Astron. Astrophys. 326, 113-129 (1997)

2. Characterization of chaos and the Ricci criterion

2.1. Difficulties with commonly used methods

Central to the idea of phase-space mixing is the concept of dynamical entropy used to quantify it. The most important of such quantities is the so called Kolmogorov-Sinai entropy. The easiest way of calculating the KS entropy is by evaluating the Liapunov exponents, which for a dynamical system defined by the vector equations

with solution are given by

where is a tangent space vector arising from the solution of the variational equations

The KS entropy is then usually given by (e.g., LL)

where the sum is taken over all positive exponents and the integral is over all possible initial conditions. For systems with simple enough phase-space and when the infinite time characteristics are required, the KS entropy calculated with the help of this formula suffices to describe the ergodic properties of a system. Positive KS entropy over a compact phase-space, or some region of it that has this property, is a sufficient condition for the presence of what is usually called chaos and the accompanying erratic behaviour which leads to the approach towards statistical equilibrium even in low dimensional systems (e.g., McCauley 1993). The evolution time-scale is usually related to the inverse of the KS entropy.

Open systems interacting via un-softened Newtonian potentials do not have compact phase-spaces however, and therefore the situation is more complicated. Here, no final state exists and one has to distinguish between the various stages of evolution i) Violent relaxation, ii) Collective (plasma type) instabilities iii) Evolution towards an isotropic rotator and iv) Kinetic evolution towards equipartition of energy, core collapse etc. Here, the distinctions are rather arbitrary and are used only for clarity - these processes may interact and influence each other. Stage three has been given very little attention in the literature because of the lack of a mechanism from traditional physics leading to such evolution (on a time-scale smaller than given by (1)). Nevertheless, the chaotic nature of the N -body problem may provide a clue as to how this might happen.

All the above processes should be characterized by phase-space instability leading to mixing, but evidently cannot be described by any quantities defined only for infinite times. One way out of this problem is to integrate either the linearized equations (5) (e.g., Goodman et al. 1993) or the full non-linear equations (3) for slightly different initial conditions (e.g., Kandrup et al. 1994 and the references therein) for short times. One drawback of such an approach, however, is that one is comparing the divergence in phase-space of different temporal states and not trajectories. Integrable systems usually have linear (in time) phase-space divergence between neighbouring states - but only on average. A simple illustration of the type of problems involved is provided by examining the behaviour of a pendulum

with linearized equation

When is negative this has a solution

That is, during this interval, the solution of the linearized equation predicts "exponential instability". Obviously, in this example, the trajectories do not diverge at all - but the states did. In addition, methods that evaluate the whole set of Liapunov exponents of a dynamical system are fairly sophisticated (e.g., Eckmann & Ruelle 1985) and are not practical for higher dimensional systems. This in practice will mean that one will have to evaluate only the largest exponent which is a measure of the maximal instability in phase-space. Now, in a multidimensional separable system it is likely that at any given time there will be some oscillations that are in the " region" - that is displaying exponential divergence in the linearized dynamics. An initial randomly oriented vector in the linear tangent space will always reorient itself along the direction of maximum expansion under the stretching effect of the flow (e.g., Wolf et al. 1985). It could therefore be possible to obtain average exponential divergence in the linearized dynamics even for separable systems. This approach therefore may be unable to distinguish between genuine phase space mixing leading to evolution and trivial phase mixing of temporal states. It may also be useful to note here that there are few strict results concerning the properties of the Liapunov exponents for higher dimensional systems (Eckmann & Ruelle 1985; Pesin 1989).

2.2. The study of motion on Lagrangian manifolds

There are a variety of ways of transforming Hamiltonian problems into the study of some metric space (see P93; Gurzadyan & Kocharyan 1994 or the articles by Gurzadyan and Pettini in Gurzadyan & Pfenniger 1994). The oldest and most well known of these hinges on the observation, apparently first made by Hertz (1900) in the course of his remarkable reformulation of classical mechanics, that the Maupertuis principle

(where T is the kinetic energy along the motion on a trajectory ) from which the equations of motion in their Lagrangian form arise is actually an expression for geodesics on a the configuration manifold M where the motion is restricted as a result of conservation laws. The fact that (10) defines geodesics is clear from the following relations:

where is the total energy and with

for particles of unit mass. Here the are elements of the metric tensor and refers to variations in the trajectory holding the energy and the end points fixed. These are thus geodesics in the energy sub-manifold of the configuration space and the are coordinates on it. If we now choose Cartesian coordinates in the enveloping space then

The metric then is

From the Jacobi equation which describes the geodesic deviation on M (e.g., Misner et al. 1973)

(where is a separation vector analogous to in (4), is the Riemann curvature operator, and the covariant derivative) one can obtain the following relation for the norm of the normal component of this deviation

where

is the two dimensional curvature in a plane defined by and where we have used the fact that . If is negative everywhere for all planes as defined above (that is for all normal to ) and if we have

for , and

for . These relations describe the linearized dynamics in the "dilating" and "contracting" spaces characteristic of the class of Anosov (1967) C-systems to which, as was mentioned earlier, large spherical N -body systems belong. In this case, averaged over M is the KS entropy. In comparing two C-systems therefore one can define the system with larger average value of to be more unstable. This system will have a larger exponentiation rate and so initial conditions will tend to mix faster along geodesics. To determine how fast the initial conditions mix in time, we note that so that if T does not vary too much during the evolution . It is clear that a system with the above characteristics cannot conserve its action variables since there is always a deviation of trajectories normal to the motion in phase-space (which is the cotangent bundle of M).

2.3. Ricci curvature and the corresponding criterion

GS have shown that the condition for C-systems is not satisfied for general gravitational ones. However, as Kandrup (1990a, 1990b) has shown, the probability of a two dimensional curvature being positive along a N -body system's trajectory decreases exponentially with increasing N. Also, in the N -body problem relation (18) implies that all orbits are unstable at all times. However, one needs much less than what is described by (18) for observable effects of instability to be detected (just of orbit becoming unstable may be enough). On the other hand, just a few orbits being unstable in general would not significantly change the physical properties of a system. We therefore need some averaged form of (16) and a corresponding instability relation instead of (18) to characterize such behaviour. Ideally such a relation should not require the evaluation of the Riemann tensor.

A natural way of proceeding is by using the Ricci (or mean) curvature of the manifold M

where are elements of the Ricci tensor and are the components of the geodesic velocity vector . The Ricci curvature is related to the two dimensional curvatures by (Eisenhart 1926)

The value of does not depend on the particular set of normal directions chosen so that can be seen as the average value of over all possible directions normal to on the configuration manifold M.

In the case when all k 's are negative, averaged over the whole manifold corresponds to the Kolmogorov entropy. In general, as was first noted by Gurzadyan & Kocharyan (1987), it will provide an "averaged" measure of irregularity. In these terms Equation (16) can be written as

where Z is now to be interpreted as the norm of a vector that is a member of a random field of vectors with uniform distribution in directions normal to and equal magnitude (El-Zant 1996a).

If is negative on a region of M then one can obtain relations for Z analogous to those obtained for in (18) with . One can then apply the criterion of relative instability of C-systems to general Hamiltonian dynamical systems in regions of their configuration manifolds where the Ricci curvatures are negative, which in this case will express the relative probability of any two systems being unstable under random perturbations. We adopt here the following definition, convenient for numerical studies of N -body systems.

Definition:

Let be some subset of the configuration manifold and be a subset of a manifold with not necessarily different from . Suppose also that the Ricci curvature is negative in both of these regions. We will say that corresponds to configurations of a dynamical system that are more unstable than those represented by if the average value of is larger in than in .

Note that: As in the case of C-systems we obtain time-scales from the relation . If we are comparing systems with different kinetic energies, the evolutionary times derived will have to be scaled accordingly. If the kinetic energies of systems are changing in the region of the dynamical systems of interest then time-scales derived from the Ricci curvature alone are not rigorous. If the logarithmic time derivative of the kinetic energy is small however then (where the bar denotes an average over the region of interest). Otherwise a fully dynamical formulation with time replacing s in Equation (22) would have to be considered (P93; Cerruti-Sola & Pettini 1995). The latter approach would also have to be used if the curvature on M is not predominantly negative. Also if the systems we are comparing consist of different numbers of particles, will have to be divided by .

The definition leaves us the choice to compare different areas of the manifold of the same dynamical system, or those of different systems. Also the averages can either be static, or taken along a computed trajectory of the system. Since the regions R in the definition above can be taken as small as we wish, the method is clearly local. This is possible because is directly related to the local geometry of the manifold where a system lives and is not an asymptotic quantity. Also, although the above formulation does not allow explicitly for dissipative forces, these can be added as time dependent perturbation to an open Hamiltonian system.

To actually calculate the value of , one contracts the Riemann tensor (which for the metric (14) is given in GS) and uses (20) to obtain

with . Here W denotes while and represent its Cartesian gradient and Laplacian respectively. From the metric (14) one can deduce that and . For an N -body system, the implied summation would be over . Moreover, if the interactions proceed through the usual Newtonian law with no direct impacts . If we now label by a, b and c the particle numbers (which run from 1 to N) and by k and l the three Cartesian coordinates of a particle, it is straightforward to obtain the following expressions for the derivatives of W

if and

if . In these equations

and .

The practical procedure of implementing the criterion described above will therefore consist of using the position and velocities of particles for the configurations of the system under study to obtain , W, and the quantities defined in (25). These are substituted into (23) to find . In this way, one can calculate the Ricci curvature for various regions of the configuration manifolds of different systems. If the Ricci curvature is found to be predominantly negative, we then use the above definition to classify systems according to their local stability properties.

There are many advantages to the above setup. For example, what is studied here is the normal deviation due random perturbations of trajectories with the same geodesic velocities on M and not the deviation of temporal states in the direction of maximal growth. Therefore what can be termed the `chaotic pendulum problem' is avoided. In fact, it can be seen from (23) that all systems possessing only one degree of freedom (those with ) have an at all times. In the terminology of classical stability theory it is said that the negativity of the Ricci curvature measures the orbital stability as opposed to the more strict Liapunov stability (Pars 1965). Because of these properties the Ricci curvature is unlikely to be negative for multidimensional integrable systems in virial equilibrium. This assertion has been checked for the special case of the two body problem with circular orbits (where the virial relation is satisfied) but obviously needs further investigation. For a system in a full statistical quasi-steady state a negative Ricci curvature must mean that the system is chaotic and mixing for all initial conditions since is constant for systems in a steady state. However, in general, the negativity of the Ricci curvature on most of a system's trajectory does not guaranty that it is chaotic but only that there is a probability of this being the case (the probability increasing with the fraction of time spent in the negative region).

Other advantages of this method are that the integration of a large number of linearized equations is avoided and space averages on compact manifolds can be compared to time averages (e.g., Casetti & Pettini 1993), thus allowing one to check for properties such as ergodicity for single particle orbits and specially constructed N -body systems for which M is compact (such as those consisting of softened particles enclosed enclosed in boxes: e.g., Lynden-Bell 1972).

© European Southern Observatory (ESO) 1997

Online publication: April 20, 1998
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