Trajectory exponential divergence

The chaotic nature of individual MD trajectories has been well appreciated. A small change in initial conditions (e.g., a fraction of an Angstrom difference in Cartesian coordinates) can lead to exponentially-diverging trajectories in a relatively short time. The larger the initial difference and/or the timestep, the more rapid this Lyapunov instability. Fig. 1 reports observed behavior for the dynamics of a butane molecule. The governing Newtonian model is the following set of two first-order differential equations ... 

Time reversibility. Newton s equation is reversible in time. Eor a numerical simulation to retain this property it should be able to retrace its path back to the initial configuration (when the sign of the time step At is changed to —At). However, because of chaos (which is part of most complex systems), even modest numerical errors make this backtracking possible only for short periods of time. Any two classical trajectories that are initially very close will eventually exponentially diverge from one another. In the same way, any small perturbation, even the tiny error associated with finite precision on the computer, will cause the computer trajectories to diverge from each other and from the exact classical trajectory (for examples, see pp. 76-77 in Ref. 6). Nonetheless, for short periods of time a stable integration should exliibit temporal reversibility. 

For regular motion, T> t) grows only linearly with time, so that the exponents are all zero. On the other hand, because chaotic flows are characterized by exponential divergences of initial nearby trajectories, a characteristic signature of such flows is the existence of at least one positive Lyapunov exponent. 

As defined above, the Lyapunov exponents effectively determine the degree of chaos that exists in a dynamical system by measuring the rate of the exponential divergence of initially closely neighboring trajectories. An alternative, and, from the point of view of CA theory, perhaps more fundamental interpretation of their numeric content, is an information-theoretic one. It is, in fact, not hard to see that Lyapunov exponents are very closely related to the rate of information loss in a dynamical system (this point will be made more precise during our discussion of entropy in the next section). 

The most important quantitative measure for the degree of chaotic-ity is provided by the Lyapunov exponents (LE) (Eckmann and Ru-elle, 1985 Wolf et. al., 1985). The LE defines the rate of exponential divergence of initially nearby trajectories, i.e. the sensitivity of the system to small changes in initial conditions. A practical way for calculating the LE is given by Meyer (Meyer, 1986). This method is based on the Taylor-expansion method for solving differential equations. This method is applicable for systems whose equations of motion are very simple and higher-order derivatives can be determined analytically (Schweizer et.al., 1988). 

Results demonstrate that when agitators are switched the slope of the pathline becomes discontinuous. We will see later in this chapter how this mechanism may produce an essentially stochastic response in the Lagrangian sense. Aref termed this chaotic advection, which he suggested to be a new intermediate regime between turbulent and laminar advection. The chaos has a kinematic origin, it is temporal—that is, along trajectories associated with the motion of individual fluid particles. Chaos is used in the sense of sensitivity of the motion to the initial position of the particle, and exponential divergence of adjacent trajectories. 

Brownian fluctuations, inertia, nonhydrodynamic interactions, etc.) to lead to exponential divergence of particle trajectories, and (2) a lack of predictability after a dimensionless time increment (called the predictability horizon by Lighthill) that is of the order of the natural logarithm of the ratio of the characteristic displacement of the deterministic mean flow relative to the RMS displacement associated with the disturbance to the system. This weak, logarithmic dependence of the predictability horizon on the magnitude of the disturbance effects means that extremely small disturbances will lead to irreversibility after a very modest period of time. 

Figure 11.10 shows that the solution differences for TC8/TC9 and TClO/TCll roughly start from the respective machine accuracies (differences of 10 for single precision after one iteration, differences of 10 ° for quadruple precision after one iteration) and increase exponentially with the same growth rate before reaching the same difference levels for all three cases. This shows that higher precision computations cannot prevent the exponential divergence of trajectories but only delay it. 

In a similar context, Nayak and Ramaswamy have shown that the maximum Liapunov exponent (MLE) rises very steeply just as the Lindemann index and thereby can detect the aforementioned transition very well [20]. Since MLE is well established to measure the exponential divergence of the distance between nearby trajectories in phase space [21,22], their numerical results seem to suggest that the phase-transition could be a consequence of strong chaos behind the dynamics. We henceforth examine the resultant phenomena, the geometry on which the dynamics is characterized, and a statistical law of associated isomerization reaction. 

The exponential divergence must stop when the separation is comparable to the diameter of the attractor— the trajectories obviously can t... 

Exponential divergence) Using numerical integration of two nearby trajectories, estimate the largest Liapunov exponent for the Lorenz system, assuming that the parameters have their standard values r = 28, a=lQ, b = 8/3. 

Exponential divergence in systems that are chaotic prevents accurate long-time trajectory calculations of their dynamics. That is, numerical errors18 propagate exponentially during the dynamics so that accuracy beyond 100 characteristic periods of motion is extraordinarily difficult to achieve thus, accurate long-time dynamics is essentially uncomputable for chaotic classical systems. This serves as additional... 

These studies indicate a direct connection between requirements of exponential divergence and adherence to statistical theories. Note, however, that the particular statistical theory obeyed by the dynamics need not be a simple analytic theory such as RRKM or phase space theory. It may appear, therefore, that the an essential simplicity of statistical theory, that is, the ability to bypass long-lived trajectory calculations in favor of an easily computed result, has been lost. This is indeed not the case, that is, it is easy to see that this approach affords a method for obtaining contributions from both the unstable (statistical) trajectory component as well the direct component with a minimum of computation. This technique, the minimally dynamic 33,34 approach, will now be sketched. 

The problems discussed above all relate to spectral properties and statistics. In fact, the classical definition of chaos is more concerned with dynamical properties one can define classical chaos in terms of the instar bility of trajectories under infinitesimal displacements, which leads to the exponential divergence of neighbouring trajectories in phase space. The Liapounov exponent is the argument of the exponential which determines the rate of this divergence, and is often taken as a measure. 

As is clear from the above, linear-response theory has stimulated many important developments yet Kubo s approach has not been uncontroversial. In particular, in a paper that cannot be found in most libraries, van Kampen [7] has criticized the assumption made in linear-response theory that, for sufficiently weak fields, the change in the phase-space density is linear in the applied perturbation. In fact, due to the exponential divergence of phase-space trajectories, even the weakest perturbation will, on a macroscopic time-scale, result in large changes in the phase-space density however, as was shown in the subsequent numerical work by Ciccotti et al. [8], on the microscopic time-scales that are usually... 

In contrast to this, if the two points are in the vicinity of an unstable equilibrium point, the difference between the trajectories will grow exponentially with time, and the trajectories will diverge. Thus, in the vicinity of an unstable equilibrium point, the small uncertainty in the initial condition will grow exponentially, with characteristic time of Any attempt to predict the system motion for a time much longer than that will fail. Notice, also, that the same argument applies to any two trajectories, as soon as they are not confined to the immediate vicinity of the stable equilibrium. If the system is unharmonic, as almost all systems are, and its trajectories are not confined to the vicinity of a stable equilibrium, the trajectories are exponentially divergent. 

Statistical or chaotic behavior resulting from exponentially diverging trajectories is a closely related subject which is attracting wide interest, A transition from regular to chaotic motion... 

The Lyapunov exponents provide a computable measure of the sensitivity to initial conditions, i.e. characterize the mean exponential rate of divergence of two nearby trajectories if there is at least one positive Lyapunov exponent, or convergence when all Lyapunov exponents are negative. The Lyapunov exponents are defined for autonomous dynamical systems and can be described by ... 

This value is a measure of the mean exponential rate of divergence (convergence) of two initially very close trajectories, i.e. when (xq,0) 0. The values from (59) are the so called Lyapunov characteristic exponents, which can be ordered by size ... 

---