Time evolution of nonadditive entropies: The logistic map

Constantino Tsallis and Ernesto P. Borges

Chaos, Solitons & Fractals, 2023, vol. 171, issue C

Abstract: 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 xt+1=1−axt2∈[−1,1](a∈[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, SBG=−∑i=1Wpilnpi, for all values of a for which the Lyapunov exponent is positive, and the nonadditive one Sq=1−∑i=1Wpiqq−1 with q=0.2445… at the edge of chaos, where the Lyapunov exponent vanishes, W 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. However, when W→∞, the overshooting gradually disappears for the most chaotic case (a=2), whereas, in remarkable contrast, it appears to monotonically diverge at the Feigenbaum point (a=1.4011…). Consequently, the stationary-state entropy value is achieved from above, instead of from below, as it could have been a priori expected. 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.

Keywords: Classical statistical mechanics; Nonlinear dynamics and chaos; Low-dimensional chaos (search for similar items in EconPapers)
Date: 2023
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Citations: View citations in EconPapers (1)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:chsofr:v:171:y:2023:i:c:s0960077923003326

DOI: 10.1016/j.chaos.2023.113431

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