Dynamics multiscale

As discussed previously, an important issue that all dynamic multiscale methodologies must deal with is the phonon spectrum mismatch between the atomistic and coarse-grained regions. In the CLS method, this effect is handled by weakly coupling the FE degrees of freedom to a Brownian motion heat bath that is set to the desired temperature. As a last remark, we need to mention that this hybrid scheme is computationally very efficient and reasonably simple to implement, so that several groups other than the original authors have adopted it. 

One of the issues in dynamical multiscale coupling is the tailoring of the time step to the different subdomains. If the same time step is used in both the atomistic and the continuum regions, computations will be wasted in the continuum model. However, if in the hand-shake region the size of the FEM elements is reduced to coincide with the individual atoms, it is difficult to tailor the time step. Therefore, the authors of the ODD method chose to use a uniform mesh for the continuum domain, so that a much larger time step could be used in the continuum model than in the atomistic one. A description of such a multiple-time-step algorithm is provided in the paper. 

Weinan, E. Vanden-Eijnden, E., in Multiscale Modelling and Simulation, LNCSE 39, 2004 (chapter Metastability, conformation dynamics, and transition pathways in complex systems)... 

Impressive, highly ordered centimetre-sized fibres are obtained whose synergistic growth mechanism based on the kinetic cross-coupling of a dynamical supramolecular self-assembly and a stabilizing silica mineralization may well be the basis of the synthetic paths used by Nature to obtain its materials with well-defined multiscale architectures in biological systems. 

An alternative mesoscale approach for high-level molecular modeling of hydrated ionomer membranes is coarse-grained molecular dynamics (CGMD) simulations. One should notice an important difference between CGMD and DPD techniques. CGMD is essentially a multiscale technique (parameters are directly extracted from classical atomistic MD) and it... 

Schreier H, Brown S (2004) Multiscale approaches to water management land-use impacts on nutrient and sediment dynamics. Scales in Hydrology and Water Management, Intern. Assoc. Hydrol. Sci, lAHS Publ, 287 61-75... 

Dynamic and Static Limitation in Multiscale Reaction Networks, Revisited... 

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. 

If the reader can use these properties (when it is necessary) without additional clarification, it is possible to skip reading Section 3 and go directly to more applied sections. In Section 4 we study static and dynamic properties of linear multiscale reaction networks. An important instrument for that study is a hierarchy of auxiliary discrete dynamical system. Let A, be nodes of the network ("components"), Ai Aj be edges (reactions), and fcy,- be the constants of these reactions (please pay attention to the inverse order of subscripts). A discrete dynamical system

dynamical system for a given network we find for each A,- the maximal constant of reactions Ai Af k ( i)i>kji for all j, and — i if there are no reactions Ai Aj. Attractors in this discrete dynamical system are cycles and fixed points. 

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

For deriving of the auxiliary discrete dynamical system we do not need the values of rate constants. Only the ordering is important. Below we consider multiscale ensembles of kinetic systems with given ordering and with well-separated kinetic constants ( (i) k(,(2) > > for some permutation cr). 

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). 

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