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Power-Law Relaxation and Cumulative Information

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Abstract

We associate the cumulative information of a system relaxing towards equilibrium with a divergent integral when a power-law relaxation holds. We discuss and illustrate numerically how this implies that a system that relaxes to equilibrium through a power-law has a cumulative information content that progressively diverges from that of its equilibrium realization to which it is relaxing. Our findings shed light on some aspects of weak ergodicity breaking and suggest that power-laws imply a form of complexity that does not require dissipation or built-in disorder.

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Notes

  1. Our attention is on systems that relax with traits of out-of-equilibrium physics, which means that each trajectory has memory. Should we evaluate S(t) along the single trajectories of m(t) (or equivalently s(t)), the information content would be intrinsically less than the S(t) of Eq. (5), weakening any form of argument based on the divergence of a time integration. Therefore, we are required to underline that Eq. (5) holds and is meaningful, representing the cumulative information content of the stochastic process, in so much that the single p n (t) are statistically independent from the same quantities at previous times.

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Acknowledgements

S. Di Sabatino is now with LPT-IRSAMC, Université Paul Sabatier—Toulouse, France. The research leading to these results was supported by funding from the Italian Ministry of Research (MIUR) through the “Futuro in Ricerca” FIRB grant PHOCOS-RBFR08E7VA. Partial funding was received through the SMARTCONFOCAL project of the Regione Lazio and through the PRIN project no. 2009P3K72Z.

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Authors and Affiliations

  1. Dipartmento di Fisica, Università di Roma “La Sapienza”, 00185, Rome, Italy

    P. Di Porto, S. Di Sabatino & E. DelRe

  2. Department of Applied Physics, California Institute of Technology, Pasadena, CA, 91125, USA

    B. Crosignani

  3. IPCF-CNR, Università di Roma “La Sapienza”, 00185, Rome, Italy

    E. DelRe

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  1. P. Di Porto
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  2. S. Di Sabatino
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  3. B. Crosignani
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  4. E. DelRe
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Correspondence to E. DelRe.

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Di Porto, P., Di Sabatino, S., Crosignani, B. et al. Power-Law Relaxation and Cumulative Information. J Stat Phys 153, 479–485 (2013). https://doi.org/10.1007/s10955-013-0840-7

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  • DOI: https://doi.org/10.1007/s10955-013-0840-7

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