Reaction-diffusion equation generalized

In addition to the fact that MPC dynamics is both simple and efficient to simulate, one of its main advantages is that the transport properties that characterize the behavior of the macroscopic laws may be computed. Furthermore, the macroscopic evolution equations can be derived from the full phase space Markov chain formulation. Such derivations have been carried out to obtain the full set of hydrodynamic equations for a one-component fluid [15, 18] and the reaction-diffusion equation for a reacting mixture [17]. In order to simplify the presentation and yet illustrate the methods that are used to carry out such derivations, we restrict our considerations to the simpler case of the derivation of the diffusion equation for a test particle in the fluid. The methods used to derive this equation and obtain the autocorrelation function expression for the diffusion coefficient are easily generalized to the full set of hydrodynamic equations. 

Transient is a C-program for solving systems of generally non-linear, parabolic partial differential equations in two variables (that is, space and time), in particular, reaction-diffusion equations within the generalized Crank-Nicolson Finite Difference Method. 

The apparent diffusion coefficient, Da in Eq. (11) is a mole fraction-weighted average of the probe diffusion coefficient in the continuous phase and the microemulsion (or micelle) diffusion coefficient. It replaces D in the current-concentration relationships where total probe concentration is used. Both the zero-kinetics and fast-kinetics expressions require knowledge of the partition coefficient and the continuous-phase diffusion coefficient for the probe. Texter et al. [57] showed that finite exchange kinetics for electroactive probes results in zero-kinetics estimates of partitioning equilibrium constants that are lower bounds to the actual equilibrium constants. The fast-kinetics limit and Eq. (11) have generally been considered as a consequence of a local equilibrium assumption. This use is more or less axiomatic, since existing analytical derivations of effective diffusion coefficients from reaction-diffusion equations are approximate. 

Remark 2.4 In the derivation of the generalized reaction-diffusion equation (2.82) we do not explicitly refer to the particular form of the waiting time PDF. Equation (2.82) is valid for arbitrary waiting time PDFs < (t) and has much wider applicability than subdiffusive transport. 

The generalized reaction-diffusion equation (2.82) can be written in a form using fractional derivatives for subdiffusive transport, where the waiting PDF of species i is given in Laplace space by (2.52), (i) 1 —. In that case... 

This section is devoted to probabilistic solutions of reaction-diffusion equations in terms of functional integrals. We will not attempt to cover the general theory and all relevant equations. Our purpose is to discuss the main ideas and principal results and give illustrating examples involving typical equations. The reader interested in the general theory and all mathematical details will find a comprehensive treatment of the subject in Freidlin s book [141],... 

If we replace Brownian motion by its simplest generalization, the persistent random walk, we obtain direction-independent reaction walks as the simplest generalization of reaction-diffusion equations. Both describe chemical reactions in the reaction-limited or activation-controlled regime. However, the activation barrier is only implicitly taken into account it is incorporated into the kinetic coefficients... 

In this section, we use Model B, see Sect. 3.4.2, to explore the effects of subdiffusion on the Turing instability. We consider the two-variable generalized reaction-diffusion equation [484],... 

Pearson has analyzed the effect of an immobile species on the Turing instability in two-variable activator-inhibitor systems for more general conditions [346]. Consider the 2+1 species system described by the following reaction-diffusion equations ... 

Spatio-temporal change of the quantities of components of chemically reactive mixtures are described by reaction-diffusion equations. These equations may be induced from reaction kinetics, when the diffusion of chemical substances is also involved, or deduced from a general theory of mixtures, when only mass-conservation is taken into account. (For theories of reactive mixtures in the deterministic, continuum context offered by the school of rational thermodynamics see, for example, Bowen (1969) or Samohyl (1982)). 

Self organization is a general phenomenon that occurs in many particle systems that are defined as active media. Such systems can be generally described by reaction-diffusion equations for their individual components i ... 

When we studied the emergence of temporal oscillations in Chapter 2, we found that it was useful to examine whether a small perturbation to a steady state would grow or decay. We now attempt a similar linear stability analysis of a system in which diffusion, as well as reaction, can occur. First, consider the general reaction-diffusion equation ... 

This equation generalizes the usual reaction-diffusion equations (first two terms on the R.H.S.) in that it describes reaction-diffusion systems in gels that may undergo volume changes in response to some stimulus, (e.g., temperature, solvent composition, illumination). However, in order to furthermore describe the transduction of chemical to mechanical energy, we have to introduce a coupling of the chemistry with a property of the gel, so that chemistry may affect... 

Equation (2.4.18) is not like the usual reaction-diffusion equations since the diffusion matrix has an antisymmetric part. This seemingly peculiar property is actually a general consequence of contracting the usual reaction-diffusion equations, for which the diffusion matrix may be a diagonal matrix of positive diffusion constants. On account of its sound physical basis, we shall use the Ginzburg-Landau equation in later chapters in preference to the A-co model. In particular, the existence of the Cj terms will turn out to be crucial to the destabilization of uniform oscillations (see Appendix A) and hence to the occurrence of a certain type of chemical turbulence. 

Generally a reactive one-phase liquid in which uniformity is not maintened by artificial mixing must be described by a Reaction-Diffusion Equation using the methods of 2.3. 

The evolution equations of such reactions schemes, has a general form called "Reaction-Diffusion Equations" ... 

The model we employ in the first study involves a general system of reaction-diffusion equations. This system includes the diffusional-thermal model [1], in which the effect of a given fluid dynamical field is employed in the transport (reaction-diffusion) equations describing the evolution of the temperature field and the concentrations of the chemical species participating in the chemical reactions. Thus the qualitative effects of thermal expansion are considered to be weak, and... 

Consider the general two-variable system of reaction-diffusion equations ... 

In the kinetics of formation of carbides by reaction of the metal widr CH4, the diffusion equation is solved for the general case where carbon is dissolved into tire metal forming a solid solution, until the concentration at the surface reaches saturation, when a solid carbide phase begins to develop on the free surface. If tire carbide has a tirickness at a given instant and the diffusion coefficient of carbon is D in the metal and D in the carbide. Pick s 2nd law may be written in the form (Figure 8.1)... 

Pollard and Newman" have also studied CVD near an infinite rotating disk, and the equations we solve are essentially the ones stated in their paper. Since predicting details of the chemical kinetic behavior is a main objective here, the system now includes a species conservation equation for each species that occurs in the gas phase. These equations account for convective and diffusive transport of species as well as their production and consumption by chemical reaction. The equations stated below are given in dimensional form since there is little generalization that can be achieved once large chemical reaction mechanisms are incorporated. 

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