Diffusion systems description

By definition chirality involves a preferred sense of rotation in a three-dimensional space. Therefore, it can only be affected by a modification of the nonscalar fields appearing in the rate equations. For a reaction-diffusion system [equations (1)] these fields are descriptive of a vector irreversible process, namely, the diffusion flux J of constituent k in the medium. According to irreversible thermodynamics, the driving force conjugate to diffusion is... 

It is clear that all the richness of reaction-diffusion systems should persist in the more accurate nonlinear integral operator model. The important difference is, however, that very rapid variations, and posssibly discontinuities in the slope of or V, should be allowed in the integral operator description. 

For the moment, a coupling like (3.87) remains speculation. Only reaction-diffusion systems as studied in section 3.2 provide a modeling description which is based on reasonably first principles. The mathematically proper approach to autonomous meanders and relative Hopf bifurcation, which are present in the reaction-diffusion setting, would then be... 

The universal description of reaction-diffusion systems near a supercritical Hopf bifurcation is provided by the complex Ginzburg-Landau equation [11]. Action of global periodic forcing on the systems described by this... 

Thus, the disperse nanofiller particle aggregation in elastomeric matrix can be described theoretically wilhin the frameworks of a modified model of irreversible aggregation particle-cluster. The obligatory consideration of nanofiller initial particle size is a feature of the indicated model application to real system description. The indicated particles diffusion in polymer matrix obeys classical laws of Newtonian liquids hydrod5mamics. The offered approach allows to predict nanoparticle aggregate final parameters as a function of the initial particles size, their contents, and other factors. 

The sensitivity of fronts to the dynamics of small perturbations about the unstable or metastable states has been studied by Brunet and Derrida [61] for pulled fronts and Kessler et al. [227] for pulled and pushed fronts. The mean-field description of reacting and diffusing systems ceases to be valid for low values of the particle density p, values that correspond to less than one particle. This fact can be incorporated into the RD equation by introducing a cutoff for the reaction term. Such a cutoff strongly affects the front velocity. Throughout this section we consider for simplicity that space and time have been rescaled such that D = r =. ... 

As in the case of homogeneous systems, there are two kinds of stochastic descriptions for reaction-diffusion systems as well the master equation approach and the stochastic differential equation method. Until now we have dealt with the first approach however, stochastic partial differential equations are also used extensively. Most often partial differential equations are supplemented with a term describing fluctuations. In particular, time-dependent Ginzburg-Landau equations describe the behaviour of the system in the vicinity of critical points (Haken, 1977 Nitzan, 1978 Suzuki, 1984). A usual formulation of the equation is ... 

The phase description can explain expanding target patterns in reaction-diffusion systems. The same method, however, breaks down for rotating spiral waves because of a phase singularity involved. The Ginzburg-Landau equation is then invoked. 

We have seen in Sects. 3.5 and 4.2 that the Ginzburg-Landau equation is appropriate as a model reaction-diffusion system for which the method of phase description is demonstrated. Some coefficients of the expansion of Qy//bt where then calculated to give, see (3.5.13a,b and 4.2.27),... 

Fig. 5.2. Schematic apparatus for determining relative stability of two stable stationary states of a reaction-diffusion system. For description see text. Prom [1]...

The models for chemically reacting media discussed above described the evolution of the system on macroscopic scales. In some instances, especially when one considers applications of nonlinear chemical dynamics to biological systems or materials on nanoscales, a mesoscopic description will be more appropriate or even essential. In this section, we show how one can construct mesoscopic models for reaction-diffusion systems and how these more fundamental descriptions relate to the macroscopic models considered previously. 

The effect of a gradual recovery can be incorporated into the kinematical description by assuming that the velocity of the normal propagation of a flat curve is a certain function Vq T) of the time T that has elapsed after the previous curve had passed the same point of the medium. The actual form of this function depends on a particular reaction-diffusion system which is described by the kinematical model. It must approach the propagation velocity Vb of a solitary excitation pulse and should become smaller for shorter time intervals T between the waves. 

In this section, we study the existence and stability of the stationary solutions of our reaction-diffusion model (3). Our goal is not to give a full description of all the possible cases, but rather to emphasize some general properties which are common to the class of reaction-diffusion systems given by a set of equations similar to Equation (3) with Dirichlet boundary conditions [104]. 

In order to solve Eq. 4.23, proper boundary conditions have to be defined. According to the system description and geometry of the package, demonstrated in Fig. 4.3, the convective flow is probably very large compared with diffusion in the flow direction, and the dye transport by diffusion may be negligible in this direction at the inflow face. Providing the main transport of dyes into the package takes place by convection, the flux at the inlet for the boundary conditions is ... 

Part of the system reacts "classically and can be described by the Goldman-Hodgkin-Katz equation for transmembrane diffusion of ions. We believe that our earlier measurements were done on chloroplasts which completely fit such a system description (fig. 3A). Such chloroplasts revealed only monophasic decay kinetics (reaction 1) after the light was turned off. 

At first we tried to explain the phenomenon on the base of the existence of the difference between the saturated vapor pressures above two menisci in dead-end capillary [12]. It results in the evaporation of a liquid from the meniscus of smaller curvature ( classical capillary imbibition) and the condensation of its vapor upon the meniscus of larger curvature originally existed due to capillary condensation. We worked out the mathematical description of both gas-vapor diffusion and evaporation-condensation processes in cone s channel. Solving the system of differential equations for evaporation-condensation processes, we ve derived the formula for the dependence of top s (or inner) liquid column growth on time. But the calculated curves for the kinetics of inner column s length are 1-2 orders of magnitude smaller than the experimental ones [12]. 

Thus far we have considered systems where stirring ensured homogeneity witliin tire medium. If molecular diffusion is tire only mechanism for mixing tire chemical species tlien one must adopt a local description where time-dependent concentrations, c r,f), are defined at each point r in space and tire evolution of tliese local concentrations is given by a reaction-diffusion equation... 

Recently an alternative approach for the description of the structure in systems with self-assembling molecules has been proposed in Ref. 68. In this approach no particular assumption about the nature of the internal interfaces or their bicontinuity is necessary. Therefore, within the same formahsm, localized, well-defined thin films and diffuse interfaces can be described both in the ordered phases and in the microemulsion. This method is based on the vector field describing the orientational ordering of surfactant, u, or rather on its curlless part s defined in Eq. (55). 

Diffuse functions are large-size versions of s- and p-type functions (as opposed to the standard valence-size functions). They allow orbitals to occupy a larger region of spgce. Basis sets with diffuse functions are important for systems where electrons are relatively far from the nucleus molecules with lone pairs, anions and other systems with significant negative charge, systems in their excited states, systems with low ionization potentials, descriptions of absolute acidities, and so on. 

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