Relaxation to equilibrium is one of the central topics of nonequilibrium statistical physics. In Markovian processes with detailed balance, the Kullback-Leibler distance from the stationary distribution decreases monotonically and serves as a Lyapunov function. Yet many real systems, described by Nakajima-Zwanzig projection methods or generalized Langevin equations, carry memory, so the future depends not only on the current state but also on history. Non-Markovianity had previously been detected through revivals of relative entropy or negative entropy production rates, but the specific mechanism causing a temporary rise in entropy remained unclear. This work fills that gap by isolating a minimal structural mechanism in finite-dimensional stochastic systems.
The author considers a generalized master equation with an exponential memory kernel K(t)=γe^{-γt}R, where R is a Markovian generator and γ controls the memory time. Such a kernel admits a Markovian embedding: an auxiliary variable Y(t) is introduced, and the system (P,Y) evolves locally in time. The embedding yields an effective generator $R_{eff}$(t), which can have negative off-diagonal elements, indicating a loss of divisibility. For the analysis, minimal models with two and three states are constructed, and the equations are solved numerically by two methods: direct integration of the integro-differential equation and via the embedded system. Parameter scans produce phase diagrams.
It was found that for short memory (γ≫λ) the dynamics is close to Markovian, and relative entropy decreases monotonically. For long memory (γ≲λ), delayed feedback from the hidden variable can temporarily reverse the direction of relaxation, causing an entropy overshoot. In the two-state model, the overshoot appears as an oscillatory transient whose amplitude grows with increasing memory time. In the three-state model, the phase diagrams show a clear separation of regimes, with the boundary matching the estimate γ~λ. The results confirm that the entropy overshoot is a robust consequence of the loss of divisibility rather than an artifact of special initial conditions.
The work offers a classical stochastic interpretation of information backflow. Unlike earlier studies that used entropy revivals as indicators of non-Markovianity, it identifies a concrete mechanism: memory generates negative effective rates that signal a loss of divisibility, which makes the standard proof of relative entropy contraction inapplicable. This clarifies why the overshoot is absent in ordinary reversible Markov chains but possible in systems with memory. The resulting structural view sets this work apart from the indicator-based approach and opens the way to studying more general memory kernels and continuous state spaces.
A natural continuation is to consider continuous state spaces, non-exponential memory kernels, and semi-Markov waiting-time distributions. It would also be interesting to compare different information measures and to determine which transient features depend on the chosen metric and which are fixed solely by the structure of divisibility. An extension to quantum stochastic dynamics is possible, where similar mechanisms may show up in open systems.
The results may be useful in biophysics, chemical kinetics, and transport theory, where non-Markovian effects matter. They also provide a simple tool for analyzing relaxation in complex systems with memory.
The next step could be to test the timescale criterion against experimental data or in more realistic models. It is also important to investigate how universal the connection between negative effective rates and entropy overshoot is for arbitrary kernels.
In stochastic thermodynamics, the question remains open as to which microscopic mechanisms break the monotonicity of entropy functionals in non-Markovian systems. This work links the loss of divisibility to information backflow, bringing us closer to understanding the general principles of nonequilibrium relaxation with memory.
In the Markovian case, relative entropy acts as a Lyapunov function that guarantees an irreversible approach to equilibrium. A loss of divisibility means the evolution cannot be split into successive Markovian steps, and then the familiar arrow of time can briefly turn backwards.