A Lindblad-Pauli Framework for Coarse-Grained Chaotic Binary-State Dynamics

TL;DR

This paper tackles the difficulty of reliably reducing chaotic bistable dynamics to binary symbols by embedding left/right statistics into a 2x2 density matrix and modeling inter-well switching with a two-rate GKSL generator, ensuring CPTP consistency. It introduces a soft-partition embedding that yields a Bloch half-disk state space with a measurable overlap c(ε) to diagnose coarse-graining quality, and provides a falsifiable diagnostic pipeline including order tests, Chapman-Kolmogorov consistency, run-length statistics, and windowed stationarity checks. Analytically, it derives closed-form GKSL evolution, steady states, and a Kraus representation equivalent to a generalized amplitude-damping channel with dephasing and rotation, while offering CPTP-preserving extensions for memory effects. A practical numerical pipeline accompanies these results, guiding validation on Duffing simulations and outlining extensions (higher-order memory, HMMs, semi-Markov, time-dependent rates) when first-order Markov closure fails, thereby delivering a rigorous, extensible framework for coarse-grained chaotic dynamics with clear diagnostic criteria and uncertainty quantification.

Abstract

Coarse-graining a chaotic bistable oscillator into a binary symbol sequence is a standard reduction, but it often obscures the geometry of the reduced state space and structural constraints of physically meaningful stochastic evolution. We develop a two-state framework that embeds coarse-grained left/right statistics of the driven Duffing oscillator into a $2\times2$ density-matrix representation and models inter-well switching by a two-rate Gorini--Kossakowski--Sudarshan--Lindblad (GKSL) generator. For diagonal states the GKSL dynamics reduces to the classical two-state master equation.The density-matrix language permits an operational ``Bloch half-disk'' embedding with overlap parameter $c(\varepsilon)$ quantifying partition fuzziness; the GKSL model is fitted to diagonal marginals treating $c(\varepsilon)$ as diagnostic. We derive closed-form solutions, an explicit Kraus representation (generalized amplitude damping with dephasing and rotation), and practical diagnostics for the time-homogeneous first-order Markov assumption (order tests, Chapman--Kolmogorov consistency, run-length statistics, stationarity checks). When higher-order memory appears, we extend the framework via augmented Markov models, constructing CPTP maps through discrete-time Kraus representations; continuous-time GKSL generators may not exist for all empirical transition matrices. We provide a numerical pipeline with templates for validating the framework on Duffing simulations. The density-matrix formalism is an organizational convenience rather than claiming quantum-classical equivalence.

Figures (6)

• Figure 1: Representative time series $x(t)$ in a chaotic bistable regime, with the two wells indicated (e.g. $x<0$ and $x>0$ regions) and stroboscopic sampling points marked. This figure justifies that the chosen parameter set exhibits inter-well hopping on the observation horizon.
• Figure 2: Example segment of the binary symbol sequence $\{S_n\}$ (or equivalently the sign of $x(t_n)$) illustrating dwell times and switching events.
• Figure 3: Population relaxation: comparison of the theoretical curve $p_L(t)=p_L^\infty+(p_0-p_L^\infty)e^{-\Gamma t}$ (Theorem \ref{['thm:pop_solution']}) to an ensemble-averaged estimate of $p_L(t)$ from simulation, started from biased initial conditions.
• Figure 4: Run-length (dwell-time) distributions testing the memoryless property of first-order Markov chains. (a) Left-well run lengths: empirical histogram (blue bars) compared to geometric distribution $\Pr(R_L=r)=(1-P_{LR})^{r-1}P_{LR}$ (red curve) with transition probability $\hat{P}_{LR}$ estimated from data. (b) Right-well run lengths: empirical histogram (orange bars) compared to geometric distribution $\Pr(R_R=r)=(1-P_{RL})^{r-1}P_{RL}$ (blue curve) with transition probability $\hat{P}_{RL}$ estimated from data. Deviations from geometric laws diagnose non-Markov or time-inhomogeneous behavior at the sampling scale. Parameters: $\alpha=1.0$, $\beta=1.0$, $\delta=0.15$, $\gamma_0=0.3$, $\omega=1.0$.
• Figure 5: Chapman--Kolmogorov consistency test comparing empirical and predicted two-step transition probabilities. (a) Empirical two-step matrix $\widehat{\mathbf{P}}^{(2)}$ estimated from symbol pairs separated by two periods. (b) Predicted two-step matrix $\widehat{\mathbf{P}}^2$ computed by squaring the one-step matrix. (c) Element-wise discrepancy $|\widehat{\mathbf{P}}^{(2)}-\widehat{\mathbf{P}}^2|$ with Frobenius norm $\Delta_{\mathrm{CK}}$. Small discrepancies indicate Chapman--Kolmogorov consistency, validating the Markov property across time scales (Section \ref{['sec:markov_tests']}). Parameters: $\alpha=1.0$, $\beta=1.0$, $\delta=0.15$, $\gamma_0=0.3$, $\omega=1.0$.

Theorems & Definitions (50)

• Remark 2.1: Optional stochastic extension
• Definition 2.2: Binary symbol sequence
• Remark 2.3: Classical diagonal states
• Definition 2.4: Soft partition / membership functions
• Definition 2.5: Pointwise embedded two-state vector
• Remark 2.6: Why the square-root embedding?
• Definition 2.7: Ensemble or window-reduced state
• Proposition 2.8: Matrix elements and overlap
• Definition 2.9: Windowed empirical measure
• Remark 2.10: Discrete Poincaré sampling