Maximum-Temperature Theorem for a Nanothermometer
Abstract
We derive a sharp upper bound on the quasi-steady temperature rise of a sensor of arbitrary shape and known thermometric volume Vs, placed anywhere inside a region enclosed by an isothermal reference surface. The bound also requires the total heat power generated in the region Q̇r and a lower bound on its thermal conductivity κscriptscriptstyle𝒱r. It is otherwise independent of the heat-source distribution, domain shape, and source--sensor distance. For Q̇r=50 pW, κscriptscriptstyle𝒱r =0.1 W m−1K−1, and Vs equal to the volume of a 60 nm-diameter sphere, the theorem permits a maximum temperature rise of approximately 2 mK. Neither source localization nor uncertain source--sensor separation can therefore account for reported kelvin-scale intracellular readings under steady-state heat conduction. The result applies to sensors based on nanodiamonds, upconversion nanoparticles, fluorescent polymer beads, and core--shell quantum dots.
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BibTeX
@misc{plakhotnik2026maximumtemperature,
title = {{Maximum-Temperature Theorem for a Nanothermometer}},
author = {Taras Plakhotnik},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24340},
eprint = {2610.24340},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24340},
}
RIS
TY - UNPB
TI - Maximum-Temperature Theorem for a Nanothermometer
AU - Taras Plakhotnik
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.24340
UR - https://arxiv.org/abs/2610.24340
AB - We derive a sharp upper bound on the quasi-steady temperature rise of a sensor of arbitrary shape and known thermometric volume V_s, placed anywhere inside a region enclosed by an isothermal reference surface. The bound also requires the total heat power generated in the region Q̇_r and a lower bound on its thermal conductivity κ_scriptscriptstyle𝒱_r. It is otherwise independent of the heat-source distribution, domain shape, and source--sensor distance. For Q̇_r=50 pW, κ_scriptscriptstyle𝒱_r =0.1 W m^−1K^−1, and V_s equal to the volume of a 60 nm-diameter sphere, the theorem permits a maximum temperature rise of approximately 2 mK. Neither source localization nor uncertain source--sensor separation can therefore account for reported kelvin-scale intracellular readings under steady-state heat conduction. The result applies to sensors based on nanodiamonds, upconversion nanoparticles, fluorescent polymer beads, and core--shell quantum dots.
ER -