Quantum Dynamics, Logistic Map, and Digit Distribution of Special Math Constants

Using the logistic map instead of the base quadratic system as in paper 53 (here), I obtain very similar quantum dynamics, this time for the function sin²(√x) instead of exp(x). When x is a small integer or a product of consecutive primes, my framework reveals new insights on the digit distribution of major math constants. I also discuss deep findings about the chaotic nature of dynamical systems and several applications including in AI.

Contents

1. Introduction.
2. Logistic map and the digit sum function
   . . .  Model comparison, with illustrations
   . . .  Normality of special math constants
   . . .  Applications and references
3. Re-balancing an uneven digit distribution
   . . .  Digit-balancing transforms
   . . .  Digit block balancing
4. Conclusion
5. Main Python code

Summary

This article is the fourth in this series. The previous ones focus on the simplest quadratic map, while this one shows how we can attain similar results with the most well-known dynamical system: the logistic map.

Orbits in chaotic dynamical systems are very sensitive to the seed. Starting with close seeds S₀ = x · 2⁻²ⁿ and S′₀ = 0, I computed ∆_k = |S_k − S′_k| and showed that ∆_n ≈ sin²(√x) with a precision of about 2n bits, also getting good approximations to ∆_k when k ≤ n. In the logistic map, S′₀ is a fixed point.

I also discussed the unique quantic behavior of the digit sum function when using special values of x. This opens up new directions to study the digit distribution of special math constants, e in particular. Most importantly, it leads to many applications including cryptography, synthetic data, pattern detection or proof automation with LLMs, agent-based modeling, and more.

The material presented here can also be used to complement a course on dynamical systems, for scientific research, or to start a PhD thesis on the subject.

Download technical paper and data generator (Python)

The PDF with many illustrations is available for free as paper 54, here. It also features fast Python code (with link to GitHub) to deal with gigantic numbers. The underlying theory is explained in detail, with several modern references. The blue links in the PDF are clickable once you download the document from GitHub and view it in any browser but may not be clickable in the GitHub “view mode”. I hope GitHub fix this issue in the future!

About the Author

Vincent Granville is a pioneering GenAI scientist, co-founder at BondingAI.io, the LLM 2.0 platform for hallucination-free, secure, in-house, lightning-fast Enterprise AI at scale with zero weight and no GPU. He is also author (Elsevier, Wiley), publisher, and successful entrepreneur with multi-million-dollar exit. Vincent’s past corporate experience includes Visa, Wells Fargo, eBay, NBC, Microsoft, and CNET. He completed a post-doc in computational statistics at University of Cambridge.