The Unexpected Link Between Randomness and Order

Author: Denis Avetisyan

New research reveals how feedback within stochastic systems can drive statistical convergence, blurring the lines between chance and predictability.

Numerical simulation reveals that two overdamped Brownian particles, subject to feedback control of their joint statistics, exhibit finite-time entanglement evidenced by a cross-covariance that grows logarithmically with time-specifically, <span class="katex-eq" data-katex-display="false">\mathrm{Cov}[x\_{1},x\_{2}]=2\kappa\sqrt{D\_{1}D\_{2}}\ln(t/t\_{0})</span>-demonstrating how outcome-space feedback can induce correlations even in systems governed by random motion. — Numerical simulation reveals that two overdamped Brownian particles, subject to feedback control of their joint statistics, exhibit finite-time entanglement evidenced by a cross-covariance that grows logarithmically with time-specifically, $\mathrm{Cov}[x\_{1},x\_{2}]=2\kappa\sqrt{D\_{1}D\_{2}}\ln(t/t\_{0})$ -demonstrating how outcome-space feedback can induce correlations even in systems governed by random motion.

This review establishes a dynamical framework where convergence, competition, and entanglement emerge from feedback interactions within the outcome space of classical stochastic processes.

The conventional understanding of statistical convergence typically relies on assumed independence, obscuring the dynamical origins of emergent order. This is addressed in ‘Feedback Driven Convergence, Competition, and Entanglement in Classical Stochastic Processes’, which proposes a framework where convergence arises from outcome-outcome feedback within stochastic systems. We demonstrate that this feedback not only explains the law of large numbers, but also reveals a fundamental connection between convergence, fluctuations, and a form of classical entanglement. Could this feedback-driven perspective reshape our understanding of randomness and collective behavior in complex systems?

Beyond Simple Convergence: The Limitations of Independent Observation

Many predictive models and statistical analyses fundamentally depend on the concept of convergence – the idea that repeated observations will eventually settle on a stable, predictable value. However, this cornerstone relies heavily on the assumption of Independent and Identically Distributed (IID) random events, a condition rarely met in natural systems. This means each observation is treated as wholly separate and unaffected by prior events, and all observations are drawn from the same probability distribution. While simplifying calculations and allowing for elegant mathematical proofs, this IID assumption often creates a disconnect between model predictions and real-world phenomena, as it fails to account for inherent dependencies, temporal correlations, or evolving distributions that characterize complex systems. Consequently, models built on this foundation may exhibit poor generalization, especially when applied to data exhibiting non-IID characteristics, highlighting the need for alternative frameworks that embrace dependency and dynamism.

The pervasive assumption of independent and identically distributed (IID) events, while simplifying many statistical models, often clashes with the interconnectedness of real-world phenomena. Systems ranging from financial markets to ecological networks exhibit inherent dependencies – past states influencing future outcomes – that the IID framework neglects. Consequently, predictions derived from these models can be significantly inaccurate, failing to anticipate cascading effects or emergent behaviors. More critically, reliance on IID hinders the development of genuine mechanistic insight; it treats symptoms rather than addressing underlying causal structures, obscuring the dynamic interplay that governs complex systems and limiting the ability to effectively intervene or forecast their evolution.

While mathematically rigorous, established frameworks for understanding convergence, such as Kolmogorov’s axiomatic probability theory, often fall short when applied to the intricacies of complex systems. These approaches excel at defining convergence – establishing the conditions under which a sequence of random variables approaches a limit – but provide limited insight into the mechanisms driving that convergence. Essentially, they detail what happens as a system settles, but not how it gets there. This lack of explanatory power hinders predictive capabilities in fields dealing with interdependent phenomena, where the path to stability is as crucial as the stable state itself, demanding new analytical tools that move beyond purely definitional approaches and embrace the dynamics of interaction.

Analysis of the correlation coefficient <span class="katex-eq" data-katex-display="false">r(t)=\mathrm{Cov}[x\_{1}(t),x\_{2}(t)]/\sqrt{\mathrm{Var}[x\_{1}]\mathrm{Var}[x\_{2}]}</span> from Brownian particle simulations confirms finite entanglement at intermediate times, which ultimately decays to independent diffusion with a scaling of <span class="katex-eq" data-katex-display="false">r(t)\sim(\kappa\ln t)/t</span>. — Analysis of the correlation coefficient $r(t)=\mathrm{Cov}[x\_{1}(t),x\_{2}(t)]/\sqrt{\mathrm{Var}[x\_{1}]\mathrm{Var}[x\_{2}]}$ from Brownian particle simulations confirms finite entanglement at intermediate times, which ultimately decays to independent diffusion with a scaling of $r(t)\sim(\kappa\ln t)/t$ .

A Dynamic View of Probability: Introducing the Convergence Field

The Convergence Field, denoted as $\Lambda\sigma$ , represents a generalization of standard statistical convergence by moving beyond the assessment of limits to an explicit representation of inter-outcome dependencies. Traditional convergence focuses on whether a sequence of events approaches a specific value with increasing trials; $\Lambda\sigma$ instead models the relationships between those outcomes as they unfold. This is achieved through a field structure, where each outcome is not treated in isolation but as a point within a space defined by its probabilistic connections to all other possible outcomes. Consequently, $\Lambda\sigma$ allows for the quantification of how the probability of one event is influenced by the realized states of other events, effectively capturing a dynamic, interconnected view of probabilistic systems.

Tracking the rate of change, denoted as $S$ , within the Convergence Field $\Lambda\sigma$ provides a method for analyzing probabilistic evolution beyond traditional asymptotic behavior. Instead of solely determining the long-term probability of an event, $S$ quantifies how that probability shifts over a defined sequence of observations. This dynamic assessment allows for the observation of transient phases and non-equilibrium states, revealing the time-dependent aspects of probabilistic systems. Specifically, $S$ measures the velocity of change in the probability distribution, indicating whether probabilities are converging rapidly, oscillating, or exhibiting other temporal patterns. This differs from asymptotic analysis, which focuses only on the ultimate limit of convergence, disregarding the pathway and speed of reaching that limit.

The Convergence Field framework departs from traditional statistical methods by natively accommodating feedback coupling between outcomes. Unlike models relying on the Independent and Identically Distributed (IID) assumption, this approach allows the probability of one event to be directly influenced by the realized outcomes of other events. This is achieved through the field’s structure, where $\Lambda\sigma$ represents a dynamic system where probabilities are not static but evolve based on interdependencies. Consequently, the framework enables the construction of more complex and realistic models capable of representing systems exhibiting state-dependent probabilities and non-Markovian behavior, offering a significant advantage in applications where outcomes demonstrably impact future probabilities.

Numerical tests of a two-outcome entanglement witness, utilizing <span class="katex-eq" data-katex-display="false">p(1-p)</span> as a convergence field, demonstrate entanglement violations-identified where the witness value falls below the separable baseline-and a clear convergence towards entanglement as the number of trials increases. — Numerical tests of a two-outcome entanglement witness, utilizing $p(1-p)$ as a convergence field, demonstrate entanglement violations-identified where the witness value falls below the separable baseline-and a clear convergence towards entanglement as the number of trials increases.

From Random Walks to Fokker-Planck: Modeling Dynamic Systems

The Kramers-Moyal expansion is a mathematical technique used to approximate the continuous limit of a discrete random walk. It achieves this by systematically expanding the solution to the master equation – which governs the time evolution of probabilities in a discrete state space – in powers of the time step $\Delta t$ . Through this expansion, the probability density function can be expressed as a partial differential equation, specifically a stochastic differential equation. The resulting equation, often taking the form of the Itô-Langevin equation, includes a drift term and a diffusion term that represent the continuous analog of the random walk’s step size and direction. The validity of this approximation relies on the assumption that the time step $\Delta t$ is sufficiently small, ensuring the higher-order terms in the expansion become negligible and a consistent continuous description is obtained.

The Fokker-Planck Equation is a partial differential equation that describes the time evolution of a probability density function $P(x,t)$ for a continuous stochastic process. It mathematically details how the probability distribution of a system changes over time due to random forces. Specifically, the equation takes the general form $\frac{\partial P}{\partial t} = -\frac{\partial}{\partial x} [A(x,t)P] + \frac{\partial^2}{\partial x^2} [B(x,t)P]$ , where $A(x,t)$ represents the drift and $B(x,t)$ represents the diffusion coefficient. By solving the Fokker-Planck Equation, one can determine the probability of finding the system in a particular state at a given time, making it essential in fields such as physics, chemistry, and finance for modeling systems subject to random fluctuations.

The Ornstein-Uhlenbeck (OU) process is a stochastic process described by the solution to the Fokker-Planck equation under specific parameter constraints, resulting in a Gaussian probability distribution that evolves towards a stable mean. Mathematically, the process is defined by $dx_t = -\theta x_t dt + \sigma dW_t$ , where θ represents the rate of mean reversion, σ is the volatility, and $dW_t$ is a Wiener process. This formulation ensures that any initial deviation from the long-term mean will decay exponentially over time, making the OU process highly suitable for modeling systems exhibiting mean reversion, such as interest rates, commodity prices, and terminal velocities. The process demonstrates dynamic convergence as the probability density function collapses around the mean value as time progresses, providing a quantifiable example of how stochastic systems can approach equilibrium.

Beyond Equilibrium: Detecting and Understanding Finite-Time Entanglement

The detection of Finite-Time Entanglement, a subtle correlation arising from dynamic interactions and cross-diffusion, relies on specialized tools like the Entanglement Witness. This phenomenon challenges the conventional understanding of entanglement, demonstrating that correlated behavior doesn’t necessarily require a system to reach long-term equilibrium. Instead of persistent entanglement, these systems exhibit transient correlations – fleeting links established through the exchange of information or energy. The Entanglement Witness provides a quantifiable measure of this transient link, identifying instances where particles, even those seemingly governed by random Brownian motion, become momentarily correlated. This ability to detect such short-lived connections opens new avenues for investigating complex systems where information processing and energy transfer occur in non-equilibrium states, offering insights beyond traditional equilibrium-based models.

The emergence of correlation within systems traditionally understood as purely random, like those exhibiting Brownian motion, challenges the conventional link between entanglement and long-term equilibrium. Recent investigations reveal that transient entanglement – a fleeting connection between particles – can arise even when a system never settles into a stable, balanced state. This phenomenon demonstrates that correlation isn’t solely a property of systems at equilibrium, but can be a dynamic, temporary feature of interaction itself. The detection of this finite-time entanglement suggests that information exchange and cooperative behavior are possible even in environments characterized by constant fluctuation, offering a new perspective on how complex systems process information and respond to change without relying on established, static order. This challenges the notion that sustained, long-lived connections are a prerequisite for cooperative phenomena.

Analysis reveals that the standard relationship between variance decay and particle number, typically expressed as 1/m, undergoes a significant modification in systems exhibiting feedback. Researchers observed that this decay is instead governed by the equation 1/[(c-1)m], where ‘c’ quantifies the strength of the feedback mechanism. This alteration indicates that feedback effects fundamentally reshape how uncertainty diminishes within the system; a stronger feedback – a larger value of ‘c’ – results in a slower rate of variance decay. Consequently, systems with pronounced feedback retain information for a longer duration than predicted by traditional Brownian motion models, highlighting the importance of considering these interactions when characterizing dynamic processes and information transfer in complex environments.

The correlation between Brownian particles, traditionally understood as fleeting and random, exhibits a nuanced temporal behavior beyond simple decay. Investigations reveal this correlation doesn’t simply diminish to zero, but rather evolves according to the function $(\kappa \ln t) / t$ , where κ is a constant defining the strength of this transient link. This indicates a finite-time correlation – a measurable connection exists between the particles for a limited duration, even as the system appears to approach equilibrium. Crucially, this correlation vanishes asymptotically as time progresses, meaning it fades to nothing in the long run; however, its temporary existence highlights that even in seemingly disordered systems, brief but significant connections can emerge, driven by the inherent dynamics of Brownian motion and influencing how information is exchanged within the system.

Numerical investigations into dynamic correlations reveal a distinctive temporal evolution of cross-covariance between Brownian particles. Simulations demonstrate that this cross-covariance increases with time according to a logarithmic function, specifically $2κD_1D_2ln(t/t_0)$ , where κ represents a constant, and $D_1$ and $D_2$ denote the diffusion coefficients of the particles. This logarithmic growth signifies that the correlation isn’t simply a product of random collisions, but arises from a persistent, albeit transient, interaction. The observed alignment between simulation results and theoretical predictions confirms the presence of finite-time entanglement, highlighting how even systems exhibiting Brownian motion can display short-lived, yet quantifiable, correlations that vanish as time progresses, but are demonstrably present at earlier stages.

The ability to detect and interpret transient correlations holds significant implications for modeling a wide range of complex systems operating far from equilibrium. Many natural and engineered systems, from biological networks to advanced materials, rely on the exchange and processing of information under dynamic, non-equilibrium conditions; understanding how correlations emerge and decay within these systems is therefore paramount. These fleeting connections, even those present in seemingly random processes like Brownian motion, can facilitate coordinated behavior and efficient information transfer. By characterizing these finite-time entanglement phenomena, researchers gain insights into the mechanisms governing collective dynamics, potentially leading to advancements in fields such as stochastic thermodynamics, sensor development, and the design of adaptive materials – all areas where harnessing non-equilibrium processes is key to achieving enhanced functionality and performance.

The study illuminates how convergence arises not as a pre-defined state, but as an emergent property of dynamic systems. This resonates with Thoreau’s observation: “It is not enough to be busy; so are the ants. The question is: What are we busy with?” The research demonstrates that simply having a stochastic process isn’t sufficient; the nature of the feedback interactions-the ‘what’ of the process-determines whether convergence, and the associated fluctuations and entanglement, will manifest. The framework reveals that structure dictates behavior, echoing the principle that a system’s internal dynamics, its ‘busy-ness,’ are key to understanding its emergent properties.

Where Do We Go From Here?

The presented framework shifts the locus of statistical convergence from an assumed truth to an emergent property-a consequence of interaction, not a prerequisite for it. This is not merely a redefinition, but a subtle inversion with potentially far-reaching consequences. The current work demonstrates this within the constrained realm of classical stochastic processes, but the architecture of feedback and resulting entanglement-like correlations suggests a path toward understanding more complex systems. The crucial question becomes: how robust is this emergent convergence when the underlying dynamics are non-Markovian, or when the ‘outcome space’ is itself evolving?

Naturally, extending this framework to quantum systems presents a tempting, though likely treacherous, path. The observed correlations, while mathematically analogous to quantum entanglement, arise from purely classical mechanisms. However, the very act of modeling these classical systems through feedback loops reveals a structural similarity to quantum measurement-a continuous ‘collapsing’ of possibilities driven by interaction. This begs the question: are the boundaries between classical and quantum descriptions more porous than conventionally assumed, or is this simply a consequence of seeking elegant, structurally similar explanations?

Any optimization, any attempt to ‘improve’ a system, inevitably creates new tension points. The drive for convergence, for predictable behavior, introduces vulnerabilities. The architecture is the system’s behavior over time, not a diagram on paper. Future work should focus not just on achieving convergence, but on characterizing the nature of those emergent tensions, and understanding how they propagate through the system-a holistic view, rather than a reductionist one.

Original article: https://arxiv.org/pdf/2601.02388.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

Beyond Simple Convergence: The Limitations of Independent Observation

A Dynamic View of Probability: Introducing the Convergence Field

From Random Walks to Fokker-Planck: Modeling Dynamic Systems

Beyond Equilibrium: Detecting and Understanding Finite-Time Entanglement

Where Do We Go From Here?

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