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Weak-Memory Dynamics in Discrete Time.

Hugues Meyer1, Kay Brandner1

  • 1University of Nottingham, University of Nottingham, School of Physics and Astronomy, Nottingham NG7 2RD, United Kingdom and Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems, Nottingham NG7 2RD, United Kingdom.

Physical Review Letters
|June 22, 2026
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Summary
This summary is machine-generated.

This study introduces a method to simplify complex discrete dynamics with memory effects into simpler first-order equations. This applies to classical and quantum systems, aiding in the analysis of stochastic Floquet dynamics.

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Area of Science:

  • Physics
  • Complex Systems
  • Mathematical Modeling

Background:

  • Discrete dynamics, often modeled by first-order recurrence relations like Markov chains, typically assume no memory effects.
  • Hidden degrees of freedom can introduce memory, necessitating higher-order discrete evolution equations.
  • Analyzing systems with memory effects in discrete dynamics is challenging.

Purpose of the Study:

  • To identify a regime in linear discrete dynamics where higher-order equations with memory effects can be simplified.
  • To develop a systematic method for reducing complex discrete evolution equations to a first-order counterpart.
  • To demonstrate the applicability of these findings to specific models in stochastic Floquet dynamics.

Main Methods:

  • Focusing on linear discrete dynamics, the study defines and analyzes a weak-memory regime.
  • Mathematical theorem formulation to systematically reduce higher-order equations to a first-order equivalent.
  • Application of the developed method to stochastic Floquet dynamics, including coarse-grained and quantum collisional models.

Main Results:

  • A well-delineated weak-memory regime is identified where higher-order discrete dynamics simplify.
  • Higher-order equations with memory effects can be systematically reduced to a unique first-order equation on the same state space over intermediate timescales.
  • The theoretical results are validated through practical examples in stochastic Floquet dynamics.

Conclusions:

  • The weak-memory regime provides a powerful framework for simplifying complex discrete dynamical systems.
  • The developed reduction method offers a valuable tool for analyzing systems with memory effects, particularly in quantum and stochastic settings.
  • This work bridges the gap between higher-order discrete dynamics and the well-understood first-order dynamics, enhancing analytical tractability.