Feedback Driven Convergence, Competition, and Entanglement in Classical Stochastic Processes

TL;DR

The paper proposes a feedback‑driven dynamical framework for statistical convergence in which the law of large numbers emerges from outcome‑to‑outcome coupling rather than i.i.d. assumptions. By introducing the convergence field $\Lambda_\sigma$ with derivative $S$, empirical frequencies become entangled through competitive drift and stochastic fluctuations, which are formalized via an Itô–Langevin and a Fokker–Planck description on the simplex; in the symmetric limit this reduces to a time‑dependent Ornstein–Uhlenbeck process with variance scaling $\mathrm{Var}[\Delta](m) \sim 1/((c-1)m)$. The authors construct variance‑based witnesses for entanglement in binary sequences and coupled Brownian trajectories, and demonstrate finite‑time cross‑diffusion and classical entanglement in two‑particle Brownian systems, all arising from the same feedback mechanism. Collectively, the results unify convergence, fluctuation, and entanglement as manifestations of a single dynamical principle, with broad implications for finite‑sample randomness, correlated dynamics, and foundational probability.

Abstract

We present a dynamical theory of statistical convergence in which the law of large numbers arises from outcome-outcome feedback rather than assumed independence. Defining the convergence field and its derivative, we show that empirical frequencies evolve through coupling, producing competition, finite-m fluctuations, and classical entanglement. Using the Kramers-Moyal expansion, we derive an Ito-Langevin and Fokker-Planck description, reducing in the symmetric regime to a time-dependent Ornstein-Uhlenbeck process. We propose variance-based witnesses that detect outcome-space entanglement in both binary sequences and coupled Brownian trajectories, and confirm entanglement through numerical experiments. Extending the formalism yields multi-outcome feedback dynamics and finite-time cross-diffusion between Brownian particles. The results unify convergence, fluctuation, and entanglement as consequences of a single feedback-driven stochastic principle.

Figures (3)

• Figure 1: Numerical test of the two–outcome entanglement witness (Eq. 67). Top: OU–driven empirical frequency $p_m$ of outcome 1. Second: Two–time witness $W_{\mathrm{pair}}$ compared with the separable baseline $W_{\mathrm{pair}}^{\mathrm{sep}}(p)$ (Eqs. 61, 64); violations indicate entanglement. Third: Windowed convergence field $\Lambda_\sigma=p(1-p)$ relative to $\Lambda_{\sigma0}=1/4$. Bottom: Entanglement regions where $W_{\mathrm{pair}}<W_{\mathrm{pair}}^{\mathrm{sep}}(p)$. The x-axis in case represents number of trials.
• Figure 2: Numerical simulation of two overdamped Brownian particles with feedback-controlled joint outcome statistics, showing the time evolution of the cross-covariance $\mathrm{Cov}[x_1(t),x_2(t)]$. The data confirm the predicted logarithmic growth $\mathrm{Cov}[x_1,x_2] = 2\kappa\sqrt{D_1D_2}\ln(t/t_0)$, demonstrating finite-time entanglement arising from outcome-space feedback.
• Figure 3: Correlation coefficient $r(t) = \mathrm{Cov}[x_1(t),x_2(t)]/\sqrt{\mathrm{Var}[x_1]\mathrm{Var}[x_2]}$ obtained from the same simulation as figure \ref{['fig:1']}. Results show the predicted asymptotic scaling $r(t) \sim (\kappa \ln t)/t$, confirming that classical entanglement between the Brownian particles is finite at intermediate times but vanishes in the long-time limit, recovering independent diffusion.