Ergodicity and mixing in gravitating systems

Abstract

When considering the ergodic problem for gravitating systems of N point masses we have to take into account the circumstance that for large enough N (i.e. for galaxies of the most usual types) the systems are practically collisionless for time scales not exceeding the Hubble time. So it is necessary to follow phase trajectories not only in the 6N-D phase space but in the 6-D phase space. Studying the 6N-D space will give no new information for completely collisionless systems. We suggest a classification of various kinds of mixing in collisionless gravitating systems based mainly on degree of non-stationarity of the smoothed gravitational field (Antonov, Nuritdinov, & Ossipkov 1973). We distinguish a compulsive mixing in violently non-stationary systems, a quasi-diffusion mixing in weakly non-stationary systems, a divergent mixing which is connected with an exponential divergence of initially close trajectories for steady-state non-integrable systems, and a circulation mixing that will be in integrable systems in the case of dependence of circulation frequencies on values of isolating integrals of motion. The general property of collisionless mixings of any kind is the increase of the quasi-entropy, i.e., an integral of any convex function of the coarse-grained distribution function (Antonov 1963; Antonov et al. 1973; Tremaine et al. 1980). So we can show that some evolutionary ways are impossible for isolated collisionless systems. For example spherical systems cannot evolve along the sequence of polytropic indices (Antonov 1990). Any mixing cannot increase the maximal value of the distribution function.