Optimal estimation of temperature in finite-sized system

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.