We used a multipolar code to create, through the dissipationless collapses of systems of 1,000,000 particles, three self-consistent triaxial stellar systems with axial ratios corresponding to those of E4, E5 and E6 galaxies. The E5 and E6 models have small, but significant, rotational velocities although their total angular momenta are zero, that is, they exhibit figure rotation; the rotational velocity decreases with decreasing flattening of the models and for the E4 model it is essentially zero. Except for minor changes, probably caused by unavoidable relaxation effects, the systems are highly stable. The potential of each system was subsequently approximated with interpolating formulae yielding smooth potentials, stationary for the non-rotating model and stationary in the rotating frame for the rotating ones. The Lyapunov exponents could then be computed for randomly selected samples of the bodies that make up the different systems, allowing the recognition of regular and partially and fully chaotic orbits. Finally, the regular orbits were Fourier analyzed and classified using their locations on the frequency map. As it could be expected, the percentages of chaotic orbits increase with the flattening of the system. As one goes from E6 through E4, the fraction of partially chaotic orbits relative to that of fully chaotic ones increases, with the former surpassing the latter in model E4; the likely cause of this behavior is that 123 308 R. O. Aquilano et al. triaxiality diminishes from E6 through E4, the latter system being almost axially symmetric. We especulate that some of the partially chaotic orbits may obey a global integral akin to the long axis component of angular momentum. Our results show that is perfectly possible to have highly stable triaxial models with large fractions of chaotic orbits, but such systems cannot have constant axial ratios from center to border: a slightly flattened reservoir of highly chaotic orbits seems to be mandatory for those systems.

The universe time-asymmetry is essentially produced by its low-entropy unstable initial state. Using quanlitative arguments, Paul Davies has demonstrated that the universe expansion may diminish the entropy gap, therefore explaining its low-entropy state, with respect to the maximal possible entropy at any time. This idea is implemented in a qualitative way in a simple homogeneous model. Some rough coincidence with observational data are found•e-mail: Castagni (a) Iafe.Edu.Ar.•Pacs Nrs. 03.65 BZ,

In this paper we improve the results of sec. VI of paper [M. Castagnino, Phys. Rev. D 57, 750 (1998)] by considering that the main source of entropy production are the photospheres of the stars. I. A ROUGH COINCIDENCE BECOMES MORE PRECISE.In paper [1] one of us reported a rough coincidence between the time where the minimum of the entropy gap ∆S = S act − S max [2], takes place and the time where all the stars will exhaust their fuel. The time where the minimum of ∆S is located was:The following numerical values were chosen: ω 1 = T N R , the temperature of the nuclear reactions within the stars (that was considered as the main source of entropy), t N R = γ −1 the characteristic time of these nuclear reactions, t 0 the age of the universe, and T 0 , the cosmic micro-wave background temperature, and making some approximations the rough coincidence was obtained.Now we have reconsider the problem and conclude that, even if nuclear reactions within the stars are a source of entropy, the parameters T N R and t N R are not the good ones to define the behavior of the term e −γt/2 ρ 1 of equation (100) of paper [1], since they do not correspond to the main unstable system that we must consider. In fact the main production of entropy in a star is not located in its core, where the temperature is almost constant (and equal to T N R ), but in the photosphere where the star radiates. The energy radiated from the surface of the star is produced in the interior by fusion of light nuclei into heavier nuclei. Most stellar structures are essentially static, so the power radiated is supplied at the same rate by these exothermic nuclear reactions that take place near the center of the star [3]. We can decompose the whole star in two branch systems [2], as explained in section VII of paper [1], where a chain of branch systems was introduced. We have two branch systems to study: the core and the photosphere. The core gives energy to the photosphere and in turn the photosphere diffuses this energy to the surroundings of the star, namely in the bath of microwave radiation at temperature T 0 . In this way, we have two sources of entropy production: the radiation of energy at the surface of the star and the change of composition inside the star (as time passes we have more helium and less hydrogen). Since the core of a star is near thermodynamic equilibrium, we neglect the second and we concentrate on the first: the radiation from the surface of the star (related with the difference between the star and the background temperatures). So the temperature of the photosphere and not the one of the core must be introduced in our formula. Thus it is better to consider the photosphere as the unstable system that defines the term e −γt/2 ρ 1 of equation (100) [1]. So we must change T N R and t N R by T P , the temperature of the photosphere and t S the characteristic lifetime of the star. Then we must change eq. (1) to:As the 90% of the stars are dwarfs with photosphere temperature T P = 10 3 K [4] and the characteristic lifetime t S = 10 9 [5]...

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