Generalized time-coarse graining and emergent Lindblad-like dynamics in classical systems

Driven Hamiltonian systems are a central model in many fields, enabling important applications ranging from robotics to quantum computing to condensed matter physics. However, their theoretical description is complex, often non-integrable, and involves multiple timescales, making them challenging to model. Various perturbative techniques have been developed to study these systems—Floquet theory for periodic drives [1], "Kamiltonian" perturbation methods in plasma physics [2], and time-coarse graining in quantum mechanics [3]. While some connections among Hamiltonian methods are known, their structural relationships and links to non-Hamiltonian approaches have been unexplored. In this work, we introduce a unified Lie-theoretic framework that reveals a common mathematical structure, extending existing connections and uncovering previously unknown relationships within different perturbation theories.

Our formulation is agnostic to the physical nature of the system, allowing techniques from one domain to be applied to others—for instance, using time-coarse graining and Lindblad-like equations in classical systems. This unified framework simplifies the application of both Hamiltonian and non-Hamiltonian perturbation methods, providing a precise comparison between them. Remarkably, it reveals that Lindblad-like operators naturally emerge in classical systems, analogous to the quantum case, and clarifies that their appearance is determined by the spectral structure and the breaking of the Lie-algebraic nature of the system, rather than its quantum nature. This work not only dispels misconceptions about the role of Lindblad terms in time-coarse graining but also offers a new approach to studying driven systems across both classical and quantum domains.

[1] - Bukov, M. et. al., Universal high-frequency behavior of periodically driven systems: from dynamical stabilization to Floquet engineering. Advances in Physics, 64(2), 139–226, (2015).

[2] - Venkatraman, J. et. al. (2022), Static Effective Hamiltonian of a Rapidly Driven Nonlinear Systems, Physical Review Letters, 129, 100601 (2022).

[3] - Bello L., et al., Systematic time-coarse graining for driven quantum systems, arXiv:2407.06068, To appear in Physical Review Applied, (2025).