Model Semantics and Simulation of Time Scale Abstractions in Collision Models

Abstract

Effective design of models requires simplification of different components of the system. For example, interactions of the system under scrutiny with its environment are modeled as idealized sources and sinks, and system behavior is derived from idealized interactions among constituent components. Some of the more detailed behavior is governed by small parameters, that produce continuous but complex nonlinear behaviors. These nonlinear continuous behaviors are often simplified to piecewise continuous forms, which then results in discontinuities at the temporal or spatial scale of interest. The resulting models are of a mixed continuous/discrete, or hybrid, nature. Physical systems theory based on continuity of power and conservation of energy is well understood. However, once discontinuous effects are included, these laws may be violated at the macroscopic level. The challenge is to endow hybrid models with semantics that do not violate the underlying physical nature of the system. This paper studies the effects and semantics of time scale abstraction, such as embodied by Newton's collision rule. The results show that hybrid modeling of physical systems can be systematic without the need for ad hoc hysteresis effects.