Available online at www.sciencedirect.com Physica A 336 (2004) 376–390 www.elsevier.com/locate/physa Liouville equation and the q-statistical formalism A.R. Plastino a; b; c ; ∗ , C. Giordano c , A. Plastino d , M. Casas b a Physics Department, University of Pretoria, 0002 Pretoria, South Africa b Department de F sica and IMEDEA, Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain c Faculty of Astronomy and Geophysics, National University La Plata and CONICET, C.C. 727, (1900) La Plata, Argentina d Physics Department, National University La Plata and CONICET, C.C. 727, (1900) La Plata, Argentina Received 3 November 2003 Abstract We revisit some aspects of the foundations of the q-non-extensive thermostatistical formalism, which are particularly relevant to its astronomical applications. We analyse some important fea- tures of escort mean values and escort distributions, in connection with the dynamical evolution of ensemble probability distributions associated with the statistical description of general classical dynamical systems. Furthermore, we compare the roles played by escort mean values and escort distributions when studying classical N -body systems at the levels of (i) the Liouville equation and (ii) the Vlasov equation. c  2003 Published by Elsevier B.V. PACS: 89.70.+c; 05.20.Jj; 05.45.-a Keywords: Information theory; Classical uids; Non linear dynamics 1. Introduction Considerable attention has been paid recently to the study of the q-generalized non-extensive thermostatistics and its applications. The q-generalized thermostatistics is based on a non-extensive entropy proposed by Tsallis [1]. This entropy has been * Corresponding author. Physics Department, University of Pretoria, Pretoria 0002, South Africa. E-mail address: arplastino@maple.up.ac.za (A.R. Plastino). 0378-4371/$-see front matter c  2003 Published by Elsevier B.V. doi:10.1016/j.physa.2003.12.053