Boltzmann-type

The Wilson parameters A,, NRTL parameters G,, and UNIQUAC parameters X all inherit a Boltzmann-type T dependence from the origins of the expressions for G, but it is only approximate. Computations of properties sensitive to this dependence (e.g., heats of mixing and liquid/hquid solubihty) are in general only qualitatively correct. 

The data in Figure 5 can now be considered in light of the conduction model developed above. As stated previously, conduction in reduced poly-I behaves like an activated process. There are two sources that potentially could be responsible for this behavior. The first is the Boltzmann type concentration dependence of the 1+ and 1- states discussed above. The number of charge carriers is expected to decrease approximately exponentially with T. The second is the activation barrier to self-exchange between 1+ and 0 sites and 0 and 1- sites. For low concentration of charge carriers both processes are expected to contribute to the measured resistance. 

Because of charge neutrality, ye- t] B and is consequently negligible. That the same thing holds for yVe (and for yV/tI) is a postulate, but a very plausible one because otherwise we would have neutrino or antineutrino degeneracy with LV 2> B. The upshot is that the chemical potentials of neutrons and protons are equal and so from Eq. (2.43) one has a simple Boltzmann-type equilibrium ratio (n/p)eq = e-fa-mptflkT = g-1.29M eV/kT (4.36)... 

Another, more intelligent guess would be to assume that has a Boltzmann-type dependence on U i. In this case the natural approximation would be... 

Kn2 is related to a (Gibbs) free energy term, AnGt by a Boltzmann-type expression (e.g., Atkins, 1998) ... 

An early (perhaps the first) example of a quantum treatment of a dissipative process is Einstein s theory of spontaneous emission.6 To describe the interaction between matter and light, Einstein assumed the Boltzmann type of kinetic equations... 

The method of the most probable distribution that underlies both Boltzmann-type and Gibbs-type formulations is outlined with admirable clarity in Chapter II of Schrodinger s book. 

The radical anion dissociation conical intersection formulation described here seems reasonably complete for a solution environment, at least for the molecule considered. Naturally, refinements of the treatments of the electronic structure and the cavities could be considered, as was mentioned near the conclusion of the Introduction. However, as was noted in the Introduction to Section 3.7.2, other environments, e.g. DNA, are also of interest. Here one could pursue the development of a Poisson-Boltzmann type of description [100] for the inhomogeneous environment. 

Thus, even in the presence of pumping, of energy Sk, from the thermal bath into the modes k, a Planck-type distribution of photons over the modes k, and hence a Boltzmann-type distribution of molecular systems over vibrational states (v, V2- -vj,..vk,..vz) results from linear systems. 

Boltzmann If the phase points of the molecules in ji-space at time tA is not of the Maxwell-Boltzmann type, then the quantity H which controls the a<-dis-tribution will for the overwhelming majority of the motions after iA assume smaller values than it had at tA. 

The same model allows direct application of the Boltzmann-type formula... 

The analysis of the effect of temperature on the mean lifetime of foam bilayers provides further evidence for the applicability of the theory of bilayer rupture by hole nucleation [399,402,403]. The experiments show that the foam bilayers become less stable with increasing temperature, due both to the Boltzmann-type thermal activation of the hole nucleation and to the decreasing work of a nucleus hole formation. 

For this two-state system, the ratio of channels in the closed to open states is given by a Boltzmann-type equilibrium expression... 

The mathematics of the problems in thermochromatography involves integration of the Boltzmann-type factor multiplied by a power of temperature. It is required to evaluate integrals of the general form... 

The two-fluid granular flow model is formulated applying the classical Eulerian continuum concept for the continuous phase, while the governing equations of the particle phase are developed in accordance with the principles of kinetic theory. In this theory it is postulated that the particulate system can be represented considering a collection of identical, smooth, rigid spheres, adapting a Boltzmann type of equation. This microscopic balance describes the rate of change of the distribution function with respect to position and time. 

Assuming that the particle velocity for a given size is known, v = v(m, r,t), one may express a Boltzmann type of transport equation for the single distribution function ... 

T source term in generalized Boltzmann type of equation representing the effects of particle coalescence, breakage and collisions J c)) collision term in the Boltzmann equation... 

At temperature above T = 0 K the population (F) of the orbitals has a Boltzmann-type distribution which is described by the Fermi-Dirac distribution function ... 

Gorsch, D. 1999 Discrete model collision operators of Boltzmann type. Journal of Computational and Applied Mathematics 104, 145-162. 

Coalson RD, Beck TL. Poisson-Boltzmann type equations numerical methods. In Schleyer PvR, ed. Encyclopedia of Computational Chemistry. Vol. 3. New York Wiley, 1998 2086-2100. 

The force constant can only be determined if deformation energies cannot, in general, be derived from the observed parameter variances. Moreover, in contrast to the Boltzmann-type argument where the parameter variance was proportional to/ it is now proportional to Qualitatively there is an inverse relationship for both models. 

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