Reaction-diffusion systems, mesoscopic

MORPHOLOGY OF SURFACES IN MESOSCOPIC POLYMERS, SURFACTANTS, ELECTRONS, OR REACTION-DIFFUSION SYSTEMS METHODS, SIMULATIONS, AND MEASUREMENTS... 

In what follows we will discuss systems with internal surfaces, ordered surfaces, topological transformations, and dynamical scaling. In Section II we shall show specific examples of mesoscopic systems with special attention devoted to the surfaces in the system—that is, periodic surfaces in surfactant systems, periodic surfaces in diblock copolymers, bicontinuous disordered interfaces in spinodally decomposing blends, ordered charge density wave patterns in electron liquids, and dissipative structures in reaction-diffusion systems. In Section III we will present the detailed theory of morphological measures the Euler characteristic, the Gaussian and mean curvatures, and so on. In fact, Sections II and III can be read independently because Section II shows specific models while Section III is devoted to the numerical and analytical computations of the surface characteristics. In a sense, Section III is robust that is, the methods presented in Section III apply to a variety of systems, not only the systems shown as examples in Section II. Brief conclusions are presented in Section IV. 

At the mesoscopic level, the reaction-diffusion system is described by a set of partial differential equations,... 

Morphology of Surfaces in Mesoscopic Polymers, Surfactants, Electrons, or Reaction-Diffusion Systems Methods,... 

There are three ways to simulate reaction-diffusion system. The traditional method is to solve partial differential equation directly. Another way is to divide system into cells, which is called cell dynamic scheme (CDS). Typical models are cellular automata (CA)[176] and coupled map lattice (CML)[177]. In cellular automata model, each value of the cell (lattice) is digital. On the other hand, in coupled map lattice model, each value of the lattice (cell) is continuous. CA model is microscopic while CML model is mesoscopic. The advantage of the CML is compatibility with the physical phenomena by smaller number of cells and numerical stability. Therefore, the model based on CML is developed. Each cell has continuum state and the time step is discrete. Generally, each cell is static and not deformable. Deformable cell (lattice) is supposed in order to represent deformation process of the gel. Each cell deforms based on the internal state, which is determined by the reaction between the cell and the environment. 

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. 

Elf, J., Doncic, A., and Ehrenberg, M. Mesoscopic reaction-diffusion in intracellular signaling, in (S. M. Bezrukov, Hans Frauenfelder, Frank Moss Eds.), Fluctuations and Noise in Biological, Biophysical, and Biomedical Systems . Proc. SPIE, 5110, 114-124, SPIE Digital Library, http //spie.org/app/Publications/, Santa Fe, NM, USA (2003). 

The equation above is the Fokker-Planck equation to estimate the evolution of the probability density in space. Various forms of the Fokker-Planck equations result from various expressions of the work done on the systems, and are used in diverse applications, such as reaction diffusion and polymer solutions (Rubi, 2008 Bedeaux et al., 2010 Rubi and Perez-Madrid, 2001). A process may lead to variations in the conformation of the macromolecules that can be described by nonequilibrium thermodynamics. The extension of this approach to the mesoscopic level is called the mesoscopic nonequilibrium thermodynamics, and applied to transport and relaxation phenomena and polymer solutions (Santamaria-Holek and Rubi, 2003). 

In order to construct mesoscopic models, we again begin by partitioning the system into cells located at the nodes of a regular lattice, but now the cells are assumed to contain some small number of molecules. We cannot use a continuum description of the dynamics in a cell as we did for the reaction-diffusion equation. Instead, we describe the reactions and motions of molecules using stochastic rules that mimic the dynamics of these processes on meso-scales. The stochastic element arises because we do not take into account the detailed motions of all solvent species or the dynamics on microscopic scales. Nevertheless, because the number of molecules in a cell may be small, we must account for the fact that this number can change by random reactive events and random motions of molecules that take them into and out of a... 

For most problems that involve the effects of molecular fluctuations on reactive dynamics, it is not necessary to revert to a full molecular dynamics description of the system. We are interested in particle number fluctuations of reactive chemical species that arise from reaction and diffusion processes and occur in, small fluid volume elements. The most appropriate scale for the consideration of fluctuations is the mesoscopic scale, the regime that lies... 

Substitution of this reaction probability matrix in the automaton mean-field equations (7) yields the Willamowski-Rbssler rate law (24). Since the full automaton dynamics is not mean field, we can now use the automaton to investigate the mesoscopic dynamics of this reacting system. In the simulations presented below we take the diffusion coefficients of all of the species to be the same thus, henceforth we dispense with the species label and refer to this common diffusion coefficient as D. 

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