Estimate of temperature and its uncertainty in small systems

TL;DR

This paper reviews how temperature and its uncertainty can be estimated in small systems using estimation theory, clarifying controversies and illustrating with a simple thermometer example.

Contribution

It applies estimation theory to analyze temperature fluctuations in small systems, providing new insights and clarifications on the subject.

Findings

Temperature uncertainty can be rigorously quantified using estimation theory.

Finite systems exhibit temperature fluctuations that can be distinguished from measurement noise.

The total observation time is crucial for accurate temperature estimation.

Abstract

The energy of a finite system thermally connected to a thermal reservoir may fluctuate, while the temperature is a constant representing a thermodynamic property of the reservoir. The finite system can also be used as a thermometer for the reservoir. From such a perspective the temperature has an uncertainty, which can be treated within the framework of estimation theory. We review the main results of this theory, and clarify some controversial issues regarding temperature fluctuations. We also offer a simple example of a thermometer with a small number of particles. We discuss the relevance of the total observation time, which must be much longer than the decorrelation time.