Phase-space feature

The pendulum phase space is not periodic in Z as was the case for the standard mapping (see Fig. 5.4). But in the vicinity of Z = 0 it qualitatively reproduces the phase-space features of the kicked rotor very... 

Finally, let us note that more "exotic" classical phase space features, such as the canton briefly alluded to in Sec. II, may also have quantum analogues both in terms metastable states that could be prepared, or in terms of intermediate states (or strncmres) that influence the dynamics. For example, numerical work on simpler models has pointed to the possibility that cantori, under certain circumstances, have even more strongly trapping effects in quantum mechanics than in classical mechanics. Conceivably, this phenomenon could be invoked in some instances to explain unusually long vibrational predissociation lifetimes. 

Although this collision rule conserves momentum and energy, in contrast to the original version of MPC dynamics, phase space volumes are not preserved. This feature arises from the fact that the collision probability depends on AV so that different system states are mapped onto the same state. Consequently, it is important to check the consistency of the results in numerical simulations to ensure that this does not lead to artifacts. 

Multiparticle collision dynamics describes the interactions in a many-body system in terms of effective collisions that occur at discrete time intervals. Although the dynamics is a simplified representation of real dynamics, it conserves mass, momentum, and energy and preserves phase space volumes. Consequently, it retains many of the basic characteristics of classical Newtonian dynamics. The statistical mechanical basis of multiparticle collision dynamics is well established. Starting with the specification of the dynamics and the collision model, one may verify its dynamical properties, derive macroscopic laws, and, perhaps most importantly, obtain expressions for the transport coefficients. These features distinguish MPC dynamics from a number of other mesoscopic schemes. In order to describe solute motion in solution, MPC dynamics may be combined with molecular dynamics to construct hybrid schemes that can be used to explore a variety of phenomena. The fact that hydrodynamic interactions are properly accounted for in hybrid MPC-MD dynamics makes it a useful tool for the investigation of polymer and colloid dynamics. Since it is a particle-based scheme it incorporates fluctuations so that the reactive and nonreactive dynamics in small systems where such effects are important can be studied. 

So far, the discussion of the dynamics and the associated phase-space geometry has been restricted to the linearized Hamiltonian in eq. (5). However, in practice the linearization will rarely be sufficiently accurate to describe the reaction dynamics. We must then generalize the discussion to arbitrary nonlinear Hamiltonians in the vicinity of the saddle point. Fortunately, general theorems of invariant manifold theory [88] ensure that the qualitative features of the dynamics are the same as in the linear approximation for every energy not too high above the energy of the saddle point, there will be a NHIM with its associated stable and unstable manifolds that act as separatrices between reactive and nonreactive trajectories in precisely the manner that was described for the harmonic approximation. 

At a higher temperature T = 11.0, for flow rates near the transition rate c, the free-energy barrier between the coiled and stretched conformation is much lower than that for T = 9.0. The chain can therefore explore the phase space and jump back and forth from the coiled to the stretched state. Similar behavior has already been observed in [59] and [60]. Figure 27 illustrates this feature. 

To show [115] that Liouville s theorem holds in any number of phase-space dimensions it is useful to restate some special features of Hamilton s equations,... 

The single Kerr anharmonic oscillator has one more interesting feature. It is obvious that for Cj = 0 and y- = 0, the Kerr oscillator becomes a simple linear oscillator that in the case of a resonance 00, = (Do manifests a primitive instability in the phase space the phase point draws an expanding spiral. On adding the Kerr nonlinearity, the linear unstable system becomes highly chaotic. For example, putting A t = 200, (D (Dq 1, i = 0.1 and yj = 0, the spectrum of Lyapunov exponents for the first oscillator is 0.20,0, —0.20 1. However, the system does not remain chaotic if we add a small damping. For example, if yj = 0.05, then the spectrum of Lyapunov exponents has the form 0.00, 0.03, 0.12 1, which indicates a limit cycle. 

At this point it is useful to conduct a thought experiment. Consider a system for which the only possible measurement is by a Stern-Gerlach machine oriented along the z-axis. In other words, assume that once the probability for coming out of the machine spin up is known, every physically predictable feature of the state is known. Then the pair (c+, c ) e would contain more information than is necessary. Only c+p and c P would have physical meaning, and because of the condition Ic+I - - c 1 = 1, even these two real numbers are dependent. Thus the phase space of this hypothetical system... 

The pattern of extreme paths, LM, and action surfaces for an overdamped periodically driven oscillator (17) are shown in Fig. 6. The figure illustrates generic topological features of the pattern in question. It can be seen from Fig. 6 that, although there is only one path to a point (q, <[),p) in phase space, several... 

Martinelli, E. Falconi, C. D Amico, A. Di Natale, C., Feature extraction of chemical sensors in phase space Sens. Actuators B 2003, 95, 132-139. 

This Wigner representation of the density pw q, p) proves particularly useful since it, satisfies a number of properties that are similar to the classical phase-space distribu tion pd(q, p). For example, if p = pure state, then fdppw = probability density in coor- dinate space. Similarly, integrating pw over q gives the probability density in ( momentum space. These features are shared by the classical density p p, q) in phase space. Note, however, that pw is not a probability density, as evidenced by if the fact that it can be negative, a reflection of quantum features of the dynamics, ) [165], 3... 

A key factor in modeling is parameter estimation. One usually needs to fit the established model to experimental data in order to estimate the parameters of the model both for simulation and control. However, a task so common in a classical system is quite difficult in a chaotic one. The sensitivity of the system s behavior to the initial conditions and the control parameters makes it very hard to assess the parameters using tools such as least squares fitting. However, efforts have been made to deal with this problem [38]. For nonlinear data analysis, a combination of statistical and mathematical tests on the data to discern inner relationships among the data points (determinism vs. randomness), periodicity, quasiperiodicity, and chaos are used. These tests are in fact nonparametric indices. They do not reveal functional relationships, but rather directly calculate process features from time-series records. For example, the calculation of the dimensionality of a time series, which results from the phase space reconstruction procedure, as well as the Lyapunov exponent are such nonparametric indices. Some others are also commonly used ... 

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