Trajectory phase space

We want to examine the relative behaviors of the two neighboring phase space trajectories, x (t) and x(t), starting from the initial conditions x (0) and x(0) = x (0) -f-Jx(0), respectively. The time evolution of their separation, x(i), may be approximated by linearizing the equations about the reference trajectory, x (t) ... 

One can apply the MC technique to the same molecular model, as explored in MD. One can use the same box and the same molecules that experience exactly the same potentials, and therefore the results are equally exact for equilibrium membranes. However, MC examples of this type are very rare. One of the reasons for this is that there is no commercial package available in which an MC strategy is combined with sufficient chemistry know-how and tuned force fields. Unlike the MD approach, where the phase-space trajectory is fixed by the equations of motion of the molecules, the optimal walkthrough phase space in an MC run may depend strongly on the system characteristics. In particular, for densely packed layers, it may be very inefficient to withdraw a molecule randomly and to let it reappear somewhere else in... 

The time-reversal symmetry of the Hamiltonian dynamics, also called the microreversibility, is the property that if the phase-space trajectory... 

On the other hand, the nonequilibrium steady states are constructed by weighting each phase-space trajectory with a probability which is different for their time reversals. As a consequence, the invariant probability distribution describing the nonequilibrium steady state at the microscopic phase-space level explicitly breaks the time-reversal symmetry. 

This point being made, we have not yet provided a description of how to follow a phase-space trajectory. This is the subject of molecular dynamics, upon which we now focus. 

Figure 3.1 Phase-space trajectory (center) for a one-dimensional harmonic oscillator. As described in the text, at time zero the system is represented by the rightmost diagram (q = b, p = 0). The system evolves clockwise until it returns to the original point, with the period depending on the mass of the ball and the force constant of the spring...

Let us consider the phase space trajectory traced out by diis behavior beginning with the position vector. Over any arbitrary time interval, die relationship between two positions is... 

These equations map out the oval phase space trajectory depicted in the figure. 

Certain aspects of this phase space trajectory merit attention. We noted above that a... 

For systems more complicated than the harmonic oscillator, it is almost never possible to write down analytical expressions for the position and momentum components of the phase space trajectory as a function of time. However, if we approximate Eqs. (3.10) and (3.12) as... 

Figure 3.2 An actual phase-space trajectory (bold curve) and an approximate trajectory generated by repeated application of Eq. (3.17) (series of arrows representing individual time steps). Note that each propagation step has an identical At, but individual Ap values can be quite different. In the illustration, the approximate trajectory hews relatively closely to the actual one, but this will not be the case if too large a time step is used...

Fig. 13.10. Cooperation of two different oscillatory modes leading to a phase space trajectory...

The new period-2 limit cycle is born with a CFM equal to + 1. This decreases rapidly as we reduce rN further. Eventually, the CFM approaches — 1 again a second period doubling occurs at tn = 0.137 307 1, where a period-2 with zp = 3.408 gives way to a period-4 solution with rp = 6.816. The initial splitting in the phase space trajectory again occurs in the vicinity of the maximum temperature rise and the double circuit gives way to four loops as shown in the sequence in Fig. 13.22. 

Certain aspects of this phase space trajectory merit attention. We noted above that a phase space trajectory cannot cross itself. However, it can be periodic, which is to say it can trace out the same path again and again the harmonic oscillator example is periodic. Note that the complete set of all harmonic oscillator trajectories, which would completely fill the corresponding two-dimensional phase space, is composed of concentric ovals (concentric circles if we were to choose the momentum metric to be (mk) 1/2 times the position metric). Thus, as required, these (periodic) trajectories do not cross one another. 

Fig. 3.2. Projection of a phase-space trajectory of the double pendulum on the plane.

This expression shows that for given ip and p p there are two branches for p. that result in two distinct Poincare sections. Therefore, naively marking a point in the (ip,Pip) plane whenever the phase-space trajectory satisfies i (f) = 0 would result in an overlay of two different sections. In order to separate properly the two sections, we mark a point only if 0, where a is the branch switch . Once selected it has to be kept fixed during the production of the corresponding Poincare section. The switch a can take the values 1. 

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