Open Systems Viewed Through Their Conservative Extensions

TL;DR

This paper investigates how open systems, viewed as parts of larger conservative systems, are influenced by their minimal extensions, revealing that only a tiny part of the original system is relevant to observed dynamics.

Contribution

It characterizes the structure of system coupling in open systems and shows that only the minimal extension of the conservative system impacts the open system's behavior.

Findings

The minimal extension relevant to the open system is often very small.

The structure of coupling determines which parts of the conservative system influence the open system.

Explains why some degrees of freedom in solids do not affect their specific heat.

Abstract

A typical linear open system is often defined as a component of a larger conservative one. For instance, a dielectric medium, defined by its frequency dependent electric permittivity and magnetic permeability is a part of a conservative system which includes the matter with all its atomic complexity. A finite slab of a lattice array of coupled oscillators modelling a solid is another example. Assuming that such an open system is all one wants to observe, we ask how big a part of the original conservative system (possibly very complex) is relevant to the observations, or, in other words, how big a part of it is coupled to the open system? We study here the structure of the system coupling and its coupled and decoupled components, showing, in particular, that it is only the system's unique minimal extension that is relevant to its dynamics, and this extension often is tiny part of the original conservative system. We also give a scenario explaining why certain degrees of freedom of a solid do not contribute to its specific heat.