Würzburg announced the result today in Science, with Nobel laureate Giorgio Parisi among the authors.

The equation is that of Kardar-Parisi Zhang, written in 1986 for surface growth, the irregular shape of burning paper, the propagation of a fire.

It was not designed for quantum mechanics, yet it works without modification on two-dimensional quantum materials.

This is not the first time it has happened. In recent years, several classical laws have turned out to hold in quantum systems, in the same form in which they were originally written, without adjustments. Ohm’s law for electrical circuits was verified in 2012 down to the atomic scale, in silicon nanowires near absolute zero. Fourier’s law of heat conduction was derived directly from the fundamental equation of quantum mechanics.

The Schrödinger equation for a free particle reduces to Fick’s second law of diffusion. The Beer-Lambert law of light absorption was derived from the same equation for single-photon states. Darcy’s law for flow through porous media was verified in superfluid helium, a quantum fluid. Five laws that share one thing: they all describe how a difference, whether of voltage, temperature, concentration, or pressure, propagates through a limit.

They are transport laws. Until now, this property seemed to belong to them alone.

But there is another family of classical laws that shows the same behaviour in the quantum world, and it has nothing to do with transport.

These are saturation laws: they describe how a system fills up to a maximum and then stops.

The Langmuir isotherm, formulated in 1916 to describe how gas molecules deposit onto a solid surface, describes without corrections the behaviour of quantum dots, nanocrystals of a few nanometres whose operation depends entirely on quantum confinement.

Quantum objects, whose adsorption onto a surface follows the same classical law from a hundred and ten years ago. And the Fermi-Dirac distribution, one of the fundamental equations of quantum mechanics, the one governing how electrons fill energy states in a metal, has the exact same mathematical form as classical saturation laws: a curve that grows, slows down, and stops at the maximum. The same structure Langmuir found in molecules on a surface, Fermi and Dirac found in electrons inside a crystal.

These laws do not merely share an idea. They share a precise form, and it can be written in a single line:

R(X) = aX / (1 + bX)

where X is the input, the difference, the gradient, the availability, and R is the system’s response. The parameter a measures local gain, b measures the strength of the limit. When the limit is still far away, meaning bX is much smaller than 1, the formula reduces to R ≈ aX and the response is linear: the harder you push, the more passes through. This is the transport regime. This is Ohm, Fourier, Fick, Darcy

When the input grows and the limit comes into play, the response can no longer remain linear. It still grows, but less and less, until it locks at the maximum value a/b. This is the saturation regime. This is Langmuir, Michaelis-Menten, Fermi-Dirac.

Linear transport and saturation are not two unrelated families. They are two readings of the same boundary response: linear when the limit has not yet intervened, saturating when the limit compresses the dynamics. The same equation, seen at two different points along the boundary.

Parisi’s equation does exactly this. It describes how an irregular boundary grows under constraints. It does so in burning paper. It does so in a two dimensional quantum material. Not because the matter is the same, but because the limit cuts the dynamics in the same way: from free growth to f iltered growth. If the reason classical laws work in quantum systems is the form of the constraint rather than the type of matter, then this is not over. The same saturation structures we know in chemistry and biology should appear in quantum systems, whenever a capacity limit of the same type exists. In biochemistry, the Michaelis-Menten equation describes how an enzyme accelerates a reaction up to a maximum and then stops because all active sites are occupied. It is the right-hand side of the same curve: growth, slowdown, halt. If a quantum system presents a boundary of the same type, the same equation can re-emerge. Not because everything is connected. But because equations do not belong to the phenomena they describe. They belong to the structures that make those phenomena possible. Two families are already confirmed: transport and saturation. The question is no longer why a classical law works in quantum systems. The question is where it will happen next.

Davide Lugli is an independent researcher. His work on Sub-Limit Dynamics and Constrained Generative Systems is published on Zenodo and documented at dailui.com. References: Weber et al., Science 2012 (atomic Ohm); Widmann et al., Science 2026 (quantum KPZ); Allen et al., CQO-11, 2019 (quantum Beer Lambert); WS₂ quantum dot adsorption, Langmuir ACS 2018