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

TL;DR

This paper develops a feedback-driven stochastic framework explaining how statistical convergence, competition, and entanglement emerge in classical processes without assuming independence, unifying these phenomena through a single theory.

Contribution

It introduces a dynamical theory of convergence based on outcome feedback, deriving new descriptions and witnesses for entanglement in classical stochastic systems.

Findings

Outcome-outcome feedback leads to convergence and entanglement.

Variance-based measures can detect outcome-space entanglement.

Numerical experiments confirm the theoretical predictions.

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.