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H-theorems for the Brownian motion on the hyperbolic plane

Author

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  • Vignat, C.
  • Lamberti, P.W.

Abstract

We study H-theorems associated with the Brownian motion with constant drift on the hyperbolic plane. Since this random process satisfies a linear Fokker–Planck equation, it is easy to show that, up to a proper scaling, its Shannon entropy is increasing over time. As a consequence, its distribution is converging to a maximum Shannon entropy distribution which is also shown to be related to the non-extensive statistics. In a second part, relying on a theorem by Shiino, we extend this result to the case of Tsallis entropies: we show that under a variance-like constraint, the Tsallis entropy of the Brownian motion on the hyperbolic plane is increasing provided that the non-extensivity parameter of this entropy is properly chosen in terms of the drift of the Brownian motion.

Suggested Citation

  • Vignat, C. & Lamberti, P.W., 2012. "H-theorems for the Brownian motion on the hyperbolic plane," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(3), pages 544-551.
    • Handle: RePEc:eee:phsmap:v:391:y:2012:i:3:p:544-551
    DOI: 10.1016/j.physa.2011.08.069
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    References listed on IDEAS

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    1. Plastino, A.R. & Plastino, A., 1995. "Non-extensive statistical mechanics and generalized Fokker-Planck equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 222(1), pages 347-354.
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