Ergodic trajectory

The second point depends on the nature of the dynamics. Say we begin with a limiting case of truly ergodic trajectories. Then I really do not need to compute the actual trajectories. I would get the very same result if I simply postulate that my final set of trajectories is the set of all possible trajectories (where by possible I mean that they conserve energy, etc.). Information theory starts from this limit, which we refer to as the prior distribution. [1,3,23]... 

It was noted by Hoover that Eqs. [65] are not guaranteed to yield ergodic trajectories, in which case a dynamical simulation based on these equations of motion would not generate a canonical distribution in H p, q). This was seen most dramatically in the example of a single harmonic oscillator coupled to a Nose-Hoover thermostat, where a distribution radically different from the correct canonical distribution was generated as a result of nonergodicity. Thus far, two different solutions to this problem based on continuous dynamics have been proposed. 

Figure 3 (A) A single ergodic trajectory visits a substantial part of phase space. (B) A swarm of nonergodic trajectories originating from widely different initial conditions fills up the entire phase space.

Another problem with microcononical-based CA simulations, and one which was not entirely circumvented by Hermann, is the lack of ergodicity. Since microcanoriical ensemble averages require summations over a constant energy surface in phase space, correct results are assured only if the trajectory of the evolution is ergodic i.e. only if it covers the whole energy surface. Unfortunately, for low temperatures (T << Tc), microcanonical-based rules such as Q2R tend to induce states in which only the only spins that can flip their values are those that are located within small... 

Central to many developments in this book is the concept of ergodicity. Let us consider a physical system consisting of N particles. Its time evolution can be described as a path, or trajectory, in phase space. If the system was initially in the state... 

Molecular dynamics is frequently portrayed as a method based on the ergodicity hypothesis which states that the trajectory of a system propagating in time through the phase space following the Newtonian laws of motion given by the equations ... 

In conclusion, it is worth reflecting on a classical trajectory study of neutral ethane [335] in which it was found that there were dynamical restrictions to intramolecular energy transfer among C—H motions and between these and C—C motions. It was pointed out [335] that this non-ergodicity might not produce results observable at present levels of experimental resolution. This is probably the situation in mass spectrometry. QET is a respected theory in mass spectrometry because, proceeding from clearly stated assumptions, it is mathematically tractable and is able to explain the currently available experimental data. 

In the limit of long times and if the ergodic assumption holds, then we have 7 = (/), where (I) is the ensemble average. As usual, we may generate many intensity trajectories one at a time, to obtain the ensemble-averaged correlation function... 

Figure 9. The probability density function of 7 = T+ /T for the case i t+(t) = / (r) 1 P(I) is a delta function on (/) = 1 /2. In the nonergodic phase, 7 is a random function for small values of a the P(J) is peaked on 7 = 0 and 7=1, indicating a trajectory which is in state off or on for a period which is of the order of measurement time T.

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