Statistical weight conditional

Now express the monomer conditional statistical weights in terms of matrix G ... 

The Hamiltonian equations for P and Q, and the variational condition for Xn provide together a formally exact set of coupled equations whose solution gives the time-evolution of the electronic states driven by nuclear motions. The present coupled equations generalize the ones previously presented in reference (21) to allow now for statistical weights in the quantal potential, which is the same for ail the initially populated states n. 

In complete equilibrium, the ratio of the population of an atomic or molecular species in an excited electronic state to the population in the groun d state is given by Boltzmann factor e — and the statistical weight term. Under these equilibrium conditions the process of electronic excitation by absorption of radiation will be in balance with electronic deactivation by emission of radiation, and collision activation will be balanced by collision deactivation excitation by chemical reaction will be balanced by the reverse reaction in which the electronically excited species supplies the excitation energy. However, this perfect equilibrium is attained only in a constant-temperature inclosure such as the ideal black-body furnace, and the radiation must then give -a continuous spectrum with unit emissivity. In practice we are more familiar with hot gases emitting dis-... 

For spectra corresponding to transitions from excited levels, line intensities depend on the mode of production of the spectra, therefore, in such cases the general expressions for moments cannot be found. These moments become purely atomic quantities if the excited states of the electronic configuration considered are equally populated (level populations are proportional to their statistical weights). This is close to physical conditions in high temperature plasmas, in arcs and sparks, also when levels are populated by the cascade of elementary processes or even by one process obeying non-strict selection rules. The distribution of oscillator strengths is also excitation-independent. In all these cases spectral moments become purely atomic quantities. If, for local thermodynamic equilibrium, the Boltzmann factor can be expanded in a series of powers (AE/kT)n (this means the condition AE < kT), then the spectral moments are also expanded in a series of purely atomic moments. 

The energy levels in the semi-empirical conformational energy calculations may be used to derive conditional probabilities using the statistical weights method described by Flory (13). 

Figure 5. Experimental s-weighted intensity function (a) compared with calculated si(s) curves for (b) random chain according to conditional probabilities derived from the statistical weights of Abe, Jernigan, and Flory ( i) with near-neighbor dependence of rotation angles, and (c) an assemblage of randomly packed spheres with average segments placed at their centroids (see text)...

In speaking of the protein as being in a state of highest statistical weight, it must be emphasized that the statement is made about the total system, protein plus solvent. At one temperature (or solvent, pH, etc.), the statistical weight of the system may be maximized if the protein is in its native state, while, at another temperature (or solvent, pH, etc.), the statistical weight may be maximized if the protein is in a denatured state. The computations reported here pertain to conditions under which, hopefully, the protein will be in its native state. 

Igumnov V.S., Statistical weight of formation fullerene in conditions catalytic interactions of methane with water the ferry, Conference ICHMS 2005, Sevastopol. 

The kinetics of these pyrolysis reactions were followed by several complementary methods under conditions as close to the product studies as possible. The most frequently-used ampule technique14 17) with gc analysis of 5 and the scavenger technique, with chloranil or Koelsch radical as scavenger 18), for very labile compounds 5 were complemented by the DSC method, in which the heat flow under conditions of linear temperature increase is analysed. It proved to be a particularly convenient and reliable technique 18- 21). Rates were followed over a temperature span of at least 40 °C with temperature control of 0.1-0.2 °C. All rate data and activation parameters were subjected to a thorough statistical analysis including statistical weights of errors. The maximum statistical errors in k were 3%, in AH 1 kcal mol-1 in AS 513 e.u. and in AG (at the temperature of measurement) g0.5 kcal mol-1. 

As soon as the reactant molecule includes numerous atoms (as is often the case in Organic Chemistry) one just cannot study the overall dynamics of the reaction. In particular, if one must renounce the investigation of the activation phase of the reaction, one must also renounce the attribution of statistical weights to individual trajectories. Then one must postulate, on the basis of either experimental information or physical intuition, initial activated states of the reactant system and study only its subsequent dynamical evolution. Thus the work is restricted to sample in a random way all the possible initial conditions with no attempt to obtain at the end theoretical values of experimental quantities. Nevertheless, this context is not too restrictive. The trajectory study of thermal unimolecular reactions allows one... 

Fig. 8. Representatives of the four conformations obtained in the M = 4 analysis and the conditional transition probabilities between them (lag time r = O.lps). Fat numbers indicating the statistical weight of each conformation, numbers in brackets the conditional probability to stay within a conformation. Flexibility in peptide angles is marked with arrows, cf. Fig. 7. Note that the transition matrix relating to this picture is not symmetric but reversible. Top left For the helix conformation the backbone is colored blue for illustrative purpose. It should be obvious from Fig. 5 that for significantly larger lag time r only two eigenvalues will correspond to metastability such that only the helical conformation and a mixed flexible and partially unfolded one remain with significantly high conditional probability to stay within...

For weighted indices, W is the statistical weight assigned to the ith sample, which accounts for the importance of the sample in the statistics statistical weight should usually satisfy two conditions ... 

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