Generalized Boltzmann Equation

Coefficient Equations.—To determine the coefficients of the expansion, the distribution function, Eq. (1-72), is used in the Boltzmann equation the equation is then multiplied by any one of the polynomials, and integrated over velocity. This gives rise to an infinite set of coupled equations for the coefficients. Only a few of the coefficients appear on the left of each equation in general, however, all coefficients (and products) appear on the right side due to the nonlinearity of the collision integral. Methods of solving these equations approximately will be discussed in later sections. 

In this equation, the generalized Boltzmann operator 2(i0)T (j0) is now implicitly defined in terms of reducible... 

If the diffusion coefficient depends on time, the diffusion equation can be transformed to the above type of constant D by defining a new time variable a = jDdt (Equation 3-53b). If the diffusion coefficient depends on concentration or X, the diffusion equation in general cannot be transformed to the simple type of constant D and cannot be solved analytically. For the case of concentration-dependent diffusivity, the Boltzmann transformation may be applied to numerically extract diffusivity as a function of concentration. 

This is sometimes called the linearized Poisson-Boltzmann equation . The general solution of the linearized Poisson-Boltzmann equation is... 

Equation (6) is the linear Poisson-Boltzmann equation. Although generally considered to be less accurate than its nonlinear counterpart, it has the advantage of being considerably easier to solve. In addition, in several cases it has been shown to give results very close to Eq. (4), even when the surface potentials are as high as one to two times the thermal voltage kT/e (i.e., 25-5 mV). Hence, Eq. (6) can yield information relevant to real colloidal systems under certain conditions. 

Boltzmann generalized Maxwell s theory and found that the probability p that a molecule will be found in a state with an energy E is given by Equations 10-15,... 

Lorenzani S (2011) Higher order slip accmding to the linearized Boltzmann equation with general boundary conditions. Philos Trans R Soc Lond Ser A 369(1944) 2228-2236... 

Before leaving this chapter, we briefly look at an important quantity known as Boltzmann s entropy, and we will examine reduced forms of the Liouville equation in generalized coordinates. 

The Boltzmann equation is generally used to obtain an expression for AS of simple mixtures (mixtures of solvent-solvent or solvent-simple solute molecules) from the number of different arrangements ft (or the thermodynamic probabilities) of the solute and solvent molecules in the system For simple systems, the volume elements of solution are modeled by a three-dimensional lattice, where solute or solvent molecules can occupy any cell within the... 

Figure 2 Illustration of a negatively charged biomolecular surface with charge density a in the presence of a mixed electrolyte. The surface may represent that of a colloidal or biophysical particle such as a membrane (plane), polynucleic acid (cylinder), or micelle (sphere) where the distance of closest approach of ions is designated x — a. n the solution of the Gouy-Chapman equation, and of the Poisson-Boltzmann equation in general, the charged surface is usually displaced from its actual position (relative to the solvent) to the plane of closest approach of nonadsorbed ions, also called the outer Helmholtz plane.

Onsager s theory can also be used to detemiine the fomi of the flucUiations for the Boltzmaim equation [15]. Since hydrodynamics can be derived from the Boltzmaim equation as a contracted description, a contraction of the flucUiating Boltzmann equation detemiines fluctuations for hydrodynamics. In general, a contraction of the description creates a new description which is non-Markovian, i.e. has memory. The Markov... 

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. 

When g = 1 the extensivity of the entropy can be used to derive the Boltzmann entropy equation 5 = fc In W in the microcanonical ensemble. When g 1, it is the odd property that the generalization of the entropy Sq is not extensive that leads to the peculiar form of the probability distribution. The non-extensivity of Sq has led to speculation that Tsallis statistics may be applicable to gravitational systems where interaction length scales comparable to the system size violate the assumptions underlying Gibbs-Boltzmann statistics. [4]... 

The Poisson-Boltzmann equation is a modification of the Poisson equation. It has an additional term describing the solvent charge separation and can also be viewed mathematically as a generalization of Debye-Huckel theory. 

Example 3. The mean free path of electrons scattered by a crystal lattice is known to iavolve temperature 9, energy E, the elastic constant C, the Planck s constant the Boltzmann constant and the electron mass M. (see, for example, (25)). The problem is to derive a general equation among these variables. 

The continuum treatment of electrostatics can also model salt effects by generalizing the Poisson equation (12) to the Poisson-Boltzmann equation. The finite difference approach to solving Eq. (12) extends naturally to treating the Poisson-Boltzmann equation [21], and the boundary element method can be extended as well [19]. 

Bather than carrying out the calculation for the general case, which yields rather unwieldy expressions, only equations sufficient to obtain certain approximations will be developed. If we multiply the Boltzmann equation, Eq. (1-39), by 1 = i%( 2)3r )) (0.9>)> the resulting equation is simply the equation of conservation of mass, since integrating unity over the collision integral gives zero ... 

J. G. Kirkwood and J. Boss, The Statistical Mechanical Basis of the Boltzmann Equation, in I. Frigogine, ed., Transport Processes in Statistical Mechanics, pp. 1-7, Interscience Publishers, Inc., New York, 1958. Also, J. G. Kirkwood, The Statistical Mechanical Theory of Transport Processes I. General Theory, J, Chem, Phys, 14, 180 (1946) II. Transport in Gases, J, Chem. Phys, 15, 72 (1947). 

Consider electrons of mass m and velocity v, and atoms of mass M and velocity V we have mjM 1. The distribution function for the electrons will be denoted by /(v,<) (we assume no space dependence) that for the atoms, F( V), assumed Maxwellian as usual, in the collision integral, unprimed quantities refer to values before collision, while primed quantities are the values after collision. In general, we would have three Boltzmann equations (one each for the electrons, ions, and neutrals), each containing three collision terms (one for self-collisions, and one each for collisions with the other two species). We are interested only in the equation for the electron distribution function by the assumption of slight ionization, we neglect the electron-electron... 

Fukui, S., and Kaneko, R., Analysis of Uitra-Thin Gas Fiim Lubrication Based on Linearized Boltzmann Equation First Report—Derivation of a Generalized Lubrication Equation In-ciuding Thermai Creep Fiow," ASME J. Tribal., Voi. 110, 1988,pp.253-262. 

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