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The Shannon entropy as a measure of diffusion in multidimensional dynamical systems

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Abstract

In the present work, we introduce two new estimators of chaotic diffusion based on the Shannon entropy. Using theoretical, heuristic and numerical arguments, we show that the entropy, S, provides a measure of the diffusion extent of a given small initial ensemble of orbits, while an indicator related with the time derivative of the entropy, \(S'\), estimates the diffusion rate. We show that in the limiting case of near ergodicity, after an appropriate normalization, \(S'\) coincides with the standard homogeneous diffusion coefficient. The very first application of this formulation to a 4D symplectic map and to the Arnold Hamiltonian reveals very successful and encouraging results.

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Figure taken from Cincotta et al. (2018)

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Notes

  1. Let us recall that Shannon entropy for a discrete-valued and discrete-time stochastic process has an analog in ergodic theory of dynamical systems; the Shannon entropy defined in (10) naturally leads to the so-called metric or Kolmogrov Sinai entropy for a given orbit of a map when considering the refinements of the partition generated by the map when \(t\rightarrow \infty \), as for instance Arnol’d and Avez (1989) and Lesne (2014) show.

  2. Note that to compute \(\langle \delta {y^2_j}\rangle \) the actions are not restricted to the unit square.

  3. The estimation of the exponent \(\beta \) and the corresponding diffusion coefficient by a power law fit seems not to work when dealing with such finite times, the value of \(\beta \) being highly sensitive to both the total motion time and \(\varDelta t\) (see however next section).

  4. Note that their intersections in action space is different.

  5. It is well known that the theoretical diffusion rate depends exponentially on \(-1/\sqrt{V_{mn}}\) where \(V_{mn}\) stands for the amplitude of the above considered resonances.

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Acknowledgements

This work was supported by grants from Consejo Nacional de Investigaciones Científicas y Técnicas de la República Argentina (CONICET), the Universidad Nacional de La Plata and Instituto de Astrofísica de La Plata. We acknowledge two anonymous reviewers for the valuable comments and suggestions that allow us to improve this manuscript.

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  1. Grupo de Caos en Sistemas Hamiltonianos, Facultad de Ciencias Astonómicas y Geofísicas, Universidad Nacional de La Plata and Instituto de Astrofísica de La Plata (CONICET-UNLP), La Plata, Argentina

    C. M. Giordano & P. M. Cincotta

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  1. C. M. Giordano
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  2. P. M. Cincotta
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Correspondence to P. M. Cincotta.

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Giordano, C.M., Cincotta, P.M. The Shannon entropy as a measure of diffusion in multidimensional dynamical systems. Celest Mech Dyn Astr 130, 35 (2018). https://doi.org/10.1007/s10569-018-9832-x

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