Weak-Memory Dynamics in Discrete Time

TL;DR

This paper investigates how weak memory effects in linear discrete-time systems can be simplified to first-order dynamics, providing a mathematical framework and practical examples for such reductions.

Contribution

It introduces a theorem that systematically reduces higher-order discrete dynamics with weak memory to a first-order form in linear systems.

Findings

Identification of a weak-memory regime where reduction is valid

Mathematical theorem formalizing the reduction process

Application to stochastic Floquet and quantum collisional models

Abstract

Discrete dynamics arise naturally in systems with broken temporal translation symmetry and are typically described by first-order recurrence relations representing classical or quantum Markov chains. When memory effects induced by hidden degrees of freedom are relevant, however, higher-order discrete evolution equations are generally required. Focusing on linear dynamics, we identify a well-delineated weak-memory regime where such equations can, on an intermediate time scale, be systematically reduced to a unique first-order counterpart acting on the same state space. We formulate our results as a mathematical theorem and work out two examples showing how they can be applied to stochastic Floquet dynamics under coarse-grained and quantum collisional models.