Boltzmann equation, application

Chen SW, Honig B. Monovalent and divalent salt effects on electrostatic free energies defined by the nonlinear Poisson-Boltzmann equation application to DNA binding reactions. J Phys Chem B 1997 101 9113-9118. 

J. Granot, Biopolymers, 22, 1831 (1983). Effect of Finite Ionic Size on the Solution of the Poisson-Boltzmann Equation Application to the Binding of Divalent Metal Ions to DNA. 

S.W. Chen andB. Honig,/. Pfjys. Chem.B, 101,9113 (1997).MonovalentandDivalentSalt Effects on Electrostatic Free Energies Defined by the Nonlinear Poisson-Boltzmann Equation Application to DNA Binding Reactions. 

Tanford, C., Kirkwood, J. G. Theory of protein titration curves. I. General equations for impenetrable spheres. J. Am. Chem. Soc. 79 (1957) 5333-5339. 6. Garrett, A. J. M., Poladian, L. Refined derivation, exact solutions, and singular limits of the Poisson-Boltzmann equation. Ann. Phys. 188 (1988) 386-435. Sharp, K. A., Honig, B. Electrostatic interactions in macromolecules. Theory and applications. Ann. Rev. Biophys. Chem. 19 (1990) 301-332. 

An interesting historical application of the Boltzmann equation involves examination of the number density of very small spherical globules of latex suspended in water. The particles are dishibuted in the potential gradient of the gravitational field. If an arbitrary point in the suspension is selected, the number of particles N at height h pm (1 pm= 10 m) above the reference point can be counted with a magnifying lens. In one series of measurements, the number of particles per unit volume of the suspension as a function of h was as shown in Table 3-3. 

In a plasma, the constituent atoms, ions, and electrons are made to move faster by an electromagnetic field and not by application of heat externally or through combustion processes. Nevertheless, the result is the same as if the plasma had been heated externally the constituent atoms, ions, and electrons are made to move faster and faster, eventually reaching a distribution of kinetic energies that would be characteristic of the Boltzmann equation applied to a gas that had been... 

Application of the Boltzmann equation to Eq. (8.33) gives the entropy of the mixture according to this model for concentrated solutions ... 

Since the middle of the 1990s, another computation method, direct simulation Monte Carlo (DSMC), has been employed in analysis of ultra-thin film gas lubrication problems [13-15]. DSMC is a particle-based simulation scheme suitable to treat rarefied gas flow problems. It was introduced by Bird [16] in the 1970s. It has been proven that a DSMC solution is an equivalent solution of the Boltzmann equation, and the method has been effectively used to solve gas flow problems in aerospace engineering. However, a disadvantageous feature of DSMC is heavy time consumption in computing, compared with the approach by solving the slip-flow or F-K models. This limits its application to two- or three-dimensional gas flow problems in microscale. In the... 

Figure 1 shows the four gas flow regimes and the applicable models. The Boltzmann equation is valid for the whole range of Am, from 0 to infinity. A simplified Boltzmann equation or the collisionless Boltzmann equation, where the right-hand side reduces to zero, is suitable when Kn is very... 

Ceecignani, C., The Boltzmann Equation and its Applications, Springer-Verlag, New York (1988). 

This theory of the diffuse layer is satisfactory up to a symmetrical electrolyte concentration of 0.1 mol dm-3, as the Poisson-Boltzmann equation is valid only for dilute solutions. Similarly to the theory of strong electrolytes, the Gouy-Chapman theory of the diffuse layer is more readily applicable to symmetrical rather than unsymmetrical electrolytes. 

Clearly, there are two quite different types of models for a gas flow the continuum models and the molecular models. Although the molecular models can, in principle, be used to any length scale, it has been almost exclusively applied to the microscale because of the limitation of computing capacity at present. The continuum models present the main stream of engineering applications and are more flexible when applying to different macroscale gas flows however, they are not suited for microscale flows. The gap between the continuum and molecular models can be bridged by the kinetic theory that is based on the Boltzmann equation. 

The solution of the Poisson-Boltzmann equation with. the application to thermal explosions) 5) D.A. Frank-Kamenetskii, "Diffusion and Heat Exchange in Chemical Kinetics, pp 202-66, Princeton Uni v-Press, Princeton, NJ (1955) (Quoted from MaSek s paper) 6) L.N. Khitrin, "Fizika Goreniya i Yzryva (Physics of Combustion and Explosion), IzdMGU, Moscow (1957)... 

The second question concerns one particular aspect of general applicability of the simple mean field equations outlined above as opposed to more sophisticated statistical mechanical descriptions. In particular, the equilibrium Poisson-Boltzmann equation (1.24) is often used in treatments of some very short-scale phenomena, e.g., in the theory of polyelectrolytes, with a typical length scale below a few tens of angstroms (1A = 10-8 cm). On the other hand, the Poisson-Boltzmann equation implicitly relies on the assumption of a pointlike ion. Thus a natural question to ask is whether (1.24) could be generalized in a simple manner so as to account for a finite ionic size. The answer to this question is positive, with several mean field modifications of the Poisson-Boltzmann equation to be found in [5], [6] and references therein. Another ultimately simple naive recipe is outlined below. 

An interesting, but probably incorrect, application of the probabilistic master equation is the description of chemical kinetics in a dilute gas.5 Instead of using the classical deterministic theory, several investigators have introduced single time functions of the form P(n1,n2,t) where P(nu n2, t) is the probability that there are nl particles of type 1 and n2 particles of type 2 in the system at time t. They use the transition rate A(nt, n2 n2, n2, t) from the state with particles of type 1 and n2 particles of type 2 to the state with nt and n2 particles of types 1 and 2, respectively, at time t. The rates that are used are obtained by assuming that only uncorrelated binary collisions occur in the system. These rates, however, are only correct in the thermodynamic limit for a low density system. In this limit, the Boltzmann equation is valid from which the deterministic theory follows. Thus, there is no reason to attach any physical significance to the differences between the results of the stochastic theory and the deterministic theory.6... 

This section contains a rather elaborate application of the method of compounding moments. The subject is the celebrated Boltzmann equation )... 

C. Cercignani, Theory and Application of the Boltzmann Equation, Scottish Academic Press, Edinburgh, London, 1975. 

Rocchia W, Alexov E, Honig B (2001) Extending the applicability of the nonlinear Poisson-Boltzmann equation Multiple dielectric constants and multivalent ions. J Phys Chem B 105 6507-6514... 

An alternative theoretical approach is the application of the Poisson-Boltzmann equation on the so-called cell model, assuming a parallel and equally spaced packing of rod-like polyions [62, 63]. This allows one to calculate at finite concentration according to ... 

An application of the Boltzmann equation to thermionics, i.e. the emission of electrons from hot bodies, especially metals, 3 may be considered. Let n... 

We have presented the first application of the newly developed method for calculating the solvation free energy to protein, which is based on the extended scaled particle theory and the Poisson-Boltzmann equation. Although the results are still preliminary, it demonstrates a possibility of obtaining the quantity theoretically, which is difficult even for the modem... 

A crucial parameter-free test of the theory is provided by its application to micelle formation from ionic surfactants in dilute solution [47]. There, if we accept that the Poisson-Boltzmann equation provides a sufficiently reasonable description of electrostatic interactions, the surface free energy of an aggregate of radius R and aggregation number N can be calculated horn the electrostatic free energy analytically. The whole surface free energy can be decomposed into two terms, one electrostatic, and another due to short-range molecular interactions that, from dimensional considerations, must be proportional to area per surfactant molecule, i.e. 

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