Dynamical instability and statistical behaviour of N-body systems

Abstract

In this paper, we argue about a synthetic characterization of the qualitative properties of generic many-degrees-of-freedom (mdf) dynamical systems (DS's) by means of a geometric description of the dynamics [Geometro-Dynamical Approach (GDA)]. We exhaustively describe the mathematical framework needed to link geometry and dynamical (in)stability, discussing in particular which geometrical quantity is actually related to instability and why some others cannot give, in general, any indication of the occurrence of chaos. The 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 analytical and numerical results obtained by us and by Pettini and coworkers concerning the FPU chain, verifying a complete agreement between the outcomes of averaging the relevant GDI's over phase space (Casetti and Pettini, 1995) and our findings (Cipriani, 1993), obtained in a more conservative way, time-averaging along geodesics. Along with the check of the ergodic properties of GDI's, these results confirm that the mechanism responsible for chaos in realistic DS's largely depends on the fluctuations of curvatures rather than on their negative values, whose occurrence is very unlikely. On these grounds we emphasize the importance of the virialization process, which separates two different regimes of instability. This evolutionary path, predicted on the basis of analytical estimates, receives clear support from numerical simulations, which, at the same time, confirm also the features of the evolution of the GDI's along with their dependence on the number of degrees of freedom, N , and on the other relevant parameters of the system, pointing out the scarce relevance of negative curvature (for N ⪢ 1) as a source of instability. The general arguments outlined above, are then concretely applied to two specific 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 system allows us to suggest a new definition of strong stochasticity, for any DS. The generalization of the concept of dynamical time-scale, t D, is at the basis of this new criterion. We derive for both the mdf systems considered the ( N , ε)-dependence of t D (ε being the specific energy) of the system. In light of this, the results obtained (Cerruti-Sola and Pettini, 1995), indeed turn out to be reliable, the perplexity there raised originating from the neglected N -dependence of t D, and not to an excessive degree of approximation in the averaged equations used. This points out also the peculiarities of gravitationally bound systems, which are always in a regime of strong instability; the dimensionless quantity L 1 = γ 1 · t D [γ 1 is the maximal Lyapunov Characteristic Number (LCN)] being always positive and independent of ε, as it happens for the FPU chain only above the strong stochasticity threshold (SST). The numerical checks on the analytical estimates about the ( N , ε)-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 practically negligible, always for the FPU chain, whereas in the case of the Gravitational N-body system, this is certainly true when the virial equilibrium has been attained. The strong stochasticity of the latter DS is clearly due to the large amplitude of curvature fluctuations. To prove the positivity of Ricci curvature, we need to discuss the pathologies of mathematical Newtonian interaction, which have some implications also on the ergodicity of the GDI's for this DS. We discuss the Statistical Mechanical properties of gravity, arguing 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 contribution comes from the Laplacian of the potential energy, whose singularity 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 (and then almost always) positive, as long as N ⪢ 1 and the system is gravitationally bound and virialized. On these grounds the equality among the above mentioned averages is restored, and the GDA to instability of gravitating systems gives fully reliable and understandable results. Finally, as a by-product of the numerical simulations performed, for both the DS's considered, it emerges that the time averages of GDI's quickly approach the corresponding canonical ones, even in the quasi-integrable limit, whereas, as expected, their fluctuations relax on much longer timescales, in particular below the SST.