Analysis surprisal

Flow can one make practical use of these observations For some time I have advocated the use of information theory. (For more details see introductory discussions in Refs. [1] and [3], surprisal analysis in Ref. [23], applications to spectra in Refs. [24] and [30], and a recent prediction of a phase transition induced by cluster impact in Ref. [31]). What this approach seeks to do is to use the minimal dynamical input that is necessary to account for the dynamical observations of interest, the point being that one very rarely has the experimental resolution to probe the individual final quantum states. The information measured is much more coarse grained. 

There are two ways to implement this program. One is to directly discuss the distribution of final states. This is known as surprisal analysis . In this simpler procedure one does not ask how this distribution came about Instead, one seeks the coarsest or most statistical" (= of maximal entropy) distribution of final states, subject to constraints. The last proviso is, of course, essential. If no constraints axe imposed, one will obtain as an answer the prior distribution. It is the constraints that generate a distribution which contains just that minimal dynamical detail as is necessary to generate the answers of interest. Few simple and physically obvious constraints are often sufficient [1, 3, 23] to account for even extreme deviations from the prior distribution. 

The distribution generated by surprisal analysis is meant to reproduce the results of actual interest, a typical example being the distribution of vibrational energy of products, which is of interest say for chemical laser action. [3] The distribution is not meant to reproduce the fully detailed distribution in classical phase space, which, as already noted, has a to be a highly correlated and complicated distribution. 

Information theory was first applied [177] to chemical reactions in an attempt to compact and classify the energy distributions of reaction products. This is achieved by surprisal analysis, where the observed product energy distribution, say for vibration, P(v ), is compared with a non-specific prior distribution P°(v ). Then, the surprisal, I(v ), is given by... 

In addition to surprisal analysis of measured product energy distributions, surprisal synthesis has been applied [178] to the prediction of energy distributions either by induction from some more limited experimental data or by deduction from some dynamical calculation. In the inductive approach to surprisal synthesis, the available experimental data is used as a constraint to compute the surprisal parameter, X, by ensuring that the entropy is maximised. This surprisal parameter then determines a more detailed distribution. In a more modest way, this approach may be used to extend incomplete product energy distributions. For example, as mentioned before, infrared chemiluminescence measurements are incapable of determining the population of products in the vibrational ground state, v = 0, and this is often induced from the surprisal analysis of the other vibrational levels. 

Figure 3 Surprisal plots (18) for the HF vibrational state distribution from the exoergic H atom abstraction reaction F + (CH,)4C - (CH,),CCH2 + HF(v). (Bottom panel) The observed (by D. J. Bogan and D. W. Setser, J. Chem. Phys. 64 586 (1976)) distribution, P(v), open dots connected by a line, and the (so called, prior) distribution, P (v) full symbols, vs. the HF vibrational energy. The prior distribution is the one expected when all products final states are equally probable (18). The observed distribution is qualitatively different from the prior one and their deviance, the surprisal, —In(P(v)/P"(v)) is plotted vs. E/Ev, where Ev is the HF vibrational energy and E is the total energy, in the upper panel. One can interpret the linear dependence of the surprisal on the HF vibrational energy as reflecting the presence of a quantity which is conserved by the dynamics. (See, for example, ref. (108)). In this sense, surprisal analysis is analogous to the search for quantum numbers that are not destroyed by the intramolecular couplings.

The open circles in (b) are from the surprisal analysis. This histograms are obtained from classical trajectory calculations on the Melius-Blint surface. The arrows show the maximum OH energies a vallable in reaction (1) at E. 

Initial vibrational distributions for HF(v) and DF(v) were measured for v = 1 to 4 and were shown to be independent of the NF2 flow rate. Occupation of the v = 5 state was observed for neither HF nor DF. Experimental distributions, P, were used to extrapolate the occupation of the v = 0 state either from a linear log Pv vs. Ev correlation (E vibrational energy) or from a surprisal analysis. Resulting distributions are given below together with (E) (the mean energy being available for the NF(a) path, (E) = AHo+Ea (assumed 1 kcal/mol) + V2 RT) and energy fractions stored in vibration, (f ), and rotation (f ) [26] ... 

Already familiar a decade ago was the information-theoretic approach [15] to the characterization of disequilibrium population distributions and the applicability of surprisal analysis to non-statistical quantum state distributions and branching ratios [16]. 

The surprisal analysis has proven particularly useful in studying trends within a similar series of reactions. Some conclusions (31) are ... 

These conclusions show that three-body behavior dominates the vibrational energy disposal of the CH X systems, as well as SiH, and GeH. The y values found for the above reactions are model I, (4.7 + 0.7) model II (10 + 1) and model III (15 + 1). Thus, with the aid of surprisal analysis, the product vibrational distributions for all of the above systems can be compactly expressed, within the experimental error, by the parameters X and Xy. The fact that the same vibrational disequilibrium is o%-tained in the above cases confirms, via information theory, that all reactions follow similar dynamics. 

Rotational surprisal analysis has been developed (81,62.72. 73,32) however, applications are rare because of the lack of reliable Initial distributions. The rotational surprisal is conditional upon fixed values of f and E, and is given by. 

The characteristic energy values for vibrational to vibrational as well as rotational to rotational energy transfer and their weak dependence on temperature relates to the phenomenon of resonance. Transfer becomes less effective when the difference in energies of the modes involved becomes larger. This follows from so-called surprisal analysis (a statistical method) and is due to decreasing cross sections when mode frequencies v become very different. According to surprisal analysis the rate constant for transitions between states labeled v and v depends Boltzmann-like on the energy difference between them ... 

Figure 6.13 Surprisal analysis of HF vibrational excitation in the F + H2 H + HF reaction. The ordinate in this figure is energy. The energies of HF vibrational states are shown on the right. Because of anharmonicity they are not quite equally spaced. Left panel energetics. Shown is the energy release along the reaction coordinate. The exoergic reaction can populate up to the v = 3 state of HF. Middle panel the observed distribution of HF vibrations, P v), solid bars [adapted from M. J. Berry,...

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