Dynamical instability and statistical behaviour of N-body systems

In this paper, we argue about a synthetic characterization of the qualitati ve properties of generic many-degrees-of-freedom (mdf) dynamical systems (D S's) by means of a geometric description of the dynamics [Geometro-Dynamica l Approach (GDA)]. We exhaustively describe the mathematical framework need ed to link geometry and dynamical (in)stabiIity, discussing in particular w hich geometrical quantity is actually related to instability and why some o thers cannot give, in general, any indication of the occurrence of chaos. T he relevance of the Schur theorem to select such Geometrodynamic Indicators (GDI) of instability is then emphasized, as its implications seem to have been underestimated in some of the previous works. We then compare the anal ytical and numerical results obtained by us and by Pettini and coworkers co ncerning the FPU chain, verifying a complete agreement between the outcomes of averaging the relevant GDI's over phase space (Casetti and Pettini, 199 5) and our findings (Cipriani, 1993), obtained in a more conservative way, time-averaging along geodesics. Along with the check of the ergodic propert ies of GDI's, these results confirm that the mechanism responsible for chao s in realistic DS's largely depends on the fluctuations of curvatures rathe r than on their negative values, whose occurrence is very unlikely. On thes e grounds we emphasize the importance of the virialization process, which s eparates two different regimes of instability. This evolutionary path, pred icted on the basis of analytical estimates, receives clear support from num erical simulations, which, at the same time, confirm also the features of t he evolution of the GDI's along with their dependence on the number of degr ees of freedom, N, and on the other relevant parameters of the system, poin ting out the scarce relevance of negative curvature (for N >> 1) as a sourc e of instability. The general arguments outlined above, are then concretely applied to two sp ecific N-body problems, obtaining some new insights into known outcomes and also some new results. The comparative analysis of the FPU chain and the gravitational N-body syst em allows us to suggest a slew definition of strong stochasticity, for any DS. The generalization of the concept of dynamical time-scale, to, is at th e basis of this new criterion. We derive for both the mdf systems considere d the (N, epsilon)-dependence of t(D) (epsilon being the specific energy) o f the system. In light of this, the results obtained (Cerruti-Sola and Pett ini, 1995), indeed turnout to be reliable, the perplexity there raised orig inating from the neglected dependence of t(D), and not to an excessive degr ee of approximation in the averaged equations used. This points out also th e peculiarities of gravitationally bound systems, which are always in a reg ime of strong instability; the dimensionless quantity L-1 = gamma(1) . t(D) [gamma(1) is the maximal Lyapunov Characteristic Number (LCN)] being alway s positive and independent of epsilon, as it happens for the FPU chain only above the strong stochasticity threshold (SST). The numerical checks on th e analytical estimates about the (N,epsilon)-dependence of GDI's, allow us to single out their scaling laws, which support our claim that, for N >> 1, the probability of finding a negative value of Ricci curvature is practica lly negligible, always for the FPU chain, whereas in the case of the Gravit ational N-body system, this is certainly true when the virial equilibrium h as been attained. The strong stochasticity of the latter DS is clearly due to the large amplitude of curvature fluctuations. To prove the positivity o f Ricci curvature, we need to discuss the pathologies of mathematical Newto nian interaction, which have some implications also on the ergodicity of th e GDI's for this DS. We discuss ing how they are related to its long range nature rather than to its short scale divergencies. The N-scaling behaviour of the single terms entering the Ricci curvature show that the dominant co ntribution comes from the Laplacian of the potential energy, whose singular ity is reflected on the issue of equality between time and static averages. However, we find that the physical N-body system is actually ergodic where the GDI's are concerned, and that the Ricci curvature associated is indeed almost everywhere land then almost always) positive, as long as N >> 1 and the system is gravitationally bound and virialized. On these grounds the e quality among the above mentioned averages is restored, and the GDA to inst ability of gravitating systems gives fully reliable and understandable resu lts. Finally, as a by-product of the numerical simulations performed, for b oth the DS's considered, it emerges that the time averages of GDI's quickly approach the corresponding canonical ones, even in the quasi-integrable li mit, whereas, as expected, their fluctuations relax on much longer timescal es, in particular below the SST. (C) 1998 Published by Elsevier Science Ltd . All rights reserved.