⚛️ high-energy theory

From MOND entropy to extended uncertainty principles: A unified framework

This paper establishes a unified framework linking generalised entropies, cutoff mechanisms, and extended uncertainty principles by deriving a novel MOND-based EUP from a recently proposed MOND entropy, demonstrating its consistency with various entropy models and its ability to generate modified Friedmann equations that encompass higher-order EUP results as limiting cases.

Original authors: Özgür Sevinç, Özgür Ökcü, Ekrem Aydiner

Published 2026-01-22

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Özgür Sevinç, Özgür Ökcü, Ekrem Aydiner

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

Imagine the universe as a giant, complex machine. For a long time, scientists have used two different rulebooks to understand how this machine works: one for the very small (quantum mechanics) and one for the very big (gravity). Usually, these rulebooks don't get along very well.

This paper is like a detective story where the authors try to find a "secret handshake" that connects these two rulebooks. They focus on a specific concept called entropy (a measure of disorder or information) and how it relates to the Uncertainty Principle (the idea that you can't know everything about a particle at once, like its position and speed).

Here is the story of their discovery, broken down into simple steps:

1. The Starting Point: A New "Rulebook" for Gravity

The authors start with a recent idea called MOND Entropy. Think of MOND (Modified Newtonian Dynamics) as a theory that tries to explain why stars in galaxies spin so fast without needing invisible "dark matter." Recently, scientists found a new way to describe the "information" (entropy) of space using this MOND idea.

The authors asked a big question: If this new MOND entropy is true, what does it say about the rules of uncertainty for particles?

2. The Detective Work: Working Backwards

Usually, scientists start with a rule about particles and try to figure out how it changes gravity. But here, the authors did the reverse. They started with the MOND Entropy (the big picture) and worked backward to find the Uncertainty Principle (the small picture) that would create it.

They called this new rule the "MOND EUP" (Extended Uncertainty Principle).

The Analogy: Imagine you find a unique pattern on a giant wall (the Entropy). You want to know what kind of tiny brushstrokes (the Uncertainty Principle) were used to paint it. By looking at the wall, they figured out exactly what the brushstrokes must look like.

3. The "Universal Translator"

The most exciting part of their discovery is that this new "MOND EUP" acts like a universal translator or a master key.

The Master Key: When they turn the "knob" on their new equation to a specific setting, it perfectly matches the rules for Rényi Entropy.
Another Setting: When they turn the knob to a different setting, it perfectly matches Dual Kaniadakis Entropy.
A Third Setting: When they look at the "rough draft" or the simplest version of their equation, it matches a previously known rule called HOEUP (Higher-Order Extended Uncertainty Principle).

The Metaphor: Think of the MOND Entropy as a Swiss Army Knife. Depending on which tool you pull out, it becomes a screwdriver, a knife, or a pair of scissors. The authors found that the "MOND EUP" is the handle that holds all these tools together. It shows that these different entropy theories aren't actually rivals; they are just different versions of the same underlying idea.

4. The "Speed Limit" of the Universe

One of the coolest things the paper points out is about "cutoffs."

In the standard world, you can theoretically get infinitely close to a particle's speed or position.
However, the new rules (EUP) suggest there is a cosmic speed limit for how precisely we can know things. It's like the universe has a "pixel size" or a minimum blur.

The paper shows that even if an entropy theory (like Rényi) doesn't explicitly say "there is a limit," the fact that it connects to this new Uncertainty Principle means it automatically has that limit built-in. It's like a car that doesn't have a speedometer, but because of how the engine is built, it physically cannot go faster than 100 mph.

5. Testing the Rules (The Friedmann Equations)

To make sure their new "Master Key" actually works, they used it to rebuild the Friedmann Equations. These are the famous equations that describe how the universe expands and evolves.

They tried three different methods to build these equations (using thermodynamics, "entropic gravity," and the idea that space "emerges" over time). In all three cases, their new MOND-based rules produced results that matched the known MOND entropy results. This confirmed that their new Uncertainty Principle is consistent with how the universe expands.

6. The Bottom Line

The paper concludes that they have built a unified framework.

They showed that a new, complex entropy (MOND) is actually the "parent" of several other known entropies.
They proved that you can derive the rules of the very small (Uncertainty Principle) from the rules of the very big (Cosmology).
They demonstrated that these different theories are all connected, like different faces of the same coin.

In simple terms: The authors found a hidden bridge between the rules of the tiny quantum world and the massive universe. They showed that a new theory about galaxy rotation (MOND) isn't just a standalone idea; it's the "boss" that organizes several other theories about how the universe stores information, and it all comes from a single, updated rule about how uncertain we can be when looking at the universe.

Technical Summary: From MOND Entropy to Extended Uncertainty Principles

Problem Statement
Standard quantum mechanics relies on the Heisenberg uncertainty principle, which faces limitations at extreme scales (Planck and cosmological). While Generalised Uncertainty Principles (GUP) address ultraviolet (UV) cutoffs (minimum length), Extended Uncertainty Principles (EUP) address infrared (IR) cutoffs (maximum length or minimum momentum). Concurrently, the gravity-thermodynamics conjecture suggests that gravitational field equations can be derived from thermodynamic principles and entropy-area relations. Recently, a novel entropy formalism derived from Modified Newtonian Dynamics (MOND) was proposed, which unifies Rényi and dual Kaniadakis entropies as limiting cases. However, a direct link between this specific MOND-derived entropy and a corresponding modified uncertainty principle had not been established. The paper addresses the question: What is the modified uncertainty principle associated with this generalised MOND entropy, and can it unify various entropy formalisms and cutoff mechanisms?

Methodology
The authors employ a dual-directional approach combining thermodynamic derivations and reverse engineering:

Forward Derivation (HOEUP to Friedmann Equations):
- Starting from a Higher-Order Extended Uncertainty Principle (HOEUP) defined as $\Delta x \Delta p \geq \frac{1}{2} \frac{1}{1 - \alpha \Delta x^2}$ , the authors derive a modified entropy-area relation ( $S_{EUP}$ ).
- Using this entropy, they derive modified Friedmann equations via three distinct frameworks:
  - The first law of thermodynamics at the apparent horizon.
  - Verlinde's entropic gravity scenario.
  - Padmanabhan's emergence of cosmic space.
- They verify the validity of the Generalised Second Law (GSL) of thermodynamics within these models.
Reverse Engineering (MOND Entropy to EUP):
- Motivated by the observation that the Friedmann equations derived from HOEUP match the limiting cases ( $\tilde{\alpha}=1$ ) of equations derived from a recently proposed MOND entropy ( $S_{MOND}$ ), the authors perform a reverse procedure.
- Starting from the differential form of the MOND entropy (which involves a hypergeometric function), they deduce the minimum change in horizon area.
- By relating the area change to position and momentum uncertainties ( $\Delta A \sim \Delta x \Delta p$ ), they derive a novel EUP relation, termed "MOND EUP."

Key Contributions and Results

Derivation of MOND EUP: The paper successfully derives a new Extended Uncertainty Principle from the MOND entropy:
$\Delta x \Delta p \geq \frac{1}{2} \left[ 1 + \left( \frac{\tilde{\gamma}\pi \Delta x^2}{4} \right)^{\tilde{\alpha}} \right]^{1/\tilde{\alpha}}$
where $\tilde{\gamma}$ is a parameter related to the MOND acceleration scale and system mass, and $\tilde{\alpha}$ is a dimensionless parameter.
Unification of Entropy Limits: The derived MOND EUP consistently reproduces known uncertainty relations in specific limits:
- For $\tilde{\alpha} = 1$ , it reduces to the standard EUP associated with Rényi entropy.
- For $\tilde{\alpha} = 2$ , it yields a new form termed Dual Kaniadakis EUP (DKEUP), corresponding to dual Kaniadakis entropy.
- In the perturbative limit (expanding the hypergeometric function), it recovers the HOEUP model, demonstrating that HOEUP is a perturbative limit of the MOND entropy.
Modified Friedmann Equations: The study confirms that the Friedmann equations derived from HOEUP via thermodynamic, entropic, and emergent space approaches are indeed the limiting cases of the equations derived from the full MOND entropy.
IR Cutoff Mechanism: The analysis reveals that while Rényi and dual Kaniadakis entropies do not explicitly imply an IR cutoff, their corresponding EUP models introduce a maximum position uncertainty ( $\Delta x_{max}$ ) and a minimum momentum uncertainty ( $\Delta p_{min}$ ). This endows the entropies with an effective IR cutoff mechanism inherited from the underlying uncertainty principle.
GSL Validity: The Generalised Second Law of thermodynamics is shown to hold within the allowed range of the apparent horizon ( $0 < \tilde{r}_A < 1/2\sqrt{\alpha}$ ), ensuring thermodynamic consistency.

Significance and Claims
The paper claims to establish a unified framework that bridges generalised entropies, cutoff mechanisms, and EUP models. The primary significance lies in the "reverse procedure," which demonstrates that a single, broader uncertainty principle (MOND EUP) can generate various specific entropy forms and their associated dynamics.

The authors posit that different generalised entropies may be viewed as different faces of a broader formalism, where the underlying modified uncertainty principle provides a complementary framework that reveals effective properties (such as IR cutoffs) not immediately apparent in the entropy expressions alone. This work suggests that modified uncertainty principles can serve as effective cutoff mechanisms for entropy forms that do not explicitly display them, potentially offering a consistent way to connect quantum gravitational corrections with cosmological dynamics. The study does not propose new experimental tests but rather offers a theoretical consolidation of existing entropy and uncertainty models.