The orchestra starts playing in unison, but a hidden conductor changes the direction of the baton for a second. In the same way, a system striving toward equilibrium suddenly takes a step back.

In Markovian processes, where the future depends only on the present, the distance to equilibrium, measured by relative entropy, decreases monotonically, like sand in an hourglass. But if a system remembers its history, that arrow of time can waver. Such revivals of entropy had been noticed before, but no one understood what minimal mechanism causes them.

The author built the simplest models with two and three states and an exponential memory kernel. It turned out that when the memory time is comparable to the relaxation time, the effective generator acquires negative off-diagonal elements. This means a loss of stochastic divisibility: the evolution can no longer be split into successive Markovian steps. Then the standard proof of monotonicity stops working, and relative entropy can temporarily grow, producing an overshoot.

In the two-state model, the overshoot appears as an oscillatory transient whose amplitude grows as the memory time increases. In the three-state model, phase diagrams clearly separate the regimes, and the boundary corresponds to the estimate γ ~ λ. This confirms that the entropy overshoot is a robust consequence of the loss of divisibility, not an artifact of special initial conditions.

The orchestra is playing in unison again. But now it is clear that the brief stumble was not an accident, but the work of memory, which for a moment returned the system to its past.