Weak-Memory Dynamics in Discrete Time

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.