Structural Theory of Information Backflow in Non-Markovian Relaxation: TC/TCL Formalism and Minimal Phase Diagrams

TL;DR

This paper presents a structural framework for understanding information backflow in non-Markovian relaxation processes using TC/TCL formalisms, with explicit phase diagrams and analysis of classical and quantum models.

Contribution

It introduces a unified theory connecting memory kernels, backflow phenomena, and transient regimes in non-Markovian dynamics, including explicit phase diagrams and model-independent analysis.

Findings

Derived conditions for absence of backflow based on divisibility and relaxation rates.

Decomposed backflow into classical and quantum contributions for classification.

Analyzed minimal models, recovering fractional relaxation as a universal behavior.

Abstract

We develop a structural theory of information backflow in minimal non-Markovian relaxation processes within the framework of nonequilibrium statistical mechanics. The approach is based on the time-convolution (TC) and time-convolutionless (TCL) projection-operator formalisms for reduced dynamics and on the doubling construction of non-equilibrium thermo field dynamics, which provides an embedding representation of dissipative evolution. We introduce a general backflow functional associated with a time-dependent information measure and derive generator-based sufficient conditions for the absence of backflow in terms of divisibility properties and effective relaxation rates. This allows a direct connection between memory kernels in generalized master equations and observable transient phenomena such as entropy overshoot and revival. Furthermore, we propose a decomposition of backflow into classical mixing and intrinsic contributions in the doubled representation, leading to a unified classification of transient regimes. Minimal classical and quantum two-state models are analyzed as analytically tractable examples, yielding explicit phase diagrams and recovering Mittag-Leffler-type fractional relaxation as a universal envelope of non-Markovian damping. The framework provides a constructive TC-to-TCL procedure for extracting effective rates and organizing memory-induced phenomena in a model-independent manner.