🌀 nonlinear sciences

Spectra as a classical phenomenon, and the Einstein classical program

This paper challenges the notion that spectra are purely unintelligible quantum phenomena by demonstrating that classical calculations of ionic crystal infrared spectra can reproduce experimental data across a wide temperature range—especially when incorporating Nernst's concept of zero-point energy—thereby advancing the "Einstein Classical Program" of deriving quantum physics from a realistic classical framework.

Original authors: Andrea Carati, Luigi Galgani, Fabrizio Gangemi

Published 2026-06-19

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Andrea Carati, Luigi Galgani, Fabrizio Gangemi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Idea: Can Classical Physics Explain "Quantum" Magic?

Imagine you are listening to a choir. In the world of modern physics (Quantum Mechanics), we are taught that a choir can only sing specific, distinct notes. If you try to sing a note in between, the universe simply won't allow it. This is why we see "spectra" (the specific colors of light emitted by materials) as a magical, purely quantum phenomenon that classical physics (the physics of Newton and everyday objects) cannot explain.

The authors of this paper are challenging that idea. They are asking: What if we could explain these "quantum" choir notes using only the rules of classical physics, without any magic?

They call this the "Einstein Classical Program." It's like trying to prove that a complex, high-tech video game is actually just a very complicated version of a simple board game, if you look at it closely enough.

The Experiment: The Crystal Choir

To test this, the researchers looked at a crystal made of Lithium Fluoride (LiF). Think of this crystal as a giant, rigid grid of atoms (like a 3D checkerboard) where the atoms vibrate back and forth. When you shine light on it, it absorbs specific frequencies, creating a "spectrum."

The Surprise:

The Quantum Approach: Scientists usually use complex quantum math to predict these vibrations. The authors found that the best quantum calculations available actually made mistakes and didn't match real-world experiments very well.
The Classical Approach: The authors ran a simulation using only classical physics (Newton's laws of motion). They treated the atoms like tiny balls connected by springs.
- At Room Temperature: The classical simulation matched the real-world data better than the quantum simulation did.
- At Very Cold Temperatures: This is where it gets tricky. Classical physics usually says that at absolute zero, everything stops moving. But the real world shows that atoms still vibrate a little bit even at near-zero temperatures (this is called "zero-point energy").

The "Zero-Point" Secret

To make their classical simulation work at very low temperatures, the authors had to make one special assumption, inspired by a physicist named Nernst from 1916: They assumed that even at absolute zero, the atoms have a tiny bit of "background energy" that never goes away.

Think of it like a car engine that never fully turns off. Even when the car is parked (absolute zero), the engine is still idling with a little bit of energy.

When they added this "idling energy" to their classical model, the simulation suddenly matched the real-world data perfectly, even at temperatures as low as 7.5 Kelvin (which is extremely cold).

The Mystery of the "Mixing"

The paper also tackles a deep puzzle about how things mix and settle down.

The Expectation: In classical physics, if you shake a box of marbles, they eventually spread out evenly (this is called "equipartition").
The Reality: In these crystals, the atoms don't just spread out randomly. They seem to keep some of their "order" for a long time, like a dance troupe that keeps a specific formation even while moving chaotically.

The authors found that the atoms are a mix of chaos (random bouncing) and order (keeping a rhythm). This "ordered" part is what allows the classical model to mimic the "quantum" behavior. It's like a crowd of people in a stadium: individually, they are moving randomly, but if they all do "The Wave," there is a clear, ordered pattern emerging from the chaos.

The Conclusion: A New Perspective

The authors conclude that:

Spectra are not exclusively quantum: You can calculate them using classical physics if you include the right assumptions (like that "idling" zero-point energy).
Classical might be better right now: Surprisingly, their simple classical model currently predicts real-world data better than the complex quantum models available today.
Einstein was right: This supports Albert Einstein's old dream that quantum physics isn't a separate, magical world, but rather a result that can be derived from a deeper, realistic theory of classical physics.

In short: The paper suggests that the "magic" of quantum spectra might just be a very complex dance of classical atoms that we haven't fully understood yet. By looking at the dance through the lens of "zero-point energy" and "ordered chaos," the classical rules work just as well as the quantum ones.

Technical Summary: Spectra as a Classical Phenomenon, and the Einstein Classical Program

Problem Statement
The paper addresses the long-standing assertion, notably by Max Born, that atomic spectra are "quantum phenomena" that are "perfectly unintelligible" from a classical standpoint. The authors challenge this dichotomy by investigating whether the infrared spectra of ionic crystals (specifically Lithium Fluoride, LiF) can be reproduced using a purely classical mechanical framework. The central problem is to determine if classical calculations, which typically assume energy equipartition, can match experimental spectral data as well as, or better than, current quantum mechanical calculations, and to explore the implications of this compatibility for the "Einstein Classical Program"—the effort to deduce quantum physics from a realistic classical theory.

Methodology
The authors employ a purely mechanical approach, calculating spectra without explicitly modeling the propagation of the electromagnetic field inside the crystal, mirroring the approximations used in the quantum calculations of the Maradudin group.

Model: The system is modeled as a crystal of point particles (ions) interacting via a short-range Buckingham potential ( $A \exp(-Br) - C/r^6$ ) and electrostatic forces. The model uses effective charges for ions to account for electronic repulsion and polarizability.
Spectral Calculation: The electric susceptibility $\chi(\omega)$ , which determines the spectrum, is derived microscopically using the Green-Kubo formula. This involves computing the time-correlation function of the macroscopic polarization $P(t)$ :
$\chi_{ij}(\omega) = \frac{V}{K} \int_0^{+\infty} e^{-i\omega t} \langle P_i(t) \dot{P}_j(0) \rangle dt$
where $K$ is the specific kinetic energy and the brackets denote a time average.
Simulation: Molecular dynamics simulations were performed on a system of 4,096 ions with periodic boundary conditions. Initial conditions were drawn from a Maxwell-Boltzmann distribution at specific energies. Time averages were used instead of phase averages, with aggregation over multiple runs to achieve simulation times on the order of nanoseconds.
Temperature Handling: The relationship between specific energy and temperature was re-evaluated. While high-temperature simulations used the standard relation $K = \frac{3}{2}k_B T$ , low-temperature results required simulations at significantly higher specific energies to match experimental data, suggesting a non-vanishing kinetic energy at absolute zero (zero-point energy) within the classical framework.

Key Contributions and Results

Classical Reproduction of Spectra: The authors demonstrate that classical computations successfully reproduce the infrared spectra of LiF.
- Room Temperature and Above: At temperatures up to 1060 K, the classical results show excellent agreement with experimental data, capturing the main peak and anharmonic shoulders. Notably, the classical calculation reproduces the exponential decay at high frequencies (due to the analyticity of classical trajectories), a feature that current quantum approximations (based on Lorentzian functions) fail to capture accurately.
- Low Temperatures: At low temperatures (down to 7.5 K), classical agreement with experimental data is achieved only if the simulations are run at specific energies corresponding to effective temperatures of 125 K and 180 K, respectively. This implies the existence of a classical zero-point energy (estimated around 120 K) that prevents kinetic energy from vanishing at absolute zero.
Comparison with Quantum Calculations: The paper compares these classical results with quantum calculations performed by the Maradudin group. The authors note that the available quantum results (limited to room temperature) show poorer agreement with experimental data than the classical results. The authors attribute the quantum discrepancies to technical approximations, such as Taylor expansions of the Hamiltonian and the neglect of certain Coulomb contributions, rather than a fundamental failure of quantum theory.
Ergodicity and Ordered Motion: The study investigates the ergodic properties of the LiF model. While the system reaches energy equipartition rapidly at specific energies above the estimated zero-point energy (120 K), the persistence of strong spectral peaks indicates that the motion is not fully chaotic. The authors suggest a coexistence of "ordered" and "disordered" motions, where certain modes (related to the integrable Toda system) retain memory of initial conditions longer than others. This partial order allows for the compatibility of equipartition (used in the classical simulation) with Planck's law (observed in the spectrum).

Significance and Claims
The paper positions these findings as a "first step" toward the implementation of the Einstein Classical Program. The authors argue that the ability to deduce quantum-like phenomena (spectra) from a realistic classical theory (classical electrodynamics of matter in bulk, extended to include radiation reaction and non-runaway conditions) challenges the notion that quantum mechanics is the only viable description of these phenomena.

Re-evaluating Born's Dictum: The results suggest that spectra are not inherently "unintelligible" classically, provided one accepts a realistic classical framework that includes zero-point energy and specific ergodic properties.
Nernst's Contribution: The paper highlights a 1916 approach by Walther Nernst, which proposed that the compatibility between the equipartition theorem and Planck's law arises from a system possessing both ordered (zero-point) and disordered (thermal) energy components. The authors link their numerical findings to this historical perspective.
Historical Context: The authors connect their work to the intellectual lineage of Enrico Fermi and the FPU (Fermi-Pasta-Ulam) problem, suggesting that the original FPU results hinted at the non-universality of mixing and thermalization, a view shared by Einstein and Fermi regarding the foundations of quantum mechanics.

In conclusion, the paper claims that classical mechanics, when applied with realistic assumptions regarding zero-point energy and the specific ergodic nature of crystal dynamics, can reproduce experimental spectra with high fidelity, potentially offering a pathway to derive quantum results from a classical realistic theory.