Spatially-extended systems

Let us start with the generalized balance equations (2.1.37). The stochastic differential equations arise due to a formal adding of the fluctuating particle sources. In a general case both the fluctuations of the diffusion flux 

These fluctuations are assumed to be local, i.e., uncorrelated in space and -correlated in time. It is generally-accepted for a stochastic variables that 

Equation (2.2.6) could also be generalized for the situation when the sources are concentration-dependent [67, 68]. As an illustration, let us generalize solution (2.2.8) of the explosive instability. Chemical reaction is described by the following equation 

Equation (2.2.18) is interpreted after Stratonovich. The correlation length H r) in (2.2.19) was taken in the exponential form [67], e.g., 

Therefore, the simplest procedure to get the stochastic description of the reaction leads to the rather complicated set of equations containing phenomenological parameters /ij (equation (2.2.17)) with non-transparent physical meaning. Fluctuations are still considered as a result of the external perturbation. An advantage of this approach is a useful analogy of reaction kinetics and the physics of equilibrium critical phenomena. As is well known, because of their nonlinearity, equations (2.1.40) reveal non-equilibrium bifurcations [78, 113]. A description of diffusion-controlled reactions in terms of continuous Markov process - equation (2.2.15) - makes our problem very similar to the static and dynamic theory of critical phenomena [63, 87]. When approaching the bifurcation points, the systems with reactions become very sensitive to the environment fluctuations, which can even produce new nonequilibrium transitions [18, 67, 68, 90, 108]. The language developed in the physics of critical phenomena can be directly applied to the processes in spatially extended systems. 

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In this book we summarize the state of the art in the study of peculiarities of chemical processes in dense condensed media its aim is to present the unique formalism for a description of self-organization phenomena in spatially extended systems whose structure elements are coupled via both matter diffusion and nonlocal interactions (chemical reactions and/or Coulomb and elastic forces). It will be shown that these systems could be described in terms of nonlinear partial differential equations and therefore are complex enough for the manifestation of wave processes. Their spatial and temporal characteristics could either depend on the initial conditions or be independent on the initial as well as the boundary conditions (the so-called autowave processes). 

Described in Section 2.1.1 the formal kinetic approach neglects the spatial fluctuations in reactant densities. However, in recent years, it was shown that even formal kinetic equations derived for the spatially extended systems could still be employed for the qualitative treatment of reactant density fluctuation effects under study in homogeneous media. The corresponding equations for fluctuational diffusion-controlled chemical reactions could be derived in the following way. As any macroscopic theory, the formal kinetics theory operates with physical quantities which are averaged over some physically infinitesimal volumes vq = Aq, neglecting their dispersion due to the atomistic structure of solids. Let us define the local particle concentrations... 

On the other hand, the stochastic treatment in terms of equation (2.2.2) is applied to the macroscopic values a(t) characterizing a system. As is generally-accepted, thermal fluctuations of macroscopic quantities being very small are not of great interest. The only exception when fluctuations appear to be important (which leads to an untrivial result) is the case when the solution of a set of equations F (c, ..., cs) = 0 finds itself near the bifurcation point [26, 34, 90], This is why we are going to consider now the spatially-extended systems [26, 67, 68],... 

The stochastic differential equation (2.2.15) could be formally compared with the Fokker-Planck equation. Unlike the complete mixing of particles when a system is characterized by s stochastic variables (concentrations the local concentrations in the spatially-extended systems, C(r,t), depend also on the continuous coordinate r, thus the distribution function f(Ci,..., Cs]t) turns to be a functional, that is real application of these equations is rather complicated. (See [26, 34] for more details about presentation of the Fokker-Planck equation in terms of the functional derivatives and problems of normalization.)... 

A more refined approach is based on the local description of fluctuations in non-equilibrium systems, which permits us to treat fluctuations of all spatial scales as well as their correlations. The birth-death formalism is applied here to the physically infinitesimal volume vo, which is related to the rest of a system due to the diffusion process. To describe fluctuations in spatially extended systems, the whole volume is divided into blocks having distinctive sizes Ao (vo = Xd, d = 1,2,3 is the space dimension). Enumerating these cells with the discrete variable f and defining the number of particles iVj(f) therein, we can introduce the joint probability of arbitrary particle distribution over cells. Particle diffusion is also considered in terms of particle death in a given cell accompanied with particle birth in the nearest cell. 

Following the approach discussed in Section 2.2.2, let us divide the whole reaction volume V of the spatially extended system into N equivalent cells (domains) [81]. However, there is an essential difference with the mesoscopic level of treatment in Section 2.2.2 a number of particles in cells were expected to be much greater than unity. Note that this restriction is not imposed on the microscopic level of system s treatment. Their volumes are chosen to be so small that each cell can be occupied by a single particle only. (There is an analogy with the lattice gas model in the theory of phase transitions [76].) Despite the finiteness of vq coming from atomistic reasons or lattice discreteness, at the very end we make the limiting transition vo - 0, iV - oo, v0N = V, to the continuous pattern of point dimensionless particles. 

In a spatially extended system, fluctuations that are always present cause the variables to differ somewhat in space, inducing transport processes, the most common one being diffusion. In the case of constant diffusion coefficients /), the system s dynamics is then governed by reaction-diffusion equations ... 

Although the hypercycle itself may be weak against parasitic molecules (i.e., those which are catalyzed but do not catalyze others), it is then discussed that compartmentalization by a cell structure may suppress the invasion of parasitic molecules [7] or that the reaction-diffusion system at spatially extended system resolves this parasite problem [8]. As chemistry of lipid, it is not so surprising that a compartment structure is formed. Still, as the origin of life, this means that more complexity and diversity in chemicals are required other than a set of information-carrying molecules (e.g., RNA). 

V0I.670 A. Dinklage, G. Marx, T. Klinger, L. Schweikhard (Eds.), Plasma Physics V0I.671 J.-R. Chazottes, B. Fernandez (Eds.), Dynamics of Coupled Map Lattices and of Related Spatially Extended Systems... 

The equivalent circuit of the basic, spatially extended, system as introduced in Ref. 147 is shown in Fig. 38. The electrochemical cell is approximated by a one-dimensional electrode with periodic boundary conditions, while the electrolyte is a two-dimensional, electroneutral... 

Figure 38. Equivalent circuit of the basic spatially extended system, x, direction parallel to the electrode z, direction perpendicular to the electrode.

For the homogeneous steady state, it makes no difference whether the ohmic resistance in the external circuit arises from the electrolyte resistance or from an external resistor deliberately introduced into the circuit. In a spatially extended system, however, these two sources of ohmic resistance have to be distinguished. This topic was studied by Mazouz et al, and their main results are summarized here. 

Durrett and Levin (4998) considered a simple lattice model occupied by three species in cyclic competition and observed that the behavior of the system in a spatially extended system with short range local interaction is different from the corresponding mean-field model. In general, cyclic competition in spatially extended systems produces a dynamical equilibrium in which all species coexist, while the mean-field model leads to either periodically oscillating total populations, or extinction of all except one of the species. 

Time-delayed feedback control has also been applied to purely noise-induced oscillations in a regime where the deterministic system rests in a steady state. It has been shown that in this way both the coherence and the mean frequency of the oscillations can be controlled in simple models [7-9, 49, 50] as well as in spatially extended systems [51-53]. 

To conclude, noise-induced front motion and oscillations have been observed in a spatially extended system. The former are induced in the vicinity of a global saddle-node bifurcation on a limit cycle where noise uncovers a mechanism of excitability responsible also for coherence resonance. In another dynamical regime, namely below a Hopf bifurcation, noise induces oscillations of decreasing regularity but with almost constant basic time scales. Applying time-delayed feedback enhances the regularity of those oscillations and allows to manipulate the time scales of the system by varying the time delay t. 

While these investigations have enlightened our basic understanding of nonlinear, spatially extended systems under the influence of time-delayed feedback and noise, they may also open up relevant applications as nano-electronic devices like oscillators and sensors. 

J. Garci a-Ojalvo and J. M. Sancho Noise in Spatially Extended Systems (Springer, New York, 1999). 

Large systems. If two oscillators can adjust their rhythms due to an interaction, then one can expect similar behavior in large populations of units. Among different models under investigation we outline regular oscillator lattices (which in the limit case provide a description of extended systems) [9], ensembles of globally (all-to-all) coupled elements, the main topic of the present contribution, random oscillator networks (small world networks, scale-free networks, etc), and spatially-extended systems [10, 44, 55]. 

Note, that for. spatially extended systems full synchroiiiiiation leads only to trivial spatial patterns, since phase and amplitude dynamics are then identical across the entire lattice. In the region of phase syiichronination, however, synchronized patch populations are typically separated by phase lags (as seen in Fig. IS.U), which when summed up over the whole lattice can give rise to complex spatio-temporal patteriLs. Most remarkably, in the... 

Garcia-Ojalvo J and Sancho J M 1999 Noise in Spatially Extended Systems (Berlin Springer)... 

Simulation methods for spatially extended systems are much more numerically challenging than are those for homogeneous systems. The governing equations, that is, the reaction-diffusion equations, are systems cff partial dif-... 

Occurrence of wave solutions in spatially extended systems is caused by an interplay of nonlinear internal dynamics combined with mass or energy transport (Kapral Showalter (1995)). TVavelling solitary pulses and front waves represent the simplest cases. Fronts are associated with bistability in the reaction-diffusion system and solitary pulse waves occur in excitable media. Near the boundary of a parameter domain where the waves appear more complex patterns may emerge due to loss of stability and bifurcations of the pulse/front waves (Krishnan et al. (1999)). Here we examine the case when multiple front waves appear and give rise to zig-zag spatiotemporal patterns. 

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