Boltzman equation

Our final form for the Boltzman equation is then obtained by substituting these last two expressions for the gain and loss terms into equation 9.29 and adding the term Vijf x, v, t) to the LHS for the case where there is an external force F Vjj is the gradient operator with respect to v and m is the hard-sphere mass) ... 

In this section we show how the fundamental equations of hydrodynamics — namely, the continuity equation (equation 9.3), Euler s equation (equation 9.7) and the Navier-Stokes equation (equation 9.16) - can all be recovered from the Boltzman equation by exploiting the fact that in any microscopic collision there are dynamical quantities that are always conserved namely (for spinless particles), mass, momentum and energy. The derivations in this section follow mostly [huangk63]. 

Euler s equation is thus recovered as a direct consequence of momentum conservation, but only via the zeroth-order approximation to the full solution to the Boltzman-equation. 

Taking the lowest order terms 6 and 5t, we have the basic collisionless form of the discrete Boltzman equation ... 

Including collisions yields the full Boltzman equation ... 

Iav89b] Lavallee, P, J.P.Boon and A.Noullez, Lattice Boltzman equation for laminar boundary flow, Complex Systems 3 (1989) 317-330. 

Radiant heat transfer from a surface is governed by the Stefan-Boltzman equation, see Volume 1, Chapter 9. 

Gur, Y. Ravina, I. Babchin, A. J., On the electrical double layer theory. II. The Poisson-Boltzman equation including hydration forces, J. Colloid Inter. Sci. 64, 333-341... 

Thermochemistry. Chen et al.168 combined the Kohn-Sham formalism with finite difference calculations of the reaction field potential. The effect of mobile ions into on the reaction field potential Poisson-Boltzman equation. The authors used the DFT(B88/P86)/SCRF method to study solvation energies, dipole moments of solvated molecules, and absolute pKa values for a variety of small organic molecules. The list of molecules studied with this approach was subsequently extended182. A simplified version, where the reaction field was calculated only at the end of the SCF cycle, was applied to study redox potentials of several iron-sulphur clusters181. 

The catalytic effect for reactions involving an ionic reactant usually shows a strong dependence on the total amphiphile concentration. The maximal effective rate constant is attained at concentrations just over the CMC. Romsted284 showed that this occurs due to the competition between the ion binding of the reactive ions (OH- in the example above) and the counterions of the amphiphile. Recently, Diekman and Frahm285 286 showed that it is possible to rationalize the kinetic data by describing the ion distribution through a solution of the Poisson-Boltzman equation. (See Fig. 5.1). 

According to the law of distribution of molecular velocities (Glasstone and Lewis, 1960), molecules in two different phases, at equilibrium, are related in translation through the Boltzman equation, stated as... 

Nicholls A, Honig B (1991) Rapid finite difference alogrithm, utililizing successive over-relaxation to solve the Poisson-Boltzman equation, J Comput Chem, 12 435-445... 

For the classical treatment of electrostatic interaction in solution the Poison-Boltzman equation (PBS) is commonly used... 

A similar approach was employed in Hecht et al. (1995) for determining the electrostatic potential near the surface of calf thymus DNA. Spin-spin interaction between an N-nitroxide derivative of 9-aminoacridine attached to DNA and free 15N-labeled nitroxides of different charges was monitored by electron-electron double resonance (ELDOR). The electrostatic potential near the surface of DNA was calculated using a nonlinear Poisson-Boltzman equation. The calculated results agreed with the experimental potentials. 

The electric potential j/ created by the micelle was calculated using a numerical integration of the Poisson-Boltzman Equation by the Runge-Kutta method on an electronic computer. We shall outline briefly the method used (14,15,16). The Poisson-Boltzman Equation can be written (for programming commodities) in the following form ... 

Calculated using a Poisson-Boltzman equation with nonelectrostatic effects modeled by a linear solvent accessible surface area dependence with B3LYP/6-31-H-G. ... 

The gas temperature will depend on the fuel used and the amount of excess air. For gaseous fuels, around 20% excess air is normally used, and 25% for liquid fuels. Radiant heat transfer from a surface is governed by the Stefan-Boltzman equation ... 

The Boltzman equation expresses the concentration of ion i belonging to the diffuse layer i d vs. its bulk concentration i b. As an example ... 

This balance equation can also be derived from kinetic theory [101], In the Maxwellian average Boltzman equation for the species s type of molecules, the collision operator does not vanish because the momentum mgCs is not an invariant quantity. Rigorous determination of the collision operator in this balance equation is hardly possible, thus an appropriate model closure for the diffusive force is required. Maxwell [65] proposed a model for the diffusive force based on the principles of kinetic theory of dilute gases. The dilute gas kinetic theory result of Maxwell [65] is generally assumed to be an acceptable form for dense gases and liquids as well, although for these mixtures the binary diffusion coefficient is a concentration dependent, experimentally determined empirical parameter. 

Boltzman equation because it does not distinguish between ions of like charge and therefore cannot account for the specificity of the distribution. In order to reflect the experimental reality, the Leodidis and Hatton model takes into account three characteristics of ions their charge, size and electrostatic free energy of hydration. 

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