Mean field approximation differential equation

CO Stripping Chronoamperometiy Before discussing experimental results, let us examine what the LH mechanism predicts for the chronoamperometric response of an experiment where we start at a potential at which the CO adlayer is stable and we step to a final potential E where the CO adlayer will be oxidized. We will also assume that the so-called mean field approximation applies, i.e., CO and OH are well mixed on the surface and the reaction rate can be expressed in terms of their average coverages dco and qh- The differential equation for the rate of change of dco with time is... 

Assuming that (13.11) makes sense in the context of the system under investigation (i.e., that physical relaxation times are in the appropriate range for the condition of local equilibrium to be satisfactorily approximated), we seek the field-type differential equation that describes asymptotic (-evolution of fields Rfx, y, z, t) toward the known metric geometrical limit. Solutions of this equation are expected to describe a wide variety of thermal, acoustic, and diffusion phenomena in nonequilibrium conditions where local thermodynamic variables retain experimental meaning. 

Chemical oscillators are described on the basis of nonlinear dynamics, in that the underlying kinetic equations under steady-state conditions are nonlinear. If one assumes that the spatial distribution of the reaction species is uniform, then these variables will only depend on time, and mathematical description in the mean field approximation for the concentration variables c,- is achieved by a set of coupled (nonlinear) ordinary differential equations (ODEs). This will be the approach applied in this chapter. 

Sophisticated numerical techniques have been devised to study this standard model within mean field approximation. They exploit that the mean field problem of a Gaussian chain in an external field can be described by a modified diffusion equation in an external field [60]. The latter leads to a partial differential equation that can be solved by efficient computational techniques. Advanced real-space, spectral, and pseudospectral algorithms have been devised to this end [28, 78-80]. ... 

Potential 14.21 signifies the mean field approximation, which follows essentially from the use of the solvation free energy in the form of Equation 14.15. The analytical first derivative with respect to the nuclear coordinates R is obtained by differentiation of the free energy... 

In addition to phase change and pyrolysis, mixing between fuel and oxidizer by turbulent motion and molecular diffusion is required to sustain continuous combustion. Turbulence and chemistry interaction is a key issue in virtually all practical combustion processes. The modeling and computational issues involved in these aspects have been covered well in the literature [15, 20-22]. An important factor in the selection of sub-models is computational tractability, which means that the differential or other equations needed to describe a submodel should not be so computationally intensive as to preclude their practical application in three-dimensional Navier-Stokes calculations. In virtually all practical flow field calculations, engineering approximations are required to make the computation tractable. 

The equation is used to describe the behaviour of an atom or molecule in terms of its wave-like (or quantum) nature. By trying to solve the equation the energy levels of the system are calculated. However, the complex nature of multielectron/nuclei systems is simplified using the Born-Oppenheimer approximation. Unfortunately it is not possible to obtain an exact solution of the Schrddinger wave equation except for the simplest case, i.e. hydrogen. Theoretical chemists have therefore established approaches to find approximate solutions to the wave equation. One such approach uses the Hartree-Fock self-consistent field method, although other approaches are possible. Two important classes of calculation are based on ab initio or semi-empirical methods. Ah initio literally means from the beginning . The term is used in computational chemistry to describe computations which are not based upon any experimental data, but based purely on theoretical principles. This is not to say that this approach has no scientific basis - indeed the approach uses mathematical approximations to simplify, for example, a differential equation. In contrast, semi-empirical methods utilize some experimental data to simplify the calculations. As a consequence semi-empirical methods are more rapid than ab initio. 

Dickman [60], who was the first to study the mean-field description of the ZGB model, used a different approach in order to circumvent the problem of AB pairs. He split up the adsorption reactions in the ZGB model, thereby differentiating between adsorption adjacent to different surface species. For example, when an A molecule adsorbs next to a Bads> immediate reaction will occur. Thus, the reaction becomes A(g) -I- - - Bads —> AB(g) - - 2. This leads in the MF approximation to differential equations with fourth-order terms for the surface coverages of A and B. The infinite rate constant kr is absent from these equations. 

We have also cast the DMC model in a set of ordinary differential equations, thus translating it to a mean-field approach with the site-approximation. Only the kinetic oscillations can be modeled in this way. To model the spatio-temporal pattern formations, diffusion terms would have to be added to the mean-field description, in order to account for the spatial dependence of the reactant concentrations. 

If one accepts the continuum, linear response dielectric approximation for the solvent, then the Poisson equation of classical electrostatics provides an exact formalism for computing the electrostatic potential (r) produced by a molecular charge distribution p(r). The screening effects of salt can be added at this level via an approximate mean-field treatment, resulting in the so-called Poisson-Boltzmann (PB) equation [13]. In general, this is a second order non-linear partial differential equation, but its simpler linearized form is often used in biomolecular applications ... 

Turbulence models containing the partial differential equations for the mean variable fields, and no differential equations for the turbulence are classified as the zero-equation models. AU models belonging to this class is based on the eddy-viscosity concept. The eddy-viscosity is furthermore related to the mean flow field via an algebraic relation. Therefore, these models are also called algebraic models. Because of their simplicity, zero-equation models have received considerable interest over the years, and have been in common use for sophisticated engineering applications during the last decades. The very simplest approximation of the turbulent effects on... 

In 1951, Hall [6] and, independently, Roothaan [7] put the Hartree-Fock equations - the ubiquitous independent particle model - in their matrix form. The Hartree-Fock equations describe the motion of each electron in the mean field of all the electrons in the system. Hall and Roothaan invoked the algebraic approximation in which, by expanding molecular orbitals in a finite analytic basis set, the integro-differential Hartree-Fock equations become a set of algebraic equations for the expansion coefficients which are well-suited to computer implementation. 

Fig. 8.31. Temperature dependence of magnetic susceptibilities for the 4TXg term at different applied fields m—mean magnetic susceptibility d—differential magnetic susceptibility (solid) a— approximate magnetic susceptibility based on the van Vleck equation.

Fig. 8.32. Temperature dependence of effective magnetic moments for the 4Tlg term at different applied fields m—based on the mean magnetic susceptibility d—based on the differential magnetic susceptibility (solid) a—based on the approximate magnetic susceptibility via the van Vleck equation.

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