Phase-space integration conditional

As a consequence, the semiclassical propagator is given as a phase-space integral over the initial conditions qo and Po, which is amenable to a Monte Carlo evaluation. For this reason, semiclassical initial-value representations are regarded as the key to the application of semiclassical methods to multidimensional systems. 

The selection of the initial conditions for classical trajectories is designed to simulate the experimental conditions of interest. The properties of interest are averages, for example, the thermal rate coefficient is an average of the reaction probabilities over all of the volume of the phase space available to the system at a particular temperature. (For a more explicit discussion of the details of these averages, see the reviews by Tmhlar and Muckerman and by Raff and Thompson. ) These averages can be written as phase space integrals of the type... 

Thus, the transition path ensemble is now represented by a distribution of initial conditions in phase space (we have, in effect, integrated out all path variables except... 

For N particles in a system there are 2N of these first-order equations. For given initial conditions the state of the system is uniquely specified by the solutions of these equations. In a conservative system F is a function of q. If q and p are known at time t0, the changes in q and p can therefore be determined at all future times by the integration of (12) and (13). The states of a particle may then be traced in the coordinate system defined by p(t) and q(t), called a phase space. An example of such a phase space for one-dimensional motion is shown in figure 3. 

As mentioned above, the time-integral of the dissipation function takes on the value of the extensive generalised entropy production, 2, over a period, t under suitable circumstances. The main requirement is that the dynamics satisfies the condition know as the adiabatic incompressibility of phase space . In this case, 2, = —JtFefiV, where is the dissipative flux caused by the field, F, p = IKk T) where T is the temperature of the corresponding initial system and V is the volume of the system. An example where such a relation can be applied is if a molten salt at equilibrium was exposed to a constant electric field. In that case the entropy production would be directly proportional to the current induced, and the FR would describe the probability that it would be observed to flow in the + ve or — ve... 

According to (9.3.17) we then require either that all f be zero or that all Fijn vanish. The first alternative cannot be correct since all the functions except fk may be chosen arbitrarily and fk is absent from the summation over i. This leaves only the alternative that all Fijk = 0. From the earlier discussion involving Eq. (9.2.20) it follows that the Pfaffian dL = dxi is integrable. We have thereby established the necessary condition for the Caratheodory theorem of Section 9.2 to hold. Given the fact that in the neighborhood of a point in phase space other points are inaccessible via solution curves of the form X, dxi — 0, the Pfaffian form is integrable. 

In many cases, instead of the PDF in Eq. (5.127), an equivalent formulation in terms of a corresponding conditional NDF Al(Vp, p Vp, p) is used. The corresponding mathematical object is such that the integral over phase space now yields... 

The calculation implied by Eq. (9) for N(E) (or Eq. (1) for k(T)) is therefore to integrate over phase space (p, q)—in practice, usually with Monte Carlo sampling methods—where each phase point (p, q) serves as the initial conditions for a trajectory that must be run (i.e., numerically integrated) to determine whether Xr is 1 or 0, i.e., whether or not this phase point contributes to the integral. Because the flux, Eq. (6), contains the factor 8[/(q)], all trajectories begin on the dividing surface/(q) = 0. 

Fig. 6.19. Evolution in the phase space at different values of the relative proportions of (initially) chaotic and periodic cell populations in a mixed suspension containing various amounts of the two types of cells, (a) Oscillations of the limit cycle type obtained for Vj = 4.5 x 10 min" when the suspension contains only cells of periodic population 2 (fj = 0, fj = l)l arrows show the direction of movement and the trajectory has been broken to indicate the part that comes behind (the portion of the curve in front corresponds to a decrease in all three variables after a peak in cAMP). (b) Period-2 oscillations obtained upon adding to periodic population 2 cells from the chaotic population 1, for which Vi = 4.396875 x 10" min" the value of the fraction of the (initially), chaotic population is = 0.5. (c) Period-4 oscillations obtained when is increased up to 0.86 notice that two of the loops of the trajectory over a period are very close to each other, which is also apparent in the bifurcation diagram of fig. 6.20. (d) Chaotic behaviour corresponding to a strange attractor when the suspension contains only cells of population 1 (Fj = 1). The curves are obtained by numerical integration of eqns (6.9) for the above-indicated values of Vj and Vj other parameter values, which hold for the two populations, are as in fig. 6.2. Variables pr and a relate to population 2 in (a)-(c), and to the homogeneous population 1 in (d) variable y is shared by the two populations. Ranges of variation for pr, a and y are 0-1,0.65-0.68 and 0-2.2, respectively. Initial conditions were a = 0.6729 and pr = 0.2446 for both populations, while 7=1.7033. The curves were obtained after a transient of 500-1000 min. The period of the oscillations shown in (a)-(c) is of the order of 8-10 min thus for F. = 0.3 and Fj = 0.7 the period is equal to 8.7 min (Halloy et al. 1990).

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