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

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 * H. Sandberg is with the School of Electrical Engineering, Royal Institute of Technology (KTH), Stockholm, Sweden, hsan@ee.kth.se.

The Observer Effect in Estimation with Physical Communication Constraints*

“…In particular, here thermal noise is inherent to the transmission medium and is not a result of dissipation in the measurement device. Hence the effect described here applies to all possible measurement devices, and not only to the types discussed in Sandberg et al (2011). Furthermore, we here also mention how the results generalize to the case when the measured system is not close to equilibrium.…”

“…This result is derived in Section 3. An earlier, much less general, version of this result was derived in Sandberg et al (2011). In particular, here thermal noise is inherent to the transmission medium and is not a result of dissipation in the measurement device.…”

Glycolytic Oscillations and Limits on Robust Efficiency

“…Below the critical point, eventually all sites will end up in the laminar phase, whereas above there is always a nonzero fraction of chaotic sites, and with increasing control parameter the fraction of laminar (nonchaotic) sites quickly diminishes. Analogies to fluid flows have been pointed out in a number of studies (20)(21)(22)(23) that indicate the potential relevance of the spatial dynamics for the long-term behavior in fluid systems. In a numerical study of pipe flow, Moxey and Barkley (24) observed that at Re ≈ 2300 turbulent puffs delocalize, and the turbulent fraction increases, which is in qualitative agreement with this picture.…”

“…3.6 still requires substantial phenomenology, because the formulas for z and p depend on assumptions about autocatalysis (q and a) and enzyme efficiencies and levels (k). It is hoped that this will encourage efforts in further unification of control theory with thermodynamics and statistical mechanics, and recent progress is encouraging (22). It also leads to rethinking how biology overcomes the "causality" limit with various mechanisms that exploit predictable environmental fluctuations (e.g., circadian rhythms) or provide remote sensing (e.g., vision and hearing), both of which can greatly mitigate hard limits such as Eq.…”

Classical and quantum stochastic models of resistive and memristive circuits

“…D ISSIPATION has long been realized as a feature, though also a resource, in the design of engineered systems, [1]- [11]. The question of how exactly to model dissipation, given that the fundamental physical dynamical equations of motion are Hamiltonian, remains of fundamental importance; the issue applies to both classical and quantum systems, [12]- [22]. Hamiltonian systems with a finite number of degrees of freedom have dynamical evolutions that preserve the canonical structure -that is, the Poisson brackets in classical theory, and the commutation relations in quantum theory.…”