H-theorems for the Brownian motion on the hyperbolic plane

C. Vignat and P.W. Lamberti

Physica A: Statistical Mechanics and its Applications, 2012, vol. 391, issue 3, 544-551

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.

Keywords: H-theorem; Brownian motion; Hyperbolic plane (search for similar items in EconPapers)
Date: 2012
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Persistent link: https://EconPapers.repec.org/RePEc:eee:phsmap:v:391:y:2012:i:3:p:544-551

DOI: 10.1016/j.physa.2011.08.069

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