Entropies from Coarse-graining: Convex Polytopes vs. Ellipsoids

Nikos Kalogeropoulos
<span title="2015-09-15">2015</span> <i title="MDPI AG"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/4d3elkqvznfzho6ki7a35bt47u" style="color: black;">Entropy</a> </i> &nbsp;
We examine the Boltzmann/Gibbs/Shannon S_BGS and the non-additive Havrda-Charvát / Daróczy/Cressie-Read/Tsallis S_q and the Kaniadakis κ-entropy S_κ from the viewpoint of coarse-graining, symplectic capacities and convexity. We argue that the functional form of such entropies can be ascribed to a discordance in phase-space coarse-graining between two generally different approaches: the Euclidean/Riemannian metric one that reflects independence and picks cubes as the fundamental cells and the
more &raquo; ... plectic/canonical one that picks spheres/ellipsoids for this role. Our discussion is motivated by and confined to the behaviour of Hamiltonian systems of many degrees of freedom. We see that Dvoretzky's theorem provides asymptotic estimates for the minimal dimension beyond which these two approaches are close to each other. We state and speculate about the role that dualities may play in this viewpoint.
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.3390/e17096329">doi:10.3390/e17096329</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/zcj224zbrnfnvima6mm2vywrq4">fatcat:zcj224zbrnfnvima6mm2vywrq4</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20151204010648/http://www.mdpi.com/1099-4300/17/9/6329/pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/17/25/1725fdf94c32dfa3ee57114596f50cbb396f7a1c.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.3390/e17096329"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> mdpi.com </button> </a>