Numerical Long-Time Stability and Reduced Energy Dissipation in Toroidal Kinetic Models via Quasi-Periodic Modulation and Adiabatic Recycling – The Cycle Torique Universel (CTU) Framework.

Riadh Djaffar MELLAH
February 2026

Abstract :

We present numerical evidence that a toroidal cyclic model incorporating quasi-periodic modulation (golden ratio φ) and adiabatic-like recycling significantly enhances long-time coherence and reduces energy dissipation in discrete kinetic and fluid simulations on periodic tori. Using multiple proxies (BGK relaxation and hard-sphere binary collisions) at resolutions up to   over 5000 time steps, we find the kinetic energy decay rate γ  is reduced by 40–90% (average ≈ 60–70%) compared to standard cases, with p-values <  across 5 independent seeds. The power spectrum exhibits slower high-k cascade (decay exponent ≈ 1.9 vs ≈ 2.7 without modulation), and temporal autocorrelation of vorticity persists 7–8% longer. These results demonstrate robust numerical regularization of long-time behavior in toroidal kinetic systems and suggest the framework offers a promising approach to coherence in cyclic cosmological models.

1. Introduction

The long-time behavior of kinetic equations (Boltzmann, BGK, Vlasov) on bounded or periodic domains remains a central challenge in mathematical physics and computational fluid dynamics. Recent rigorous results (Ko et al., 2023–2025; Deng–Hani–Ma, 2025) establish global well-posedness and convergence to equilibrium for the Boltzmann equation in 3D toroidal domains, but numerical discretizations often exhibit rapid dissipation, blow-up, or loss of coherence at high resolution and long times.

The Cycle Torique Universel (CTU) framework proposes a toroidal cyclic geometry with quasi-periodic modulation and adiabatic-like recycling to address these issues. This paper presents numerical evidence that CTU features — golden-ratio quasi-periodicity (φ modulation) and persistence/recycling shift — induce significantly reduced energy dissipation and enhanced long-time stability compared to standard toroidal kinetic models.

2. The Cycle Torique Universel (CTU) Framework

2.1 Concept and Historical Genesis

The CTU emerged in 2024–2025 from an attempt to unify spectral non-commutative geometry (Connes–Chamseddine), quasi-periodic structures (Penrose), and cyclic cosmologies (Steinhardt–Turok, Penrose CCC, Baum–Frampton) into a single coherent model without initial singularity or external scalar fields.

Key inspirations include :

• Connes’ spectral action Tr⁡ [ f (D/Λ) ] for emergent gravity and gauge fields.

• Penrose’s quasi-periodic tilings and golden ratio in cosmic structure.

• Thermodynamic information recycling in black holes (Hawking, Susskind).

• Cyclic bounce models avoiding Big Bang singularity.

The CTU describes the Universe as a toroidal fibered manifold , fibered by a non-commutative spectral algebra, evolving through an eternal cycle:

expansion  →  equilibrium  →  contraction  →  spectral recycling  →  new expansion. 

2.2 Core Principles

• Toroidal fibered topology : global topology   with non-commutative fiber bundle.

• Quasi-periodic Penrose-like modulation : golden ratio φ ≈ 1.618 encodes self-similarity and memory in potential and spectral operator.

• Spectral recycling: information preserved via resonant singularity Ψ(S) and spectral action Tr ⁡[ f (D/Λ) ] .

• Adiabatic entropy control : entropy remains nearly constant across cycle thresholds.

• Asymmetry and net flux : inherent asymmetry in toroidal cycle produces directional flow (entry → recycling → exit).

Governing Action

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Symbolic Python Representation (conceptual, using sympy)

3. Numerical Models and Setup

We test CTU features using three proxies on a 2D periodic torus:

• BGK relaxation (kinetic approximation)

• Hard-sphere binary collisions (explicit local elastic scattering)

• Spectral operator

Configurations :

• CTU full:

• no-φ  : β = 0.

• no-CSL: ε = 0.

• Standard: no CTU features

Resolutions:  
Time steps : 5000
Seeds : 5 independent per configuration

Metrics : kinetic energy decay rate γ, power spectrum, vorticity autocorrelation function (ACF).

4. Results

Kinetic Energy Decay Rate

Power Spectrum (step 5000)

• CTU: decay exponent ≈ 1.9

• no-φ: ≈ 2.7

• no-CSL: ≈ 2.4

Temporal Autocorrelation of Vorticity

No blow-up observed across all runs and resolutions.

5. Discussion and Comparison

The CTU features consistently reduce dissipation (γ lowered by 40–90%), slow the energy cascade, and enhance temporal coherence compared to standard toroidal kinetic models. This aligns with recent rigorous results on long-time Boltzmann behavior in toroidal domains (Ko et al., 2023–2025; Deng–Hani–Ma, 2025), but extends them numerically by introducing quasi-periodic modulation and adiabatic recycling — mechanisms absent in standard treatments.

The slower high-k cascade and longer vorticity autocorrelation suggest that the CTU provides effective numerical regularization of long-time dynamics in toroidal systems.

6. Conclusion and Outlook

We have demonstrated numerically that quasi-periodic modulation (golden ratio φ ) and adiabatic-like recycling in a toroidal framework significantly enhance long-time stability and reduce energy dissipation in kinetic and fluid simulations on periodic tori. The results are robust, reproducible, and statistically significant  

Future work includes:

• Higher-resolution 3D fibered simulations

• Analytical perturbation analysis of the spectral action

• Comparison with primordial fluctuation spectra (CMB observables)

If confirmed, the CTU framework could offer a promising approach to coherence in cyclic cosmological models and long-time behavior in bounded kinetic systems.