Generalized Entropies and Black Hole Area Quantization from Landauer's Principle

This paper investigates black hole area quantization by applying Landauer's principle to discrete entropy changes, demonstrating how the resulting area spectrum parameters and their asymptotic behaviors vary across generalized entropy models like Barrow, modified Rényi, and Kaniadakis entropies compared to the standard Bekenstein–Mukhanov limit.

The Gist

Imagine a black hole not as a swirling vortex of darkness, but as a giant, cosmic hard drive. In the world of physics, this hard drive stores information about everything that falls into it. For a long time, scientists have wondered: Is this storage continuous (like a smooth ramp), or is it made of tiny, indivisible blocks (like steps on a staircase)?

This paper explores the idea that black hole "steps" are real and quantized. The authors use a clever rule from information theory called Landauer's Principle to figure out exactly how big these steps are.

Here is a simple breakdown of their journey:

1. The Golden Rule: Erasing a Bit Costs Energy

Think of Landauer's Principle like a "tax" on deleting data. If you have a computer and you want to erase one single bit of information (a 0 or a 1), you must spend a tiny, specific amount of energy to do it. You can't cheat the system; the universe demands a receipt for every deletion.

The authors apply this rule to black holes. They imagine the black hole's surface area (the "hard drive") jumping up one step at a time. They ask: "If the black hole moves from step $n$ to step $n+1$, how much 'information' is being added or erased?"

They decide that every single step up the ladder corresponds to the cost of erasing exactly one bit of information. This simple rule acts as a ruler to measure the size of the steps.

2. The Standard Case: The Perfect Staircase

First, they tested this rule on the classic, standard theory of black holes (Bekenstein-Hawking entropy).

• The Result: The "tax" rule perfectly matched the old, famous predictions. It confirmed that the steps are evenly spaced.
• The Analogy: Imagine a staircase where every step is exactly the same height. As you climb higher and higher (getting to a massive black hole), the steps still exist, but compared to the total height of the staircase, the difference between one step and the next becomes so tiny that it looks like a smooth ramp to the naked eye. This explains why we don't see "pixelation" in large black holes.

3. The Twisted Cases: Deformed Staircases

The paper then asked: "What if the rules of the universe are slightly different?" They tested three different "twisted" versions of entropy (how we count information) that scientists have proposed to account for quantum gravity effects.

A. The Fractal Staircase (Barrow Entropy)

Imagine a staircase where the steps get slightly smaller as you go up, or the shape of the stairs is "fractal" (rough and bumpy).

• The Finding: The size of the "tax" (the step height) changes depending on which step you are on. It's not a fixed ruler anymore; the ruler itself stretches and shrinks.
• The Outcome: Even though the steps change size, if you climb high enough, the steps still become so small relative to the total height that they look smooth. The "pixelation" disappears at the macro scale.

B. The Split Staircase (Modified Rényi Entropy)

This version of the math creates a staircase with two different paths:

• Path A (The Dangerous Path): As you climb, the steps get weird. At a certain point, the math breaks down, the step size becomes negative (which makes no physical sense), and the staircase collapses. This path is a dead end.
• Path B (The Safe Path): The steps get smaller and smaller as you climb, eventually leveling off at a maximum height. The black hole can't get infinitely big; it hits a ceiling.
• The Outcome: Only the "Safe Path" works. On this path, the steps eventually become invisible at large scales, just like the standard case.

C. The Stretchy Staircase (Modified Kaniadakis Entropy)

This version introduces a "stretch factor" (a parameter called $\kappa$).

• The Problem: If you keep this stretch factor fixed, the steps don't get small enough as you climb. Instead of looking like a smooth ramp at the top, the staircase stays "chunky" forever. The steps remain visible even for giant black holes, which contradicts our everyday observation of smooth physics.
• The Fix: The authors suggest that the "stretch factor" shouldn't be a fixed number. Instead, it should shrink as the black hole gets bigger. If the stretch factor shrinks fast enough, the steps finally become smooth again.

The Big Picture

The paper concludes that Landauer's Principle is a powerful tool. It acts like a universal "quality control" check for theories about black holes.

• It confirms the standard theory works.
• It helps us spot which "twisted" theories are broken (like the dangerous path in the Rényi case).
• It tells us what conditions must be met for a new theory to make sense in the real world (like the stretch factor needing to shrink in the Kaniadakis case).

In short, by treating the black hole's surface as a series of information bits that cost energy to change, the authors provided a clear way to test if new, complex theories of the universe actually hold up when you look at them up close.

Technical Summary

Technical Summary: Generalized Entropies and Black Hole Area Quantization from Landauer's Principle

Problem Statement
The paper addresses the problem of black hole horizon area quantization. While the Bekenstein–Mukhanov approach posits that the horizon area is a classical adiabatic invariant with a discrete spectrum ($A_n = \gamma \ell_P^2 n$), the dimensionless parameter $\gamma$ is traditionally fixed by assuming a specific degeneracy of microscopic states (e.g., $W=k^n$). The authors seek an alternative physical criterion to determine $\gamma$ without relying on specific microscopic degeneracy assumptions. Furthermore, they aim to investigate how this quantization scheme behaves when applied to generalized entropy functionals (Barrow, Modified Rényi, and Modified Kaniadakis entropies) that arise from quantum gravitational effects or non-extensive statistical mechanics. A key objective is to analyze the macroscopic behavior of these spectra, specifically whether the relative spacing between adjacent area levels ($\Delta A_n / A_n$) vanishes as $n \to \infty$, ensuring consistency with the classical continuum limit.

Methodology
The authors employ Landauer's principle as the fundamental physical criterion. Landauer's principle states that the erasure of one bit of information incurs a minimum entropy cost of $\Delta S = k_B \ln 2$. The core methodology involves:

1. Discrete Identification: Identifying the entropy difference between two consecutive area levels ($n$ and $n+1$) with the elementary entropy increment required to erase one bit: $\Delta S = S(n+1) - S(n) = k_B \ln 2$.
2. Parameter Determination: Using this condition to solve for the parameter $\gamma$ (or its level-dependent equivalent) for each entropy functional under consideration.
3. Macroscopic Analysis: Calculating the ratio $\Delta A_n / A_n$ for large $n$ to determine if the spectrum approaches a continuum limit (vanishing relative spacing) or exhibits divergent behavior.

Key Contributions and Results

• Standard Bekenstein–Hawking Entropy:
  Applying the Landauer prescription to the standard entropy $S_{BH} = k_B A / (4\ell_P^2)$ reproduces the standard Bekenstein–Mukhanov result. The parameter is fixed at $\gamma = 4 \ln 2$. The resulting area spectrum is $A_n = 4 \ell_P^2 (\ln 2) n$. Crucially, the relative spacing behaves as $\Delta A_n / A_n = 1/n$, which vanishes as $n \to \infty$, recovering the expected macroscopic behavior.

• Barrow Entropy:
  For Barrow entropy, which introduces a fractal deformation parameter $\Delta$ ($0 \le \Delta \le 1$), the parameter $\gamma$ becomes dependent on the area level $n$ and $\Delta$. The derived expression is $\gamma_B = 4 [\ln 2 / ((n+1)^{(1+\Delta)/2} - n^{(1+\Delta)/2})]^{2/(2+\Delta)}$.

  • Result: While the spectrum is no longer uniformly spaced, the relative spacing $\Delta A_n / A_n$ still vanishes as $n \to \infty$ (scaling as $\sim 1/n$). This indicates that the fundamental discreteness is preserved, but the macroscopic limit remains insensitive to the underlying level structure.

• Modified Rényi Entropy (MRE):
  The MRE is derived from Tsallis statistics via a logarithmic transformation involving a parameter $\lambda = 1-q$. The analysis reveals two distinct branches based on the sign of $\lambda$:

  • Singular Branch ($\lambda > 0$): The parameter $\gamma_R$ develops a pole at a finite level $n_c = 1/(2\lambda - 1)$ and becomes negative for $n > n_c$. Consequently, this branch does not define a physically valid (positive) area spectrum beyond the pole.
  • Regular Branch ($\lambda < 0$): No divergence occurs; $\gamma_R$ remains positive for all $n$. In this branch, the area spectrum approaches a finite asymptotic value ($A_n \to 4\ell_P^2/|\lambda|$) as $n \to \infty$. The relative spacing vanishes as $\sim 1/n^2$, ensuring a consistent macroscopic regime.

• Modified Kaniadakis Entropy (MKE):
  For the MKE, characterized by a deformation parameter $\kappa$, the authors perform a small-$\kappa$ expansion. The resulting parameter $\gamma_\kappa(n)$ includes a correction term proportional to $\kappa^2 n^2$.

  • Result: If $\kappa$ is treated as a fixed fundamental constant, the relative spacing $\Delta A_n / A_n$ does not vanish in the large $n$ limit; instead, it grows linearly with $n$ due to the $\kappa^2 n^2$ term.
  • Resolution: The authors suggest that a consistent macroscopic regime can only be recovered if $\kappa$ is interpreted as an effective, scale-dependent parameter $\kappa(n)$ that decreases sufficiently fast with $n$ (specifically $\kappa(n)\sqrt{n} \ll 1$) to suppress the deformation at large scales.

Significance and Claims
The paper claims that Landauer's principle provides a robust and useful framework for analyzing generalized entropic extensions of the Bekenstein–Mukhanov area spectrum. By bypassing assumptions about microscopic degeneracy, the principle directly links information-theoretic costs to the geometric structure of the black hole horizon.

The authors emphasize that this approach:

1. Successfully recovers the standard Bekenstein–Mukhanov result as a reference limit.
2. Distinguishes between regular and singular branches in generalized entropy models (as seen in the Modified Rényi case), thereby filtering out physically inconsistent spectral structures.
3. Clarifies the conditions required for generalized models to possess a consistent macroscopic limit. Specifically, it highlights that for certain deformations (like Kaniadakis), a fixed deformation parameter may prevent the recovery of the classical continuum, necessitating a scale-dependent interpretation of the deformation parameter.

The work does not propose new experimental tests but rather offers a theoretical consistency check for various quantum gravity-inspired entropy models using information-theoretic principles.