Relative Entropy Revivals Caused by Memory-Induced Loss of Divisibility in Classical Non-Markovian Dynamics

Relative-entropy revivals and negative entropy-production rates are established signatures of memory effects in non-Markovian dynamics. The present paper addresses a more specific question: what minimal dynamical mechanism produces entropy overshoot in classical stochastic relaxation? We define entropy overshoot as a transient increase of the Kullback--Leibler divergence from a stationary distribution during a relaxation process that ultimately returns to equilibrium. Starting from finite-state generalized master equations with memory kernels, we analyze the effective time-local generator governing the observed dynamics. The standard monotonic decay of relative entropy is recalled as a reference property of reversible Markov dynamics. We then show that memory can destroy stochastic divisibility by producing transiently negative effective rates. Once the evolution ceases to be divisible into intermediate Markov steps, the usual relative-entropy contraction argument no longer applies, and entropy overshoot can occur. The contribution of the paper is therefore not the introduction of another witness of non-Markovianity, but the identification of a minimal structural mechanism and its phase structure. The mechanism is illustrated with two- and three-state stochastic models, where the competition between the intrinsic relaxation time and the memory time separates monotonic relaxation from overshoot. Numerical phase diagrams show that the overshoot region expands when memory persists on the relaxation timescale. These results clarify the relation between classical information backflow, negative entropy-rate behavior, and memory-induced loss of divisibility. They also distinguish the present mechanism-based interpretation from previous work that used relative-entropy revivals or entropy-production rates primarily as indicators of non-Markovianity.