Due to the second principle of thermodynamics, the time dependence of entropy for all kinds of systems under all kinds of physical circumstances always thrives interest.

The logistic map +1=1−2∈[−1,1](∈[0,2]) is neither large, since it has only one degree of freedom, nor closed, since it is dissipative. It exhibits, nevertheless, a peculiar time evolution of its natural entropy, which is the additive Boltzmann–Gibbs-Shannon one, =−∑=1ln, for all values of  for which the Lyapunov exponent is positive, and the nonadditive one =1−∑=1−1 with =0.2445… at the edge of chaos, where the Lyapunov exponent vanishes,  being the number of windows of the phase space partition. We numerically show that, for increasing time, the phase-space-averaged entropy overshoots above its stationary-state value in all cases._x000D_

However, when →∞, the overshooting gradually disappears for the most chaotic case (=2), whereas, in remarkable contrast, it appears to monotonically diverge at the Feigenbaum point (=1.4011…). Consequently, the stationary-state entropy value is achieved from above, instead of from below, as it could have been a priori expected._x000D_

These results raise the question whether the usual requirements – large, closed, and for generic initial conditions – for the second principle validity might be necessary but not sufficient.

C. Tsallis, E.P. Borges, Time evolution of nonadditive entropies: The logistic map, Chaos, Solitons & Fractals 171 (2023) 113431.