Modeling Defined Contrasting Perspectives

2 Modeling Defined Contrasting Perspectives As is perhaps already evident, the idea conveyed by the word modeling is ambiguous. This ambiguity is ultimately linked to fundamental questions concerning 

As another example, this time drawn from the realm of classical thermodynamics, we may consider the thermal state of a neutron star, a high-temperature superconductor or a dense gas. In each case, there is little doubt as to the validity of thermodynamics itself On the other hand, if we wish to make progress in the description of the dense gas, for example, the laws of thermodynamics by themselves do not suffice. This is where modeling in fhe sense that it will be used primarily in this book comes in. In addition to the fundamental laws that apply to all thermodynamic systems, we must characterize those features of the problem that are nonuniversal. That is, we require an equation of state which has nowhere near the same level of generality as the laws of thermodynamics themselves. Again, 

These examples, and others like them, allow us to discern three distinct levels of model building, though admittedly the boundary between them is blurred. In particular, the level of such modeling might be divided into (i) fundamental laws, (ii) effective theories and (iii) constitutive models. Our use of the term fundamental laws is meant to include foundational notions such as Maxwell s equations and the laws of thermodynamics, laws thought to have validity independent of which system they are applied to. As will be seen in coming paragraphs, the notion of an effective theory is more subtle, but is exemplified by ideas like elasticity theory and hydrodynamics. We have reserved constitutive model as a term to refer to material-dependent models which capture some important features of observed material response. 

Once the degrees of freedom have been identified, a dynamics of these degrees of freedom is constructed. To continue with our elaboration of the sense in which both elasticity and hydrodynamic theories serve as paradigmatic examples of such thinking, we note that in the case of elasticity (we have yet to say precisely which elastic constitutive model we have in mind) the characterization of the system is in terms of kinematic quantities such as displacements and strains which are themselves surrogates for the full atomic-level description of the system. Similarly, in the hydrodynamic context, velocities and strain rates replace an atom by atom description of the system. What all of these examples have in common is their reliance on a truncated description of material response in which the underlying 

---

In the previous subsection, we used the example of diffusion to illustrate the proliferation of temporal scales in one of the central problems in the study of materials. The present discussion has a similar aim in that we will briefly review fhe features of plasticity that place modeling demands at many different spatial scales. Though plasticity is also an area of immense importance, the conceptual foundations for its analysis both at the macroscopic level as well as from a reductionist perspective are not nearly as mature as is the study of diffusion. Recall that at the macroscopic scale in the context of diffusion we have the time-honored diffusion equation while at the microseopic scale we have the machinery of transition state theory as the basis of a well-defined scheme for informafion passage. By way of contrast, the macroscopic equations of plasticity are not nearly as robust as the diffusion equafion and there is no clear path for... 

By contrast, all parameters (whether adjustable or uniquely determined) are expressed in terms of the molecular level structural features in statistical mechanical theories, such as the diffusion model of P D. [12-17] These parameters are defined by the local structure of, and interactions in, the polymer-penetrant system. Most of them describe features at the scale of polymer chain segments. The trade-off for this apparently more molecular level perspective is that physical assumptions whose general applicability and validity are questionable have to be made in defining the model. 

---