Planck Law from a Classical Free Energy Extremum Involving Fisher Information

This paper derives the Planck law for black-body radiation using a classical variational principle that extremizes a generalized free energy functional incorporating Fisher information, relying solely on a minimal threshold assumption for photon emission rather than quantum mechanics.

The Gist

The Big Idea: A Classical Puzzle Solved with a New Tool

For over a century, physicists have believed that the way hot objects glow (like the sun or a lightbulb filament) is described by Planck's Law. This law was traditionally thought to be the "smoking gun" of quantum mechanics—the proof that energy comes in tiny, discrete packets called "quanta."

This paper argues something surprising: You don't actually need quantum mechanics to get this result.

The author, Carlos Gomez-Uribe, claims that if you look at a hot object using only classical physics (the kind of physics that describes rolling balls and flowing water) but add two specific ingredients, you get the exact same glowing pattern that Planck discovered.

The Two Ingredients

To make this work, the author uses a mathematical tool called Fisher Information. Think of this not as a physical force, but as a measure of "sharpness" or "clarity."

1. The "Threshold" Rule (The Bouncer):
   Imagine a crowded dance floor (the hot object) where people are bumping into each other (thermal fluctuations). Usually, these bumps are small and harmless.

   • The Rule: The author proposes a simple rule: A "photon" (a packet of light) is only emitted if a bump is strong enough to knock a specific energy threshold ($\hbar\omega$) over the edge.
   • The Analogy: Think of a bouncer at a club. If a person's energy is too low, they just get bumped around on the dance floor. But if they have enough energy to pay the "cover charge" (the threshold), they get to jump out the door and emit light. Small bumps don't count; only big ones do.

2. The "Sharpness" Penalty (Fisher Information):
   In classical physics, we usually just count how much energy things have. This author adds a new rule: The system "dislikes" being too fuzzy or spread out. It prefers to be sharp and localized.

   • The Analogy: Imagine trying to balance a stack of cards. If the stack is too wobbly (fuzzy), it costs you "energy" to keep it together. The system naturally tries to find the most stable, "sharpest" shape possible.
   • The author combines this "sharpness cost" with the "entropy" (disorder) of the system. By balancing the desire to be disordered (hot) with the desire to be sharp (localized), the math naturally settles into the exact pattern of Planck's Law.

How It Works (The "Goldilocks" Balance)

The paper uses a method called Variational Principle. Imagine you are trying to find the perfect temperature for a cup of coffee. You want it hot enough to be enjoyable, but not so hot that it burns your tongue.

• The Setup: The author creates a "Free Energy" formula. This formula has two competing parts:
  1. Entropy: The tendency to spread out and be chaotic (like heat).
  2. Fisher Information: The tendency to stay sharp and localized (like a specific shape).
• The Magic: The author adjusts the "weights" of these two parts based on the ratio of the "threshold energy" to the "thermal heat."
• The Result: When the math finds the "perfect balance" (the minimum energy state), the resulting distribution of energy is exactly the Planck distribution.

What This Means (and What It Doesn't)

What the paper claims:

• You can derive the famous formula for black-body radiation without assuming that energy levels are "quantized" (discrete steps) inside the atom.
• You don't need to assume the existence of "photons" as particles inside the system.
• The only "quantum" thing you need to admit is the threshold rule: that light is only emitted when a fluctuation is big enough to pay the $\hbar\omega$ price tag.
• The paper suggests that the "zero-point energy" (the energy an object has even at absolute zero) emerges naturally from this balance between "sharpness" and "disorder," rather than from a mysterious quantum vacuum.

What the paper does NOT claim:

• It does not say quantum mechanics is wrong. It says Planck's Law might be a result of classical thermodynamics plus a simple threshold rule, rather than a result of deep quantum weirdness.
• It does not propose new medical treatments, new technologies, or immediate engineering applications.
• It does not claim to explain why the threshold exists, only that if it exists, the rest of the math follows classically.

The "Kinetic" Side Story

The paper also offers a second way to look at this, called a Kinetic Derivation.

• The Analogy: Imagine a bucket with a hole. Water (energy) leaks in randomly. Most of the time, the water level rises slowly. But occasionally, a huge wave hits the bucket, pushing the water level high enough to splash over the rim (emit a photon).
• Once the water splashes over, it creates a "cascade" of splashes until the water level drops back below the rim.
• The paper shows that if you count these "splash events" using classical probability, you get the same Planck distribution.

Summary

This paper suggests that the glowing of hot objects might not be a mystery of the quantum world, but rather a natural outcome of classical physics where:

1. Light is only emitted when a thermal bump is big enough (Threshold).
2. The system naturally finds a balance between chaos and sharpness (Fisher Information).

If this is true, it means the "quantum" behavior we see in black-body radiation might actually be a classical phenomenon that we just haven't looked at the right way before.

Technical Summary

Technical Summary: Planck's Law from a Classical Free Energy Extremum Involving Fisher Information

Problem Statement
The paper addresses the derivation of Planck's law for black-body radiation, traditionally considered a foundational result requiring quantum mechanics (discrete energy levels, quantized oscillators, or Bose-Einstein statistics). The ultraviolet catastrophe highlighted a failure of classical electrodynamics to explain the spectral energy density $\rho(\omega, T)$. While previous classical attempts (e.g., Stochastic Electrodynamics) have relied on background zero-point fields or specific dynamical models, this work seeks to derive the Planck distribution using a purely classical variational principle over continuous probability densities in configuration space, without invoking quantized oscillator states or ensembles over discrete energy levels.

Methodology
The author proposes a generalized free energy functional, $F_\gamma[p]$, defined over a probability density $p(x)$ in configuration space. This functional interpolates between classical thermodynamics and quantum variational principles by incorporating three terms:

1. Expected Potential Energy: $\langle U \rangle$.
2. Shannon Entropy: Weighted by a function $f(\gamma)$ and the energy scale $\hbar\omega$ rather than the thermal scale $k_B T$.
3. Fisher Information: $J(p) = \int (\nabla p)^2 / p \, dx$, weighted by a function $g(\gamma)$ and the quantum kinetic term $\hbar^2/8m$.

The dimensionless parameter $\gamma = \hbar\omega / k_B T$ governs the weights $f(\gamma)$ and $g(\gamma)$. The derivation relies on two primary physical assumptions:

• Threshold Activation: Photon emission occurs only when a thermal fluctuation delivers energy $E \geq \hbar\omega$. Fluctuations below this threshold contribute to background noise but do not trigger emission.
• Gaussian Ansatz: The variational search is restricted to normalized Gaussian distributions with zero mean, which are known to extremize both entropy and Fisher information for quadratic (harmonic) potentials.

The specific scaling functions are derived as follows:

• $f(\gamma) = \gamma$: This weights the entropy term by the emission threshold energy $\hbar\omega$, reflecting that only a fraction $e^{-\gamma}$ of thermal kicks are effective for emission.
• $g(\gamma) = \left(\frac{2}{e^\gamma - 1}\right)^2 - 1$: This term is motivated by the ratio of failed (sub-threshold) kicks to successful ones, ensuring the Fisher term acts as a geometric constraint that transitions from penalizing localization (thermal smoothing) to favoring it (quantum localization) as temperature decreases.

Key Results
By extremizing the functional $F[p]$ with respect to the variance $\sigma^2$ of the Gaussian distribution, the paper derives the optimal variance:
$$\sigma^2 = \frac{\hbar}{2m\omega} \left( 1 + \frac{2}{e^\gamma - 1} \right)$$
From this variance, the expected potential energy is calculated as:
$$\langle U \rangle = \frac{\hbar\omega}{2} + \frac{\hbar\omega}{e^\gamma - 1}$$
Assuming a factorized phase-space distribution $p(x, v) = p(x)p(v)$ (a classical idealization), the total energy per oscillator is identified as $\langle E \rangle = 2\langle U \rangle$. The radiative energy, defined as the energy difference above the zero-temperature limit, yields the exact Planck factor:
$$\langle E_{rad} \rangle = \frac{\hbar\omega}{e^\gamma - 1}$$
Multiplying this by the mode density $n_\omega = \omega^2 / \pi^2 c^3$ recovers the full Planck spectral energy density $\rho(\omega, T)$.

The paper also presents a complementary kinetic derivation (detailed in Appendix A) based on "threshold-activated thermal emission cascades." In this model, an above-threshold fluctuation opens a window for repeated emission attempts, producing a geometrically distributed burst size. This stochastic reasoning yields the same Planck distribution without invoking the variational principle.

Significance and Claims
The paper claims that Planck's law can emerge from classical thermodynamic principles supplemented by minimal constraints, specifically:

1. Minimal Quantum Input: The only quantum element required is the experimentally established threshold condition that emission requires energy $\hbar\omega$. The framework does not derive the existence of photons or the relation $E=\hbar\omega$ but treats them as constraints.
2. Information Geometry: The inclusion of Fisher information serves as a geometric regularizer in a classical setting, encoding the competition between thermal fluctuations and localization. This suggests that the mathematical structure often associated with quantum mechanics (e.g., the Schrödinger equation as a free energy extremum) may have a classical information-theoretic origin.
3. Alternative Foundation: The work reframes black-body radiation not as a direct consequence of energy quantization, but as the natural equilibrium outcome of classical systems governed by threshold constraints and variational structure.
4. Experimental Implications: The kinetic cascade model suggests that thermal radiation might exhibit non-Poissonian statistics (photon bunching) in low-temperature, single-mode systems, offering a potential avenue for experimental verification of the classical activation mechanism.

The author positions this work as a complement to existing classical approaches (such as Stochastic Electrodynamics) but distinguishes it by relying on thermodynamic variational principles and information geometry rather than explicit background fields or stochastic differential equations.