Deterministic path

The fluctuational trajectory away from a stationary state to a given point in concentration space (x, y) may differ from the deterministic path from that point back to the stationary state, for systems without detailed balance. Of course, the free energy change must vanish for a closed loop in the space of A,B,X,Y) but need not vanish for a closed loop in the restricted space of (x,y). 

This proves that the stationary solution of the master equation, Sn x,y), is a Lyapunov function for the deterministic path from (x, y) to the stable stationary state. 

The symbol ii appears in the master equation, (19.8). The deterministic path of the stable limit cycle is located on the ridge of the crater. The exponential in the stationary distribution of the master equation in the eikonal limit is an excess work related to thermodynamic functions, see Chaps. 2 7. 

Coming to the role of the intrinsic parameters, the general trend is that bimodality is enhanced for the parameter range for which the deterministic evolution displays two widely separated time scales. On the other hand -and this leads us to the role of initial conditions- to "probe such a time scale difference the system has to start from a state located sufficiently before the inflexion point of the deterministic potential (cf. Fig. 4) Otherwise it undergoes a rapid relaxation to the final state following essentially the deterministic path. 

Equation (2.67) also follows from the Langevin equation (2.35) if the fluctuations are neglected by putting q (j ) = 0, or letting e - 0. Therefore (2.67) also describes the deterministic path. 

The stochastic spread of the distribution about the deterministic path results from the B terms in Eqs (8.121) and (8.122) in analogy with the D term in Eq. (8.113). 

In this section, we will briefly summarize the transition path sampling method for deterministic paths which has been developed by David Chandler and co-workers based on earlier ideas of Pratt [235]. This method is not only able to calculate the hopping rate (and therefore also the diffusion coefficient) between two stable sites (here the intersections of Silicalite). For a more complete discussion about this simulation technique, the reader is referred to refs. [227,232,234]. 

It should be realized that unlike the study of equilibrium thermodynamics for which a model is often mapped onto Ising system, elementary mechanism of atomic motion plays a deterministic role in the kinetic study. In an actual alloy system, diffusion of an atomic species is mainly driven by vacancy mechanism. The incorporation of the vacancy mechanism into PPM formalism, however, is not readily achieved, since the abundant freedom of microscopic path of atomic movement demands intractable number of variational parameters. The present study is, therefore, limited to a simple spin kinetics, known as Glauber dynamics [14] for which flipping events at fixed lattice points drive the phase transition. Hence, the present study for a spin system is regarded as a precursor to an alloy kinetics. The limitation of the model is critically examined and pointed out in the subsequent sections. 

Both g and g are non-deterministic, in the sense that multiple arcs originating from the same node carry the same symbol. A given word in the resulting formal language therefore need not originate from a unique path. 

We will first follow the decay paths taken during several individual runs, just to see how they can vary. Then we will examine the behavior of larger samples to find actual values for cpf and cellular automata models are stochastic, the results for and small samples will likely differ significantly from the deterministic values cited above. The differences between the observed and the deterministic values will normally decrease as the sample size is increased. We will also examine the observed lifetimes Xf and tp of the decays of the Si and T i states (Chapter 7) and compare the values found with the corresponding deterministic values. 

Then the path probability from (7.3) consists of a product of such delta functions. Due to the singular nature of such a path probability it is more convenient to view the entire deterministic trajectory as represented by its initial state z0. In this case the transition path ensemble from (7.10) reduces to a distribution of initial conditions z0 yielding pathways connecting srf with 2%... 

It can be shown [173] that the average time for the system to approach S5 is much smaller then the average escape time and thus the optimal escape paths found from the statistical analysis of the escape trajectories is independent of the initial conditions on the attractor and provides an approximation to the global minimum of the corresponding deterministic control problem. 

Imagine the path of a flying ball. This is a deterministic curve. But rain and wind are random disturbances which have to be smoothed and included in the resulting path. 

Deterministic dynamics of biochemical reaction systems can be visualized as the trajectory of (ci(t), c2(t), , c v(0) in a space of concentrations, where d(t) is the concentration of ith species changing with time. This mental picture of path traced out in the N-dimensional concentration space by deterministic systems may prove a useful reference when we deal with stochastic chemical dynamics. In stochastic systems, one no longer thinks in terms of definite concentrations at time t rather, one deals with the probability of the concentrations being xu x2, , Wy at time t ... 

With suitable definitions of search functions, EA methods can also be used to locate more features on the PES than just low-energy local and global minima. Chaudhury et al. [144,145] have implemented methods for finding first-order saddle points and reaction paths, applying them to LJ clusters up to n=30. It remains to be tested, however, if these method can be competitive with deterministic exhaustive searches for critical points for small systems [146], on the one hand, and with the large arsenal of methods for finding saddles and reaction paths between two known minima for larger systems [63], on the other hand. 

That is the very beauty of quantum mechanics. It is a perfectly deterministic theory for calculating the development of a physical system, once you know it at a given time. But because only the absolute square of the wave-function is accessible to experiment, it is not possible for us to know the state precisely at such an initial time, and hence all the further development of the system is to the human observer clouded with imcertainty relations and similar indeterminacy, despite the fact that the system itself knows perfectly well what it is doing and follows a uniquely given path of development in time and space. 

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