Boltzmann equation variational solution

Consider a mixture of acoustic-mode (rL) and ionized-impurity (r,) scattering. For tL t, we would expect r 0 = 1.18 and for r, tl, rn0 = 1.93. But for intermediate mixtures, r 0 goes through a minimum value, dropping to about 1.05 at 15% ionized-impurity scattering (Nam, 1980). For this special case (sL = i, s, = — f), the integrals can be evaluated in terms of tabulated functions (Bube, 1974). For optical-mode scattering the relaxation-time approach is not valid, at least below the Debye temperature, but rn may still be obtained by such theoretical methods as a variational calculation (Ehrenreich, 1960 Nag, 1980) or an iterative solution of the Boltzmann equation (Rode, 1970), and typically varies between 1.0 and 1.4 as a function of temperature (Stillman et al., 1970 Debney and Jay, 1980). 

No experimental results are available for the nucleic acids, with or without methyl substitution, to test the theories, but we can compare the results for thymine to three theoretical estimates based on the linearized Poisson-Boltzmann equation. The AM1-SM2 and PM3-SM3 values are —16.5 and -20.1 kcal/mol, respectively. Using charges and force field parameters from the AMBER,347 CHARMM, and OPLS molecular mechanics force fields and a solute dielectric constant of 1, Mohan et al.i calculated solvation energies of -19.1, -10.4, and -8.4 kcal/mol. The wide variation is disconcerting. In light of such wide variations with off-the-shelf parameters, the SMx approach based on parameters specifically adjusted to solvation energies appears to be more reliable. 

A novel data analysis procedure is described, based on a variational solution of the Schrddinger equation, that can be used to analyze gas electron diffraction (GED) data obtained from molecular ensembles in nonequilibrium (non-Boltzmann) vibrational distributions. The method replaces the conventional expression used in GED studies, which is restricted to molecules with small-amplitude vibrations in equilibrium distributions, and is important in time-resolved (stroboscopic) GED, a new tool developed to study the nuclear dynamics of laser-excited molecules. As an example, the new formalism has been used to investigate the structural and vibrational kinetics of C=S, using stroboscopic GED data recorded during the first 120 ns following the 193 nm photodissociation of CS2. Temporal changes of vibrational population are observed, which can... 

Dr. Brooks informed me of recent considerations, by himself and Mr. Calame, which, unfortunately, he could not present at the Symposium. They used the variational principle as applied to the Boltzmann equation directly, rather than to equations of approximation methods. This way, the advantages of the positivity of the true solutions are preserved. 

Finite element modeling of DNA functionalized electrodes was applied to calculate the interfacial potential, and used to identify conditions for maximum potential change with target hybridization [35], Using different models such as the Donnan potential model [34] and numerical solution of the Poisson-Boltzmann equation for a three-dimensional model, the authors estimate a maximum potential variation of -17 mV for 100% h3djridization efficiency at the optimized DNA probe density of 3 x 10 cm even at low ionic strength. 

The determination of the constants Cs, t, and is a complicated problem requiring a complete solution of the Boltzmann equation, including the kinetic boundary layer. Exact solutions have been found only for certain modeled Boltzmann equations, like the BGK equation, in flows with a simple geometry (e-g- stationary shear flow along a flat plate in a semi-infinite space, the so-called Kramers problem). Approximate results have been obtained by using variational methods and moment expansions. ... 

The SMC Method —A modified Monte Carlo treatment is used for the solution of the Boltzmann equation in the resonance region. The energy variation of a neutron history is treated exactly and the space-angle transport is approximated by a flat-source collision probability formulation. 

The Bloch-Griineisen formula is a special case of a more general expression. Within a variational solution of the Boltzmann equation we can, to a good approximation, write the resistivity Pei-ph that is limited by the scattering of conduction electrons by phonons as... 

S. L. Brenner and R. E. Roberts,/. Phys. Chem., 77,2367 (1973). A Variational Solution of the Poisson-Boltzmann Equation for a Spherical Colloidal Particle. 

L. Martinez, A. Hernandez, A. Gonzalez, and F. Tejerina,/. Colloid Interface Sci., 152, 325 (1992). Use of Variational Methods to Establish and Increase the Ranges of Application of Analytic Solutions of the Poisson-Boltzmann Equation for a Charged Microcapillary. 

The final class of methods that we shall consider for calculating the electrostatic compone of the solvation free energy are based upon the Poisson or the Poisson-Boltzmann equatior Ihese methods have been particularly useful for investigating the electrostatic properties biological macromolecules such as proteins and DNA. The solute is treated as a body of co stant low dielectric (usually between 2 and 4), and the solvent is modelled as a continuum high dielectric. The Poisson equation relates the variation in the potential (f> within a mediu of uniform dielectric constant e to the charge density p ... 

Variational approaches provide a general way to derive many of the partial differential equations of physics. They are also a starting point for devising efficient numerical solution techniques. We summarize the variational formulation of electrostatics next and show how this leads to flexible numerical techniques for solving the Poisson equation. Similar ideas will be presented for the Poisson-Boltzmann (PB) equation in later sections. 

When a mechanical part is made from a polymer, and when it is to be used as a loadcarrying component, obviously it is not necessarily always going to be subject to a constant stress as in the creep test. It generally has to be designed to withstand some history of stress variation. How will the polymer respond to the stress history Can its response be predicted Fortunately, for hnear viscoelastic behavior, predicting the response is possible, because of the principle of superposition of solutions to linear differential equations. The student, of course, remembers that if y,(Ji ) and y2(x) are both solutions of an ordinary differential equation for y x), then the sum y (x) + yj (x) is also a solution. This is the basis of the Boltzmann Superposition Principle for linear polymer behavior. 

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