Geometric study for the Legendre duality of generalized entropies and its application to the porous medium equation
Department of Systems Science, Osaka University, 1-3 Machikane-yama, Toyonaka, 560-8531, Osaka, Japan
Corresponding author: a email@example.com
Revised: 12 February 2009
Published online: 13 May 2009
We geometrically study the Legendre duality relation that plays an important role in statistical physics with the standard or generalized entropies. For this purpose, we introduce dualistic structure defined by information geometry, and discuss concepts arising in generalized thermostatistics, such as relative entropies, escort distributions and modified expectations. Further, a possible generalization of these concepts in a certain direction is also considered. Finally, as an application of such a geometric viewpoint, we briefly demonstrate several new results on the behavior of the solution to a nonlinear diffusion equation called the porous medium equation.
PACS: 89.70.Cf – Entropy and other measures of information / 02.40.Hw – Classical differential geometry / 05.90.+m – Other topics in statistical physics, thermodynamics, and nonlinear dynamical systems
© EDP Sciences, Società Italiana di Fisica, Springer-Verlag, 2009