Boltzmann statistical weight

The Boltzmann statistical weight for each rotamer is multiplied by (1 + aPt(cos 8)), with a being an adjustable parameter, before a particular rotamer is selected. Values of a, between zero and unity, will give rise to an order parameter for segment orientations. Average of Pt(cos 8) of 0.0 to 0.5. 

Allegra further generalized these considerations for polymers in various solvent media, and showed that the Boltzmann statistical weight for an ensemble of identical chains of length iV, within the Gaussian approximation is... 

This means that particle configurations where at least two particles overlap, i.e., have a distance r smaller than the diameter cr, are forbidden. They are forbidden because the Boltzmann factor contains a term, exp(—oo) 0, that leads to a vanishing statistical weight. Hence we have an ensemble of... 

We assume that exploring all possible forms for the fields corresponds to exploring the overall usual phase space. To determine the partition function Z the contributions from all the p+ r) and P- r) distributions are summed up with a statistical weight, dependent on p+ r) and p (r), put in the form analogous to the Boltzmann factor exp[—p (F)]], where the effective Hamiltonian p (F)] is a functional of the fields. The... 

For a particular conformation c of a molecule, the positions of all (united) atoms in space as well as the chain conformers are known. The potential energy of this conformation is therefore just the sum of the contributions, as given by equation (9) for all the united atoms and a particular energy quantity per gauche bond in the chain. The statistical weight for this conformation is proportional to the Boltzmann factor containing this segment potential ... 

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-... 

In the case of the NR and PR calculations, the statistical weights c, ly, and co are determined from the respective values of 2 derived from the potential energy maps. Values of the statistical weight parameters determined in this manner take explicit consideration of the relative size of the domains for each state, as denoted by the so-called "entropy factor In the case of the FR calculations, for which the absence of potential energy maps precludes computation of z values, the statistical weight parameters are given as simple Boltzmann factors ( 0 = 1). c Calculated for = 1 throughout. 

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. 

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