A Lindblad-Pauli Framework for Coarse-Grained Chaotic Binary-State Dynamics
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914219889262592 |
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| author | Qiu, Yicong Zheng, Qiye |
| author_facet | Qiu, Yicong Zheng, Qiye |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_17200 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Lindblad-Pauli Framework for Coarse-Grained Chaotic Binary-State Dynamics Qiu, Yicong Zheng, Qiye Chaotic Dynamics Quantum Physics 37D45, 37M10, 60J20, 81P68 G.1.10; I.6.5 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. |
| title | A Lindblad-Pauli Framework for Coarse-Grained Chaotic Binary-State Dynamics |
| topic | Chaotic Dynamics Quantum Physics 37D45, 37M10, 60J20, 81P68 G.1.10; I.6.5 |
| url | https://arxiv.org/abs/2512.17200 |