Phase space systems regularity

Figure Al.2.7. Trajectory of two coupled stretches, obtained by integrating Hamilton s equations for motion on a PES for the two modes. The system has stable anhamionic synmretric and antisyimnetric stretch modes, like those illustrated in figrne Al.2.6. In this trajectory, semiclassically there is one quantum of energy in each mode, so the trajectory corresponds to a combination state with quantum numbers nj = [1, 1]. The woven pattern shows that the trajectory is regular rather than chaotic, corresponding to motion in phase space on an invariant torus.

In a perfect crystal at 0 K all atoms are ordered in a regular uniform way and the translational symmetry is therefore perfect. The entropy is thus zero. In order to become perfectly crystalline at absolute zero, the system in question must be able to explore its entire phase space the system must be in internal thermodynamic equilibrium. Thus the third law of thermodynamics does not apply to substances that are not in internal thermodynamic equilibrium, such as glasses and glassy crystals. Such non-ergodic states do have a finite entropy at the absolute zero, called zero-point entropy or residual entropy at 0 K. 

In tetra-atomic molecules that are linear, the number of degrees of freedom is F =7, which again considerably complicates the analysis. Nevertheless, the dynamics of such systems can turn out to be tractable if the anharmonicities may be considered as perturbations with respect to the harmonic zero-order Hamiltonian. In such cases, the regular classical motions remain dominant in phase space as compared with the chaotic zones, and the edge periodic orbits of subsystems again form a skeleton for the bulk periodic orbits. 

Figure 51. Classical phase-space structures of a modified kicked rotor system. Note that the regular islands are transporting islands. [From J. B. Gong, H. J. Worner, and P. Brumer, Phys. Rev. E. 68, 026209 (2003).]...

Besides the one little regular island at 0 and — 2 there are undoubtedly more regular islands in the phase space of the double pendulum at E = 2. We missed them by our rather coarse choice of initial conditions. As indicated in Fig. 3.3(b), their total area in phase space is probably very small. Nevertheless, Fig. 3.3(b) illustrates an important feature of the phase space of most physical systems the phase space contains an intricate mixture of regular and chaotic regions. The system is said to exhibit a mixed phase space. 

As illustrated by Figs. 3.3(a) and (b), Poincare sections are a very powerful tool for the visual inspection and classification of the dynamics of a given Hamiltonian. The double pendulum illustrates that for autonomous systems with two degrees of fireedom a Poincare section can immediately suggest whether a given Hamiltonian allows for the existence of chaos or not. Moreover, it tells us the locations of chaotic and regular regions in phase space. 

In this section we provide an analjdical proof for the absence of regular period-1 islands in the phase space of the positively kicked onedimensional hydrogen atom. This proof may also serve as a template to prove the absence of elliptic points for other important atomic physics systems such as the stretched hehum atom discussed in Chapter 10. 

Prom the physical point of view the absence of stable islands means that all the phase-space probability eventually ionizes. Since all the atomic physics systems investigated to date possess a mixed phase space that shows regular islands embedded in a chaotic sea, the absence of stable islands in the kicked hydrogen atom is a very unique property. 

In this case the system is called integrable. From the first equation we have /, = const, for all i that is, /, are constants of motion. Thus / , (/) = 0//o/8/,- are also constants, and we have

phase space of the system is on a regular orbit (torus). 

For chaotic orbits some of the conserved quantities in Hq are no longer conserved, and the state of the system tends to move in the phase space, rather than be bound to tori. But, as we have seen, there still exist regular orbits in the phase space as is shown by the KAM theorem. Sometimes the regular orbits (also called KAM tori ) can be topological obstacles for the drift motion of chaotic orbits. 

The spectrum of Lyapunov exponents provides fundamental and quantitative characterization of a dynamical system. Lyapunov exponents of a reference trajectory measure the exponential rates of principal divergences of the initially neighboring trajectories [1], Motion with at least one positive Lyapunov exponent has strong sensitivity to small perturbations of the initial conditions, and is said to be chaotic. In contrast, the principal divergences in regular motion, such as quasi-periodic motion, are at most linear in time, and then all the Lyapunov exponents are vanishing. The Lyapunov exponents have been studied both theoretically and experimentally in a wide range of systems [2-5], to elucidate the connections to the physical phenomena of importance, such as transports in phase spaces and nonequilibrium relaxation [6,7]. 

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