On Lossless Approximations, the Fluctuation-Dissipation Theorem, and Limitations of Measurements

TL;DR

This paper uses control theory to explore the relationship between lossless and dissipative systems, deriving fundamental measurement limitations and explaining key physical principles like fluctuation-dissipation and Onsager relations.

Contribution

It establishes that dissipative systems can be approximated by lossless systems over long times and explains measurement limitations through this framework, linking physical laws to system approximations.

Findings

Lossless systems are dense in dissipative systems.

Active systems can be approximated by nonlinear lossless systems with initial energy.

Deterministic back action can be mitigated, stochastic back action depends on temperature.

Abstract

In this paper, we take a control-theoretic approach to answering some standard questions in statistical mechanics, and use the results to derive limitations of classical measurements. A central problem is the relation between systems which appear macroscopically dissipative but are microscopically lossless. We show that a linear system is dissipative if, and only if, it can be approximated by a linear lossless system over arbitrarily long time intervals. Hence lossless systems are in this sense dense in dissipative systems. A linear active system can be approximated by a nonlinear lossless system that is charged with initial energy. As a by-product, we obtain mechanisms explaining the Onsager relations from time-reversible lossless approximations, and the fluctuation-dissipation theorem from uncertainty in the initial state of the lossless system. The results are applied to measurement devices and are used to quantify limits on the so-called observer effect, also called back action, which is the impact the measurement device has on the observed system. In particular, it is shown that deterministic back action can be compensated by using active elements, whereas stochastic back action is unavoidable and depends on the temperature of the measurement device.