1. C. Tsallis, J. Stat. Phys. 52, 479 (1988).
2. C. Tsallis, in Nonextensive Statistical Mechanics and Thermodynamics, eds. S. R. A. Salinas and C. Tsallis, Braz. J. Phys. 29, 1 (1999) [accessible at http://sbf.if.usp.br/WWWpages/Journals/BJP/Vol29/Num1/index.htm ]. The present review is an extended and updated version of the one just quoted.
3. F. Takens, Structures in dynamics—Finite dimensional deterministic studies, eds. H. W. Broer, F. Dumortier, S. J. van Strien, and F. Takens (North-Holland, Amsterdam, 1991), p. 253. [In his words: “The values of pi are determined by the following dogma: if the energy of the system in the ith state is Ei and if the temperature of the system is T then: pi = exp/kT/Z(T), here Z(T) =_i exp/kT, (this last constant is taken so that _i pi = 1). This choice of pi is called Gibbs distribution. We shall give no justification for this dogma; even a physicist like Ruelle disposes of this question as “deep and incompletely clarified”.”]
4. N. Krylov, Nature 153, 709 (1944). [In his words: “In the present investigation, the notion of ergodicity is ignored. I reject the ergodical hypothesis completely: it is both insufficient and unnecessary for statistics. I use, as starting point, the notion of motions of the mixing type, and show that the essential mechanical condition for the applicability of statistics consists in the requirement that in the phase space of the system all the regions with a sufficiently large size should vary in the course of time in such a way that while their volume remains constant—according to Liouville’s theorem—their parts should be distributed over the whole phase space (more exactly over the layer, corresponding to given values of the single-valued integrals of the motion) with a steadily increasing degree of uniformity. (...) The main condition of mixing, which ensures the fulfillment of this condition, is a sufficiently rapid divergence of the geodetic lines of this Riemann space (that is, of the paths of the system in the n-dimensional configuration space), namely, an exponential divergence (cf. Nopf1).”]. For full details on this pioneering approach see N.S. Krylov, Works on the Foundations of Statistical Physics, translated by A. B. Migdal, Ya. G. Sinai, and Yu. L. Zeeman, Princeton Series in Physics (Princeton University Press, Princeton, 1979).
5. R. Balescu, Equilibrium and Non-equilibrium Statistical Mechanics (Wiley, New York, 1975), p. 727. [In his words: “It therefore appears from the present discussion that the mixing property of a mechanical system is much more important for the understanding of statistical mechanics than the mere ergodicity. (...) A detailed rigorous study of the way in which the concepts of mixing and the concept of large numbers of degrees of freedom in.uence the macroscopic laws of motion is still lacking.”]