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

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.

💡 Research Summary

The paper presents a comprehensive structural theory of information backflow in minimal non‑Markovian relaxation processes, unifying the time‑convolution (TC) and time‑convolutionless (TCL) projection‑operator formalisms with the doubled‑Hilbert‑space construction of non‑equilibrium thermo‑field dynamics (NETFD). The authors first introduce the reduced dynamics of a system interacting with an environment, describing it either by a TC master equation with an explicit memory kernel K(t‑s) or by a TCL master equation with a time‑local generator G(t). While the TC form makes memory effects transparent, the TCL form encodes the same physics in a generator that is more convenient for extracting effective rates and for analytical work.

A central contribution is the definition of an information‑backflow functional
(N_I = \int_0^\infty \Theta(\dot I(t)),\dot I(t),dt),
where (I(t)) is any scalar information measure (von‑Neumann entropy, relative entropy, trace distance, etc.) and (\Theta) is the Heaviside step function. (N_I) quantifies the total amount of increase of the chosen information quantity during the relaxation; (N_I=0) signals monotonic decay, while (N_I>0) indicates genuine backflow.

The authors then derive sufficient conditions for the absence of backflow. In the quantum case, if the TCL generator can be written at all times in the time‑dependent Gorini‑Kossakowski‑Sudarshan‑Lindblad (GKSL) form
(G(t)\rho = -i