Boltzmann equations numerical problems

Understandably, it is much more common to see analyses of problems based on Eq. (32) since for simple geometries the solution can be written down in closed form, expressed in terms of simple functions. For plane surfaces, for example, the solutions are elementary hyperbolic functions while for an isolated spherical surface the Debye-Huckel potential expression prevails. For two charged spherical surfaces the general solution can be written down as a convergent infinite series of Legendre polynomials [16-19]. The series is normally truncated for calculation purposes [16] K For an ellipsoidal body ellipsoidal harmonics are the natural choice for a series representation [20]. (The nonlinear Poisson Boltzmann equation has been solved numerically for a ellipsoidal body... 

The problem is to relate v (z) to the surface potential - v (0) or the surface charge density a° = a(O)) and the volume fraction profiles of the components. Early versions t-2) of a polyelectrolyte adsorption model neglected the volume of the small Ions and solved (numerically) the Poisson-Boltzmann equation 13.5.6). A more sophisticated, yet simpler, approach was proposed by Bflhmer et al. who accounted for the Ion volume by adopting a multilayer Stem model, see fig. 5.17. This Is a straightforward extension of the monolayer Stern model discussed in sec. 3.6c. The charges of the ions and the segments are assumed to be located on planes in the centres of the lattice layers. The lattice is thus con-... 

The MSA is fundamentally connected to the Debye-Hiickel (DH) theory [7, 8], in which the linearized Poisson-Boltzmann equation is solved for a central ion surrounded by a neutralizing ionic cloud. In the DH framework, the main simplifying assumption is that the ions in the cloud are point ions. These ions are supposed to be able to approach the central ion to some minimum distance, the distance of closest approach. The MSA is the solution of the same linearized Poisson-Boltzmann equation but with finite size for all ions. The mathematical solution of the proper boundary conditions of this problem is more complex than for the DH theory. However, it is tractable and the MSA leads to analytical expressions. The latter shares with the DH theory the remarkable simplicity of being a function of a single screening parameter, generally denoted by r. For an arbitrary (neutral) mixture of ions, this parameter satisfies a simple equation which can be easily solved numerically by iterations. Its expression is explicit in the case of equisized ions (restricted case) [12]. One has... 

Full evaluation of equation (2.4) thus requires knowledge of the charge distribution at the electrode - electrolyte interface, a problem that has been explored in various works.For example, Dickinson and Compton recently used numerical modelling to solve the Poisson - Boltzmann equation, which describes the electric field in an electrolyte solution under thermodynamic equilibrium, for hemispherical electrodes. The simulations revealed a transition between two classical limits a planar double layer as predicted by the Gouy - Chapman model and the spherical double layer associated with a point charge (Coulomb s Law). This is illustrated in Fig. 2.2, in which the dimensionless charge density, Q ( FrqjRTEQEg) is plotted as a function of the dimensionless hemispherical electrode radius,... 

As matrix inversion is computationally very costly, so this particular technique is limited to one-dimensional problems. Also, the generalization of Eq. (6.102) to space-dependent dielectric constant creates extra numerical difficulties while solving Poisson-Boltzmann equation. 

A more general approach to the diffusion problem is needed. The essential concepts behind the development of general relationships regarding diffusion were given more than a century ago, by Maxwell [39] and Stefan [40]. The Maxwell-Stefan approach is an approximation of Boltzmann s equation that was developed for dilute gas mixtures. Thermal diffusion, pressure diffusion, and forced diffusion are all easily included in this theory. Krishna et al. [38] discussed the Maxwell-Stefan diffusion formulation and illustrated its superiority over the Pick s formulation with the aid of several examples. The MaxweU-Stefan formulation, which provides a useful tool for solving practical problems in intraparticle diffusion, is described in several textbooks and in numerous publications [7,41-44]. 

Among all the non-numerical approximation methods, the effectiveness of the variational methods is perhaps most surprising [11]. This method serves to determine characteristic values of linear operators. Since the time variable can be eliminated from almost every reactor equation by transforming it into a characteristic value problem, the variational method should have wide applications in reactor theory. Its use has been limited, so far, because Boltzmann s operator is not self adjoint or normal. Whether this limitation is a necessary one, remains to be seen. The reason for the great accuracy of the variational principle in simple problems of quantum mechanics is that any function which is positive everywhere and has a single maximum can be so well approximated by any other similar function. Thus... 

The two methods of computer simulation are known by the labels of Molecular Dynamic (MD) and Monte Carlo (MQ simulation. In the fimt the evolution of an assembly of N molecules is followed by numerical solution of Newton s equations of motion. The system is one of fixed N, V, and U and so is the simulation of a micro-canonical ensemble, but since the sequence of states is that of real time both equilibrium and dynamic information can be obtained. In the second method a sequence of states is generated such that each state occurs with a probability proportional to its Boltzmann factor, exp(-%(i )/fcT0. The sequence is (usually) specified by fixed values of N, V, and T, and so the ensemble represented is canonical. The ordering of the steps of the sequence is arbitrary (that is, it contains no information) and so only thermodynamic properties can be calculated. The principles and practice of these techniques are described elsewhere " both have been used to study the liquid-gas surface and here we describe only the special problems which these studies involve. 

There are basically two ways to overcome such problems. Firstly, one can thermally excite the system so that it can escape from local minima and continue to search the surrounding conformational space. This is the principle behind molecular dynamics simulations, which generate the trajectory of a molecule in time by numerically integrating Newton s equations of motion. This technique is discussed in another section of the present volume. It is also the principle behind Monte Carlo simulations, which build up a thermodynamic ensemble of molecular conformations based on their Boltzmann probabilities. Application of this approach to nucleic acids is discussed below. 

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