Emergent Quantum Statistics in Stochastic Pilot-Wave Dynamics

Christian Revoire
Independent Researcher — January 2026

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🔬 Overview

This repository contains high-performance Python/Numba simulations of a local hidden-variable model inspired by walking droplets (Couder–Fort–Bush).

The goal is to investigate whether key quantum statistical properties, such as Born’s rule and fermionic exclusion, can emerge dynamically from local wave–particle dynamics with feedback, without imposing quantum axioms by hand.

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📊 Interpretation & Scripts & Key Results

1️⃣ Emergence of Born’s Rule (1D)

Objective:

Demonstrate the statistical emergence of Born's Rule. That particle guided by a local pilot-wave with feedback dynamically relaxes toward the Born probability density ρ(x) ≈ |ψ(x)|². By statistical convergence and Ergodicity, from singles particle dynamics

Physical Model:

This model couples a stochastic point particle to a complex scalar field (pilot wave):

• The wave evolves according to a complex Ginzburg-Landau equation (Schrödinger-like).
• The particle is guided by the local phase gradient of the field (Langevin dynamics).
• The particle acts as a moving source, continuously interacting and fueling its own pilot wave (feedback).
• The system is in a state of free expansion (diffusion).

Key Result:

• The simulation demonstrates that the particle's statistical distribution ρ(x) dynamically conforms to the shape of the spread wave packet |ψ|², with a high correlation and a low error L1.

• In other words, the probability density of the particle's position converges towards the intensity |ψ|² of the field, validating the dynamical "quantum relaxation" towards Born's Rule without axiomatic postulates.

• It supports the emergence of Born's Rule from a purely deterministic, and realistic dynamics.

📁 Script: src/simulation_1D_Born.py

⏱ Runtime: ~2–10 minutes (depending on params and CPU cores).

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2️⃣ Emergent Pauli-Like Exclusion (1D)

Objective:

This simulation investigates whether fermionic-like exclusion effects can emerge dynamically from a local pilot-wave dynamics, without imposing antisymmetry, exchange rules, or quantum statistics by hand.

The model extends a previously validated single-particle framework demonstrating dynamical convergence toward the Born rule (ρ ≈ |ψ|²). Here, the focus is placed on two-particle correlations.

Physical Model:

• Each particle continuously emits and interacts with its own complex guiding field ψ₁(x,t), ψ₂(x,t).
• Particles dynamics is governed by the phase gradient of an effective guiding field constructed from the sum of these individual fields.

Key assumptions:

• ❌ No antisymmetrization of trajectories
• ❌ No explicit exclusion principle
• ❌ No fermionic statistics imposed
• ✅ Only local field–particle coupling and stochastic diffusion are included.

Key Result:

Ensemble-averaged statistics reveal:

• A depletion of the pair correlation function at short distances: g(r) << 1
• A correlation hole analogous to the Fermi hole
• A suppression of joint configurations along x₁ = x₂
• Single-particle densities still obey ρ ≈ |ψ|²

These results suggest that Pauli-like exclusion may emerge as a consequence of local pilot-wave dynamics, rather than as a fundamental postulate.

📁 Script: src/simulation_1D_Pauli.py

⏱ Runtime: ~10–50 minutes (depending on params and CPU cores).

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3️⃣ Born Rule & Polarization in 2D

Objective:

Extend the pilot-wave model to 2D and analyze statistical convergence and to reveals local vectorial structures of the field (vortices).

Key Result:

• The particle correctly samples the 2D field, with empirical density ρ(x,y) ≈ |(x,y)|²
• Although ψ is a scalar field, analysis of the probability current reveals local vectorial structures (vortices), suggesting a possible connection with intrinsic angular momentum (spin)

📁 Script:

Two implementations are available:

• CPU Version: src/simulation_2D_Born_&_polarization.py

  Reference implementation (slow, not recommended for high N_runs with high correlation)

  ⏱ Runtime: ~1-5 hours (depending on params and CPU cores)

• GPU Version (Recommended): notebooks/simulation_2D_Born_&_polarization_GPU_Colab.ipynb

  High-performance implementation using Taichi Lang. (Coded for Google Colab GPU)

  

  ⏱ Runtime: ~20–100 minutes (depending on params)

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🛠️ Installation

This code relies on Numba for JIT compilation and Joblib for parallel execution.

Install dependencies:

pip install numpy matplotlib scipy numba joblib tqdm

or

pip install -r requirements.txt

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⚙️ Model Parameters

Key physical parameters (diffusion Dψ​, coupling α, memory γ, Dispersive frequency ω, Amplitude source, stochastic noise) are defined inside src/config.py. The default values are tuned to reach the quantum relaxation regime.

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▶️ Usage

Run 1st simulation:

src/simulation_1D_Born.py

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⚠️ Scientific Disclaimer

This repository presents exploratory numerical simulations in the context of quantum foundations.

The results reported here do not claim to reproduce quantum mechanics in full generality.

All conclusions are model-dependent and subject to further validation, falsification, or reinterpretation.