Averaging over the initial distribution

So far, the discussion has presumed that the ion-pair was formed with a unique initial separation, r0. If the initial distribution extends over a range of distances, tc(r0), in which ut(r0) is normalised to unity, then the average scavenging probability is 

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If the system evolves according to some stochastic scheme, each initial point can lead to a multitude of trajectories. We note, however, that as long as each trajectory is initiated from an equilibrium distribution, (8.46) can still be rewritten as an average over the initial distribution ... 

The solution of eqn. (44) for a coulomb potential with boundary conditions (45) and (46) for either initial conditions (48) or (49) has only been developed in recent years. Hong and Noolandi [72] showed that the solution of the Debye—Smoluchowski equation is related to the Mathieu equation. Many of the details of their analysis are discussed in the Appendix A, Sect. 4, and the Appendix eqn. (A.21) is the Green s function (fundamental solution), which is the probability that a reactant B is at r given that it was initially at r0. This equation is developed as the Laplace transform. To obtain the density of interest p(r, ), with either condition, the Green s function has to be averaged over the initial distribution, as in eqn. (A.12), and the Laplace transform inverted. Alternatively, the density p(r, ) can be found from the inverse Laplace transform of the linear combination of independent solutions (A.17) which satisfy the boundary and initial conditions. This is shown in Fig. 10. For a Boltzmann initial condition, Hong and Noolandi [72] found... 

Equation (350) merely states that the probability of scavenging an ion-pair whose initial distribution is iv(r0) is equal to the average over the initial distribution of the probability that the ion-pair with initial separation r0 will recombine. 

The bar over cp means that this quantity is averaged over the initial distribution of the interion distances prepared by the forward electron transfer. In IET this averaging is done implicitly and automatically, justifying the straightforward calculation of this quantity. 

Following the simulations, the time-dependent observables of interest are obtained via an ensemble average over the initial distribution. For example, the average energy content of the k h normal mode is given by... 

For the purposes of the present treatment, we wish to rewrite this trajectory average as an average over the initial, equilibrium distribution. If the system evolves according to deterministic (e.g., Hamiltonian) dynamics, each trajectory is uniquely determined by its initial point, and (8.46) can be written without modification as an average over the canonical phase space distribution. 

The first way is to obtain the transition probability density by the solution of Eq. (2.6) with the delta-shaped initial distribution and after that averaging it over the initial distribution Wo(x) [see formula (2.4)]. 

Assuming that the initial mass function is invariable, we may calculate the average production of the various star generations, born with the same metaUicity, and estimate their contribution to the evolution of the galaxy (see Appendix 4). The abundances produced by a whole population are not as discontinuous and irregular as those shown in the table of individual yields (Table A4.1). This is because the latter are averaged over the mass distribution. 

The interpretation of the above expressions is rather remarkable. The centroid constraints in the Boltzmann operator, which appear in the definition of the QDO from Eqs. (19) and (20), cause the canonical ensemble to become non-stationary. Equally important is the fact that the non-stationary QDO, when traced with the operator 9. (or P) as in Eq. (37), defines a dynamically evolving centroid trajectory. The average over the initial conditions of such trajectories according to the centroid distribution [ cf Eq. (36) ] recovers the stationary canorrical average of the operator (or ). However, centroid trajectories for individual sets ofirritial conditions are in fact dynamical objects and, as will be shown in the next section, contain important information on the dynamics of the spontaneous fluctuations in the canonical ensemble. 

According to (9.3) this equation can be interpreted as follows. Define a time-dependent vector qn(t) by stipulating that at t = 0 it has the components qn, and for t > 0 it evolves according to (9.1). Then the average of Q at time t equals the average of q (t) over the initial distribution ... 

It was remarked in Refs. 1 and 7 that in the semiclassical model, if the initial phase 0o is not known, one can take a statistical average over the initial phases, with uniform distribution ... 

We can conclude by repeating our initial remark the amoimt of experimental data is still not sufficient and the further work is necessary. Moreover, in most actual studies carried out in the gas phase only the average cross-sections may be determined and this necessitates the averaging over the statistical distribution of velocities and of impact parameters in all theoretical treatments. From this point of view, crossbeam experiments, where the dependence of quenching cross-sections on the relative velocity of colliding particles (and even on the impact parameters, if the angular distribution of products is studies) may be determined, would be of highest interest. 

STY relate averages, such as (R ) and tangent vectors. (Even this auxiliary distribution is expressed only as an infinite series expansion.)... 

It is convenient to choose an equally-spaced grid instead of the more usual random sampling [41] so that the average over the initial rotational distribution can later be done by numerical quadratures. For instance, in our studies of Li -N collisions at impact energies E = 2eV through lOeV, a, and cos fi, were each sampled in 33 equally-spaced steps, for a total of 1024 distinct orientations the impact parameter b was sampled in steps of 0.1 A in the interval [0, 6.0 A], which spanned 180°s 6 2° [62]. 

The above calculation of dcr fdil, together with that of the corresponding TCFs (see below), is repeated similarly for all the tabulated values of a, /3, and 7. In accordance with eq. (4.3.4), the contributions from all branches / are accumulated, and then the average over the initial rotational distribution is performed by standard quadratures, e.g, Simpson and/or Gauss-Legendre depending on the tabulation previously chosen for a, /3j and 7. 

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