Macroscopic Limits of Microscopic Models

Abstract

The definitive, peer reviewed and edited version of this article is published in: [Rohan Abeyaratne], [2014] AbstractMany physical systems are comprised of several discrete elements, the equations of motion of each element being known. If the system has a large number of degrees of freedom, it may be possible to treat it as a continuous system. In this event, one might wish to derive the equations of motion of the continuous (macroscopic) system by taking a suitable limit of the equations governing the discrete (microscopic) system. The classical example of this involves a row of particles with each particle connected to its nearest neighbor by a linear spring, its continuum counterpart being a linearly elastic bar; see Figure 1.In a typical undergraduate engineering subject on, say Dynamics, the transition from a discrete system to a continuous system is usually carried out through a formal Taylor expansion of the terms of the discrete model about some reference configuration. The aim of this paper is to draw attention to the fact that a macroscopic model derived in this way should be examined critically in order to confirm that it provides a faithful representation of the underlying microscopic model. We use a specific (striking) example to make this point. In this example, a simple solution of the discrete model can be stable or unstable depending on the state of the system. However the corresponding solution of the continuous system is always unstable! We go on to show how the dispersion relations of the two models can be used to identify the source of the discrepancy and to suggest how one might modify the continuous model.

On the equivalence between continuum and car-following models of traffic flow

“…Thus for the optimal velocity model, the linear stability condition becomes Ψ 2 s (v 0 , s 0 , 0) < 0, which is impossible. Such an inconsistency was observed in (Abeyaratne, 2014), where a modified version of (21). However, the modification may not apply to general models.…”

“…According to (Abeyaratne, 2014), both roots of the equation have negative imaginary parts if and only if b 2 > 0 and 4b 1 b 2 d 2 − 4d 1 b 2 2 > d 2 2 ; i.e., Ψ v (v 0 , s 0 , 0) < 0, and…”

Multiband homogenization of metamaterials in real-space: Higher-order nonlocal models and scattering at external surfaces

“…Homogenization of wave scattering in the nonlinear regime is largely unexplored, though we mention a study of wave scattering at an interface between a phase-transforming solid and a linear elastic solid [AK92]. On the other hand, we also note that the linear dispersion behavior is important even in nonlinear problems, for instance to govern the effective dissipation [DB06] or stability [Abe14].…”

Construction and Analysis of a Discrete Heat Equation Using Dynamic Consistency: The Meso-scale Limit

“…Our analysis of these modified heat equations leads us to the conclusion that these efforts have no fundamental groundings in an overall consistent physical theory. Further, they are related to other efforts involving the examination of macroscopic limits of microscopic mathematical models (Abeyaratne (2014); Ascher (2020)) and the validity of expansions in a (supposedly) small parameter (Mazanov et al (1974)). An accurate, but colorful way of characterizing this situation is to look at it from the perspective of a "wacka-mole" situation where the things required to resolve one issue provides the opportunity for the creation of another unresolved issue.…”