Prehistory probability distribution

The experiments are based on analog electronic circuits designed in the usual way [112,126] to model the system of interest, and then driven by appropriate external forces. Their response is measured and analysed digitally to create the statistical quantity of interest which, in the present case, was usually a prehistory probability distribution [60,124]. We again emphasize that such experiments provide a valid test of the theory, and that the theory should in this case be universally applicable to any system described by (16), including natural systems, technological ones, or the electronic models studied here. Some experiments on a model of (17) are now described and discussed as an illustrative example of what can already be achieved. 

Figure 9. Bottom The prehistory probability distribution of the radiation intensity I (in arbitrary units) for dropout events in a semiconductor laser. Top The PPD for a Brownian particle, obtained from simulations [83],...

We emphasize that the question of stability of a CA under small random perturbations is in itself an important unsolved problem in the theory of fluctuations [92-94] and the difficulties in solving it are similar to those mentioned above. Thus it is unclear at first glance how an analogy between these two unsolved problems could be of any help. However, as already noted above, the new method for statistical analysis of fluctuational trajectories [60,62,95,112] based on the prehistory probability distribution allows direct experimental insight into the almost deterministic dynamics of fluctuations in the limit of small noise intensity. Using this techique, it turns out to be possible to verify experimentally the existence of a unique solution, to identify the boundary condition on a CA, and to find an accurate approximation of the optimal control function. 

A statistical analysis of the fluctuational trajectories is based on the measurements of the prehistory probability distribution [60] ph(q, t qy, tf) (see Section IIIC). By investigating the prehistory probability distribution experimentally, one can establish the area of phase space within which optimal paths are well defined, specifically, where the tube of fluctuational paths around an optimal path is narrow. The prehistory distribution thus provides information about both the optimal path and the probability that it will be followed. In practice the method essentially reduces to continuously following the dynamics of the system and constructing the distribution of all realizations of the fluctuational trajectories that transfer it from a state of equilibrium to a prescribed remote state. 

To find the boundary conditions on the CA, we analyze the prehistory probability distribution Ph(q, t qf, f/) of the escape trajectories. The corresponding distribution is shown in the Fig. 16. It can be inferred by the inspection of how the ridge of the most probable escape path merges the CA that most of the escape trajectories pass close to the saddle cycle of the period 5 embedded into the CA. 

Figure 16. The prehistory probability distribution of the escape trajectories for the parameters as in Fig. 13. The circles, squares, and triangles show single periods of the saddle cycies of periods 5 (S5), 3 (S3), and 1 (SI), respectively [173].

Figure 8. Fluctuational behavior measured and calculated for an electronic model of the nonequilibrium system (17) with A = 0.264, D — 0.012. The man figure plots the prehistory probability density (pk x,t]Xf,0) and posthistory distribution to/from the remote state Xf — —0.63, t — 0.83, which lies on the switching line. In the top plane, the fluctuational (squares) and relaxational (circles) optimal paths to/from this remote state were determined by tracing the ridges of the distribution [62],...

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