IL NUOVO CIMENTO NOTA BREVE VOL. 109 B, N. 3 Marzo 1994 Some Critical Remarks on Relaxation in N-Body Systems. P. CIPRIANI(1)(3) and G. PUCACCO(~)(3) (~) Dipartimento di Fisica, Universit5 di Roma I ,La Sapienza,, - 1-00185 Roma, Italia (2) Dipartimento di Fisic~ Universitd di Roma Tor Vergata - 1-00133 Rom(~ Italia (3) ICRA, Universitd di Roma I ,,La Sapienza,, - Roma, Italia (ricevuto il 25 Novembre 1993; approvato il 7 Dicembre 1993) Summary. -- We analyse in detail the origin of some controversial points concerning the relationships between the chaotic behaviour of dynamics and the relaxation properties of Hamiltonian systems with many degrees of freedom (m.d.f.), showing that most claims existing in the literature and applied, in particular, to the N-body self-gravitating systems, are, at least, unjustified. PACS 05.45 - Theory and models of chaotic systems. PACS 98.10 - Stellar dynamics. Introduction. - It is a known statement of rigorous Statistical Mechanics, that, when in a physical system gravity is the dominant interaction (e.g. self-gravitating collisionless N-body systems), then it is not possible to make a well-founded thermodynamics: one of the main problems is related to the operations needed to construct the thermodynamic limit and to the difficulties encountered in demonstrating the equivalence between microcanonical and canonical Gibbs ensembles, and then the corresponding orthodicity. So, it is a very ambitious task to try to describe the approach to the equilibrium for such systems. Nevertheless, in the last years [1,2] there have been many attempts to describe this process, from a purely theoretical point of view, in the framework of the ergodic theory of dynamical systems (DS), and also to attack it numerically. A detailed account of the main controversial points can be found in [3], but here we would like to stress what is, in our opinion, the origin of this debate: the unjustified extrapolation of rigorous results, whose validity is restricted to some abstract DS, to realistic Hamiltonian ones, whose properties are by far more complex. As one of the main points of those works is given by the identification of the instability (or Lyapunov) time and of the relaxation time, then, the aims of this paper are to point out that the relationships between the instability time scale, as measured from the exponential divergence of trajectories, the mixing time, if it can be estimated in some ~averaged, analytical way, and the ~relaxation~, time (when it is possible to speak about one relaxation time) are in general highly non-trivial; then, a very cautious treatment of these concepts is needed in order to avoid contradictions and misunderstandings. A careful treatment of this topic in the case of m.d.f. Hamiltonians of interest in Solid State Physics has been carried out by Pettini[4], and his results clearly support some of our conclusions. 325