Simulations hierarchy

The different simulation hierarchies (QM, atomistic MD, and CG simulations) can be used to address phenomena or properties of a given system at several levels... 

Hierarchical Structure. In order to be better able to simulate the hierarchical nature of many real-world complex systems, in which agent behavior can itself be best described as being the result of the collective behavior of some swarm of constituent agents. Swarm is designed so that agents themselves can be swarms of other agents. Moreover, Swarm is designed around a time hierarchy, Thus, Swarm is both a nested hierarchy of swarms and a nested hierarchy of schedules. 

Recently a hierarchy of methods has been developed which covers the mapping of polymers to a mesoscopic level as well as the reintroduction of the atomistic structure [43-45]. Section 6 (Kremer, Murat, Hahn) gives some very first attempts to bridge the gap from microscopic to mesoscopic [43,44] and thereafter to the semi-macroscopic regime [45] within a simulation scheme. 

This is the isothermal bulk modulus. Thus we can use our simulation data in Figure 5.1 and calculate a modulus for a hard sphere system. Equations (5.14) to (5.16) form an interesting hierarchy of equations ... 

L, et al. 2001. Hierarchy of simulation models in predicting molecular recognition mechanisms from the binding energy landscapes structural analysis of the peptide complexes with SH2 domains. Proteins 45(4) 456-470. 

As noted in the Molecular Simulation of Structure and Properties section, there have been no fundamental principle-based mathematical models for Nafion that have predicted new phenomena or caused property improvements in a significant way. This is due to a number of limitations inherent in one or the other of the various schemes. These shortcomings include an inability to sufficiently account for chemical identity, an inability to simulate and predict the long-range structure as would be probed by SAXS or TEM, and the failure to simulate structure over different hierarchy levels. Certainly, advances in this important research front will emerge and be combined with advances in experimentally derived information to yield a much deeper state of understanding of Nafion. 

From a software point of view, it is desirable to have a well-structured hierarchical description for the different biomacromolecules in the simulation system. Such a hierarchy should provide data structures and access functions on the atomic, residue, molecule, and system levels. This allows routines for the evaluation of energy terms to be set up at the level of residue pairs. Experience [14] suggests that this setup is advantageous since it provides a route to easily and intuitively implement the computational algorithms sketched below. 

There can be any number of types of sites on a surface. For example, in the simulation of a crystal growth process we might specify that a surface consists of step sites and terrace sites. The number of sites of each type may be characteristic of the crystal surface, for example, the mis-cut orientation of a crystal face. We denote each surface site type as a phase these phases reside in a particular surface (2D) domain. Surface species occupy the surface sites (i.e., populate the surface phases), which is the next step down the hierarchy. 

In such a representation of an infinite set of master equations for the distribution functions of the state of the surface and of pairs of surface sites (and so on) will arise. This set of equations cannot be solved analytically. To handle this problem practically, this hierarchy must be truncated at a certain level. In such an approach the numerical part needs only a small amount of computer time compared to direct computer simulations. In spite of very simple theoretical descriptions (for example, mean-field approach for certain aspects) structural aspects of the systems are explicitly taken here into account. This leads to results which are in good agreement with computer simulations. But the stochastic model successfully avoids the main difficulty of computer simulations the tremendous amount of computer time which is needed to obtain good statistics for the results. Therefore more complex systems can be studied in detail which may eventually lead to a better understanding of such systems. 

In order to quantify the structure-property relations at each scale, a multiscale hierarchy of numerical simulations was performed, coupled with experiments, to determine the internal state variable equations of macroscale plasticity and damage... 

Actually, all methods of closure involve some type of modeling with the introduction of adjustable parameters that must be fixed by comparison with data. The only question is where in the hierarchy of equations the empiricism should be introduced. Many different systems of modeling have been developed. The zero-equation models have already been introduced. In addition there are one-equation and two-equation models, stress-equation models, three-equation models, and large-eddy simulation models. Depending on the complexity of the model and the problem... 

A computer simulation of promotion practices at a hypothetical corporation provides a convincing demonstration of the cumulative effects of small-scale bias.22 The simulation modeled an organization with an 8-level pyramidal hierarchy, in which each level was staffed with equal numbers of men and... 

Obviously, the spectrum of mesoscale, particle-based tools is too vast to be covered in a single paper. Therefore, in this and the subsequent sections, I mainly elaborate on MC methods to illustrate various aspects of multiscale modeling and simulation. Below, the modeling hierarchy for stochastic well-mixed chemically reacting systems is first outlined, followed by a brief introduction to MC methods. 

A hierarchy of models can often be derived from a more detailed model under certain assumptions. This approach was discussed above in the case of deterministic, continuum models (see Fig. 3a). Such hierarchical models can be valuable in multiscale modeling. Let us just mention two cases. First, one could use different models from a hierarchy of models for different situations or length scales. This approach plays a key role in hybrid multiscale simulation discussed extensively below. Second, one could easily apply systems tasks to a simpler model to obtain an approximate solution that is then refined by employing a more sophisticated, accurate, and expensive model from the hierarchy. 

Despite this last observation, for this type of simulation and modelling research, two main means of evolution remain the first consists in enlarging the library with new and newly coded models for unit operations or apparatuses (such as the unit processes mentioned above multiphase reactors, membrane processes, etc.) the second is specified by the sophistication of the models developed for the apparatus that characterizes the unit operations. With respect to this second means, we can develop a hierarchy dividing into three levels. The first level corresponds to connectionist models of equilibrium (frequently used in the past). The second level involves the models of transport phenomena with heat and mass transfer kinetics given by approximate solutions. And finally, in the third level, the real transport phenomena the flow, heat and mass transport are correctly described. In... 

As another approach, one could think in terms of simulations or develop two-dimensional analogues of the semi-emplrlcal equations of state discussed in sec. I.3.9d. Models that fit into the picture of adsorbate mobility ignore variations parallel to the surface, i.e. adopt the mean field approach. Such models have a hierarchy similar to that of the FFG or two-dimensional Van der Waals equations, where any effect that lateral interaction may have on the distribution is also disregarded. 

The dynamic behavior of the carbon cycle and other complex systems may tend toward conditions of no change or steady state when exchanges are balanced by feedback loops. For example, model simulations of historical and projected effects of anthropogenic CO2 and CH4 emissions are usually based on an assumed carbon-cycle steady state before the onset of human influence. It is important to understand that the concept of steady state refers to an approximate condition within the context of a particular time-dependent frame of reference. Sundquist (1985) examined this problem rigorously using eigenanalysis of a hierarchy of carbon-cycle box models in which boxes were mathematically... 

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