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Optimal Estimation of Temperature

Shaoyong Zhang, Zhaoyu Fei, Xiaoguang Wang

TL;DR

The work addresses estimating temperature in finite-sized systems where fluctuations are non-negligible. It develops a rigorous estimation-theoretic framework, revealing that the uniform minimum-variance unbiased estimator (UMVUE) for temperature and its powers is intimately connected to entropy forms, with $S_B = \ln [\sigma(E)\epsilon]$ and $S_G = \ln \Omega(E)$. It derives a refined energy–temperature uncertainty relation (ETU) and demonstrates meaningful sample-size dependence, including large-$N$ Gaussian limits via the CLT and a $1/N$ scaling in ETU, and extends the analysis to repeated sampling in Hill’s nanothermodynamics. The framework provides a path toward high-precision nanothermometry, enabling experimental tests of finite-size temperature fluctuations in many-body systems.

Abstract

Temperature of a finite-sized system fluctuates due to the thermal fluctuations. However, a systematic mathematical framework for measuring or estimating the temperature is still underdeveloped. Here, we incorporate the estimation theory in statistical inference to estimate the temperature of a finite-sized system and propose optimal estimation based on the uniform minimum variance unbiased estimation. Treating the finite-sized system as a thermometer measuring the temperature of a heat reservoir, we demonstrate that different optimal estimation of parameters yield different formulas of entropy, e.g., optimal estimation of inverse temperature (or temperature) aligns with the Boltzmann entropy (or Gibbs entropy). The optimal estimation leads to a achievable energy-temperature uncertainty relation and exhibits sample-size dependence, coinciding with their counterparts in nanothermodynamics. The achievable bound and the non-Gaussian distribution of temperature enable experimental testing in finite-sized systems.

Optimal Estimation of Temperature

TL;DR

The work addresses estimating temperature in finite-sized systems where fluctuations are non-negligible. It develops a rigorous estimation-theoretic framework, revealing that the uniform minimum-variance unbiased estimator (UMVUE) for temperature and its powers is intimately connected to entropy forms, with and . It derives a refined energy–temperature uncertainty relation (ETU) and demonstrates meaningful sample-size dependence, including large- Gaussian limits via the CLT and a scaling in ETU, and extends the analysis to repeated sampling in Hill’s nanothermodynamics. The framework provides a path toward high-precision nanothermometry, enabling experimental tests of finite-size temperature fluctuations in many-body systems.

Abstract

Temperature of a finite-sized system fluctuates due to the thermal fluctuations. However, a systematic mathematical framework for measuring or estimating the temperature is still underdeveloped. Here, we incorporate the estimation theory in statistical inference to estimate the temperature of a finite-sized system and propose optimal estimation based on the uniform minimum variance unbiased estimation. Treating the finite-sized system as a thermometer measuring the temperature of a heat reservoir, we demonstrate that different optimal estimation of parameters yield different formulas of entropy, e.g., optimal estimation of inverse temperature (or temperature) aligns with the Boltzmann entropy (or Gibbs entropy). The optimal estimation leads to a achievable energy-temperature uncertainty relation and exhibits sample-size dependence, coinciding with their counterparts in nanothermodynamics. The achievable bound and the non-Gaussian distribution of temperature enable experimental testing in finite-sized systems.
Paper Structure (11 sections, 1 theorem, 73 equations, 6 figures)

This paper contains 11 sections, 1 theorem, 73 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Estimation of temperature in single sampling
  3. Estimation of temperature for large-$N$ systems
  4. Estimation of temperature in repeated sampling
  5. Conclusion and Outlook
  6. Unbiased estimator of $\beta^{k}$
  7. The density of states of an analogous low-temperature ideal Fermi gas
  8. Relative biases of $\beta$ and $T$ in a two-level system
  9. Unbiased estimator of $\beta^{\alpha}$ for $\alpha\in\mathbb{R}$
  10. The approximate expression of $\hat{\beta}_{B}$ and $\hat{T}_{G}$ in large-$N$ limit
  11. Estimation of temperature for the classical ideal gas

Key Result

Theorem 1

If $\hat{\theta}$ is an unbiased estimator of $\theta$ and $T$ is a sufficient statistic for $\theta$, The expectation of $\hat{\theta}$ conditional on $T$, i.e., $\tilde{\theta}= \langle\hat{\theta} | T\rangle$ is 1. a vaild estimator for $\theta$ (not dependent on $\theta$) 2. unbiased 3. of less

Figures (6)

  • Figure 1: Relative bias of $\beta$ varies as $N$ when $E_{\text{max}}=\infty$. Here, $\sigma(E)=e^{\sqrt{N(E-E_{\text{min}})}}$ is analogous to the density of states of the low-temperature ideal fermi gas (see Appendix \ref{['fermi']}), where the value of $E_{\text{min}}$ does not affect the results.
  • Figure 2: Relative bias of $\beta$ varies as $N$ when $E_{\text{max}}$ is finite. Here, $\sigma(E)$ is the density of states of an $N$ non-interacting two-level system (see Appendix \ref{['twolevel']}) and we set the spacing of energy levels $\varepsilon=1$.
  • Figure 3: Relative bias of $T$ varies as $N$ when $E_{\text{max}}$ is finite. Here, the density of states is the same as FIG.\ref{['ERROR BETA1']} (see Appendix \ref{['twolevel']}). (a) $T>0$. (b) $T<0$.
  • Figure 4: ETU varies as $1/N$. Here, $\sigma(E)=E^{\frac{3N}{2}-1}$ is the density of states of the classical ideal gas (see Appendix \ref{['classical']}). We set the inverse temperature $\beta=1$.
  • Figure 5: Probability distributions $P$ of the estimators $\hat{T}_{G,M}$ and $\hat{T}_{B,M}$ for different $M$, where the vertical lines indicate the mean values of the corresponding distributions. Here, $\sigma_{M}(E)=E^{\frac{3NM}{2}-1}$ is the density of states of the classical ideal gas bib62. We set temperature $T=1$ and particle number $N=6$.
  • ...and 1 more figures

Theorems & Definitions (1)

  • Theorem 1