Linear surprisal

The H + X2 reactions give non-linear surprisal plots for vibrational and translational energy disposal and are approximately quadratic in form. By including an additional minimal-momentum transfer or Frank—Condon-like constraint [250, 251], the translational distributions are reproduced by a surprisal that is Gaussian in momentum. A corresponding vibrational form can be derived [241]. 

Many semi-classical and quantum mechanical calculations have been performed on the F + H2 reaction, mainly being restricted to one dimension [520, 521, 602]. The prediction of features due to quantum-mechanical interferences (resonances) dominates many of the calculations. In one semi-classical study [522], it was predicted that the rate coefficient for the reaction F (2P1/2) + H2 is about an order of magnitude smaller than that for F(2P3/2) 4- H2, which lends support to the conclusion [508] that the experimental studies relate solely to the reaction of ground state fluorine atoms. Information theory has been applied to many aspects of the reaction including the rotational energy disposal and branching ratios for F + HD [523, 524] and has been used for transformation of one-dimensional quantum results to three dimensions [150]. Linear surprisal plots occur for F 4- H2(i> = 0), as noted before, but non-linear surprisal plots are noted in calculations for F + H2 (v < 2) [524],... 

Initial analyses of the product vibrational energy distributions for X + HX suggested that they could be described by a linear surprisal plot. However, more recent work [545] suggests that there is a deviation... 

Elementary processes in chemical dynamics are universally important, besides their own virtues, in that they can link statistical mechanics to deterministic dynamics based on quantum and classical mechanics. The linear surprisal is one of the most outstanding discoveries in this aspect (we only refer to review articles [2-7]), the theoretical foundation of which is not yet well established. In view of our findings in the previous section, it is worth studying a possible origin of the linear surprisal theory in terms of variational statistical theory for microcanonical ensemble. 

The discovery, the linear surprisal, due to Kinsey, Bernstein, and Levine is about a rule on microcanonical rate constants ( /( /)) or the associated product distribution (p/(s/)) experimentally observed in a chemical reaction, in which a final state, for instance, in a vibrational level of a given energy Ej is specified. A statistically estimated product distribution pj (s ) corresponding to Pj(Ej) is called the prior distribution, which is usually evaluated in terms of the volume of a relevant classical phase space and is frequently represented in terms of energy parameters. Their remarkable finding [2-5] is an exponential form... 

Figure 17. Linear surprisal in the reaction of F + HBr —> HF(v ) + Br. Black triangles indicate an experimental energy disposal, while the white triangles represent a statistical inference. The linear surprisal is given by the circles connected by a straight line. (Reproduced from Ref. 42 with permission.)...

The information theoretic logic behind the MEP is perfectly self-contained within itself. On the other hand, the linear surprisal (LS) are often valid even in collision processes of energy conserved small systems, which are seemingly far... 

Here, we show, in terms of a variational principle of statistical mechanics, that a temperature proportional to Xj-1 can be naturally defined to characterize the first-order feature (fluctuation) around the peak position of a distribution in a state space that is projected onto an appropriate coordinate. It is also shown that the necessity of the so-called prior distribution can naturally result along with its clear physical meaning. Several new features of the linear surprisal theory are uncovered through the analysis. 

On the basis of the above physical situation presupposed, the entire state space is now represented in three independent modes (1) a mode of the reaction coordinate R, which is eventually led to translational or dissociative state (2) an internal state, that is, a vibrational mode under focus (denoted by A), which is to be observed as a linear surprisal and (3) all other remaining internal modes (collectively denoted as B). A critical region (denoted by SR) is supposed to exist somewhere on the R coordinate beyond the transition state. At SR the modes A and B are practically separated and uniquely identified. For instance, the path Hamiltonian at... 

We next consider an experimental situation in which the final energy disposals are projected only onto the A mode by integrating information over all the B modes. Since the linear surprisal is usually formulated on the basis of accumulated product distributions rather than reaction rates [2-7], we hence begin with a standard statistical theory for the ratio of a population produced in the product site ... 

As seen in the reaction path Hamiltonian, Eq. (80), the energies A and B are measured from zero. On the other hand, the linear surprisal Eq. (72) does not care about the origin in the energy coordinate. For the sake of simplicity,... 

We have discussed a variational structure in a microcanonical ensemble and shed a new light on a possible physical origin of the linear surprisal. There can be various classes of variational structures depending on systems under study. In the... 

In this review we have investigated only one aspect of the gigantic feature of the linear surprisal theory. To our regret, people s interest in the linear surprisal seems to have been almost lost these days, and we are afraid that it is regarded only as an empirically fitting formula. However, as we have discussed above, it is a general theory that should be examined from one s viewpoint to link elementary process and statistical mechanics. We are happy if this review stimulates such studies. 

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