Multiscale relaxation

Multiscale ensembles of reaction networks with well-separated constants are introduced and typical properties of such systems are studied. For any given ordering of reaction rate constants the explicit approximation of steady state, relaxation spectrum and related eigenvectors ( modes ) is presented. In particular, we prove that for systems with well-separated constants eigenvalues are real (damped oscillations are improbable). For systems with modular structure, we propose the selection of such modules that it is possible to solve the kinetic equation for every module in the explicit form. All such solvable networks are described. The obtained multiscale approximations, that we call dominant systems are... 

Our goal is the general theory of static and dynamic limitation for multiscale networks. The concept of the limiting step gives, in some sense, the limit simplification the whole network behaves as a single step. As the first result of our chapter we introduce further detail in this idea the whole network behaves as a single step in statics, and as another single step in dynamics even for simplest cycles the stationary rate and the relaxation time to this stationary rate are limited by different reaction steps, and we describe how to find these steps. 

In our approach, we analyze not only the steady-state reaction rates, but also the relaxation dynamics of multiscale systems. We focused mostly on the case when all the elementary processes have significantly different timescales. In this case, we obtain "limit simplification" of the model all stationary states and relaxation processes could be analyzed "to the very end", by straightforward computations, mostly analytically. Chemical kinetics is an inexhaustible source of examples of multiscale systems for analysis. It is not surprising that many ideas and methods for such analysis were first invented for chemical systems. 

Another general effect observed for a cycle is robustness of stationary rate and relaxation time. For multiscale systems with random constants, the standard deviation of constants that determine stationary rate (the smallest constant for a cycle) or relaxation time (the second in order constant) is approximately n times smaller than the standard deviation of the individual constants (where n is the cycle length). Here we deal with the so-called order statistics. This decrease of the deviation as n is much faster than for the standard error summation, where it decreases with increasing n as... 

RELAXATION OF MULTISCALE NETWORKS AND HIERARCHY OF AUXILIARY DISCRETE DYNAMICAL SYSTEMS... 

As it is demonstrated, dynamics of this system approximates relaxation of the whole network in subspace = 0. Eigenvalues for Equation (45) are —k, (i < n), the corresponded eigenvectors are represented by Equations (34), (36) and zero-one multiscale asymptotic representation is based on Equations (37) and (35). 

Is there a constant that limits the relaxation time The general answer for multiscale system is 1/t is equal to the minimal reaction rate constant of the dominant system. It is impossible to guess a priori, before construction of the dominant system, which constant it is. Moreover, this may be not a rate constant for a reaction from the initial network, but a monomial of such constants. 

We study a multiscale system with a given reaction rate constants ordering, kj >kj > >kj. Let us suppose that the network is weakly ergodic (when there are several ergodic components, each one has its longest relaxation time that can be found independently). We say that kj, l r n is the ergodicity boundary k if the network of reactions with parameters kj, kj, .., kj (when. . kj — 0)... 

If there is neither the first obstacle, nor the second one, then The possibility of these obstacles depends on the definition of multiscale ensembles we use. For example for the log-uniform distribution of rate constants in the ordering cone >kj > >kj (Section 3.3) the both obstacles have nonzero probability, if they are topologically possible. However, if we study asymptotic of relaxation time at e 0 for A , = skj i for given values of kj, kj, ..., kj -i, then for sufficiently small e>0 the second obstacle is impossible. 

The Mittag-Leffler function [44-46] can be viewed as a natural generalization of the exponential function. Within fractional dynamics, it replaces the traditional exponential relaxation patterns of moments, modes, or of the Kramers survival. It is an entire function that decays completely monotoni-cally for 0 < a < 1. It is the exact relaxation function for the underlying multiscale process, and it leads to the Cole-Cole behavior for the complex... 

Christen and van Gunsteren ° have developed a novel multiscale method that they call multigraining , which aims to use the CG model to enable both relaxation of large molecular systems and sampling of slow processes with concurrent atomic detail representation of the results. In this method, both an atomistic and a CG model of a molecule are used simultaneously. Each molecule in the simulation has... 

The kinematic viscosity of the fluid is related to the relaxation parameter through the relation v = c (t — l/2)i5(, which is obtained by means of a multiscale Chapman-Enskog analysis [5]. 

Variations of the average size of cationic argon clusters excited by electron impact at t — 0, as predicted by a time multiscale approach combining non-adiabatic molecular dynamics trajectories until internal conversion to the ground electronic state is achieved, followed by classical MD up to 100 ps and finally a kinetic approach based on phase space theory for sequential monomer evaporation. The results of approximate treatments with harmonic densities of states, or with the neglect of the classical MD relaxation stage, are also shown. 

Coupling Atomistic and Continuum Length Scales in Heteroepitaxial Systems Multiscale Molecular-Dynamics/Finite-Element Simulations of Strain Relaxation in Si/Si3N4 Nanopixels. 

Many multiscale methods have been developed across different disciplines. Consequently, much needs to be done in the fundamental theory of multiscale numerical methods that applies across these disciplines. One method is famous in structural materials problems the quasi-continuum method of Tadmor, Ortiz, and Philips. It links the atomistic and continuum models through the finite element method by doing a separate atomist structural relaxation calculation on each cell of the finite element method mesh, rather than using empirical constitutive information. Thus, it directly and dynamically incorporates atomistic-scale information into the deterministic scale finite element method. It has been nsed mainly to predict observed mechanical properties of materials on the basis of their constituent defects. 

In order to cover the fidl range of multiscale dynamics, one needs a relatively large dimension of the sheet in our bond fluctuation top-down) approach as in studying the multiscale dynamics of polymer chains. As mentioned above, a composite consists of a number of components represented by particles, chains, and sheets. Multiscale characteristics of each component, for example, particles, chains, and sheets, are further modified when a large number of these constituents are placed in a simulation box, that is, by their concentrations (volume fractions). Interaction and physical constraints at higher concentrations introduce multiple relaxation times for composites to reach equilibrium or steady state. In order to carry out a systematic investigation and draw meaningful conclusions, we restrict ourselves to constituents... 

If one uses a standard real space SOR relaxation algorithm on a given scale, invariably the solution process will stall after the short wavelength modes of the error have been removed. This led to the development of multiscale methods, discussed next. 

The same conceptual formulation was used for both the kinematic model of filling and the kinematic model of isovolumic relaxation and similar hypotheses regarding DD can be made. Indeed, the fact that the same type of conceptual and mathematical modehng works when applied to different physiology problems underscores the multiscale power of kinematic modeling [66]. 

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